The $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } n^\text{th}$ expansion of $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } f$ around $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } x_0$ is given by $$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } f(x) = \tpar{ \sum_{k=0}^n \frac{f^{(k)}(x_0)}{k!}(x-x_0)^k } + \tpar{ \frac{f^{(n+1)}(\xi_x)}{(n+1)!}(x-x_0)^{n+1} }$$ for $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \xi_x$ between $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } x_0$ and $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } x$

Say we have a set of $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } n+1$ points $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } x_0, x_1, x_2, \dots, x_n$ with known values $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } f(x_0), f(x_1), \dots, f(x_n)$. Then we can create a unique polynomial which goes through these points. The construction is illustrated as follows. Say we have $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } x_0, x_1, x_2$ with values $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } y_0, y_1, y_2$. The constructed polynomial is $$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } P(x) = \frac{ (x - x_1)(x - x_2) }{ (x_0 - x_1)(x_0 - x_2) } \cdot y_0 + \frac{ (x-x_0)(x-x_2) }{ (x_1-x_0)(x_1-x_2) } \cdot y_1 + \frac{(x-x_0)(x-x_1)}{(x_2-x_0)(x_2-x_1)} \cdot y_2$$ The idea is that when, for instance, $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } x = x_2$ then the $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } (x - x_2)$ multiplier in the first and third term will cause these terms to become zero, and the fraction in the second term will be one by symmetry of numerator and denominator, making the function as a whole produce $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } y_2$. This gives a degree-at-most-$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } n$ polynomial. The error term for this polynomial is $$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } E(x) = \frac{ f^{(n+1)}(\xi_x) }{(n+1)!} (x-x_0)(x-x_1)\cdots (x-x_n)$$ for some $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \xi_x$. Note how similar it looks to Taylor error terms! Worth memorizing this one probably.

Let $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } P_{a, b, c}$ be the interpolating polynomial agreeing with points $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } x_a, x_b, x_c$, etc. Choose $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } i, j$; then we have the identity $$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } P_{0,1,\dots,n}(x) = \frac{ (x - x_i)P_{0,1,\dots,{i-1},{i+1},\dots,n}(x) + (x - x_j)P_{0,1,\dots,{j-1},{j+1},\dots,n}(x) }{ x_j - x_i }$$ Now say we have input points $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } x_0, x_1, x_2, x_3$ and know $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } f(x_0), \dots, f(x_3)$. We define $$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \begin{align*} P_0(x) &= f(x_0) \\ P_1(x) &= f(x_1) \\ P_2(x) &= f(x_2) \\ P_3(x) &= f(x_3) \end{align*}$$ Now for any $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } x$ we can use the mentioned identity to compute the table

$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } P_0(x)$
$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } P_1(x)$	$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } P_{0,1}(x)$
$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } P_2(x)$	$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } P_{1,2}(x)$	$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } P_{0,1,2}(x)$
$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } P_3(x)$	$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } P_{2,3}(x)$	$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } P_{1,2,3}(x)$	$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } P_{0,1,2,3}(x)$

where each column is computed based on the previous. The result is an approximation $$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } P_{0,1,2,3}(x) \approx f(x)$$ Note that $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } P_{0,1,2,3}$ is exactly the Lagrange interpolating polynomial agreeing with the input points.

Again say we know $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } f(x_0), \dots, f(x_3)$. Define $$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \begin{align*} f[x_0] &= f(x_0) \\ f[x_1] &= f(x_1) \\ f[x_2] &= f(x_2) \\ f[x_3] &= f(x_3) \end{align*}$$ and then $$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } f[x_n, \dots, x_{n+d}] = \frac{ f[x_{n+1}, \dots, x_{n+d}] - f[x_n, \dots, x_{n+d-1}] }{ x_{n+d} - x_n }$$ We can build up these values as successive columns of the table

$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } f[x_0]$
$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } f[x_1]$	$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } f[x_0,x_1]$
$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } f[x_2]$	$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } f[x_1,x_2]$	$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } f[x_0,x_1,x_2]$
$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } f[x_3]$	$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } f[x_2,x_3]$	$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } f[x_1,x_2,x_3]$	$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } f[x_0,x_1,x_2,x_3]$

Once the table is computed we can use the diagonal to express the interpolating polynomial $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } P$ as $$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \begin{align*} P(x) &= f[x_0] \\ &+ f[x_0, x_1] (x - x_0) \\ &+ f[x_0, x_1, x_2] (x - x_0)(x - x_1) \\ &+ f[x_0, x_1, x_2, x_3] (x - x_0)(x - x_1)(x - x_2) \end{align*}$$

Given $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } n+1$ datapoints $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } (x_i, f(x_i), f'(x_i))$, the Hermite polynomial $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } H_{2n+1}$ is the unique polynomial of degree at most $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } 2n + 1$ which agrees with $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } f$ and whose derivatve agrees with $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } f'$ on all of $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } x_0 \dots x_n$

Given $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } n+1$ datapoints $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } (x_0, f(x_i))$, we can approximate each subinterval $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } [x_i, x_{i+1}]$ with a single cubic polynomial $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } S_i$ (ie, $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } S_i$ is of the form $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } a x^3 + b x^2 + c x + d$). Each $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } S_i$ is required to agree with $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } f$ at its endpoints, and also agree with its neighbors on value, derivative, and second derivative at its endpoints (eg, $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } S_0(x_1) = S_1(x_1)$ and the same for derivatives). The piecewise composition of all $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } S_i$ is denoted $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } S$. We then choose to impose one of the following two conditions. The first is $$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } S''(x_0) = S''(x_n) = 0$$ in which case we say $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } S$ has a “free” or “natural” boundary; or, $$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } S'(x_0) = f'(x_0) \quad \t{and} \quad S'(x_n) = f'(x_n)$$ in which case we say $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } S$ has a “clamped” boundary. Imposing a boundary condition constrains $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } S$ just enough to induce a unique solution. The induced interpolating function:

Forward difference formula as follows. Also called backwards difference for $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } h < 0$ $$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } f'(x) = \frac{ f(x+h) - f(x) }h - \frac h 2 f''(\xi(x))$$ for $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \xi(x) \in [x, x + h]$. Three-point endpoint formula $$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } f'(x) = \frac 1 {2 h} \left( {-3}f(x) + 4f(x+h) - f(x+2h) \right) + \frac{h^2}3 f^{(3)}(\xi(x))$$ for $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \xi(x) \in [x, x + 2h]$ Three-point midpoint formula $$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } f'(x) = \frac 1 {2 h} ( f(x + h) - f (x - h) ) - \frac {h^2} 6 f^{(3)}(\xi(x))$$ for $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \xi(x) \in [x - h, x + h]$ Five-point endpoint formula and five-point midpoint formula also exist.

Say we have a value $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } M$ we’re approximating by some $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } N_1(h)$, where $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } h$ can be anything (typically some small “step” parameter) $$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } M \approx N_1(h)$$ Say we know the exact error term. Say, for instance, $$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } M = N_1(h) + a h + b h^2 + c h^3 + \cdots$$ Gives an $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } O(h)$ approximation to $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } M$. (Here $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } O(h)$ is as $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } h \to 0$) Plugging in $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } h/2$ we get $$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } M = N_1(h/2) + \frac 1 2 a h + \frac 1 4 b h^2 + \frac 1 8 c h^3 + \cdots$$ Now subtracting the first equation from twice the second gives $$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } M = \tbrak{ 2N_1(h/2) - N_1(h) } + 0 a h - \frac 1 2 b h^2 - \frac 3 4 c h^3 - \cdots$$ Note that the $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \text{ah}$ term has been killed. This means that the approximation $$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } M \approx 2 N_1(h/2) - N_1(h) =: N_2(h)$$ is $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } O(h^2)$. We have “upgraded” our approxmation from $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } O(h)$ to $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } O(h^2)$. We may repeat the process on $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } N_2$ to produce an $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } O(h^3)$ approximation, and so on. (To do this, we would consider $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } N_2(h/2)$ and combine it with $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } N_2(h)$ in a way to eliminate the $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } h^2$ term.)

This is just formulas for approximating integrals. Typically we make approximations based off of equally-spaced nodes $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } x_0, x_1, \dots, x_n$ with $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } x_i = h + x_{i+1}$. One approximation is the Trapezoidal rule, using two input nodes: $$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \int_{x_0}^{x_1} f(x) \id x = \frac h 2 (f(x_0) + f(x_1)) - \frac{h^3}{{12}} f''(\xi)$$ for $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \xi \in (x_0, x_1)$. Another is Simpson’s rule, using three input nodes: $$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \int_{x_0}^{x_2} f(x) \id x = \frac h 3 ( f(x_0) + 4f(x_1) + f(x_2) ) - \frac{h^5}{{90}} f^{(4)}(\xi)$$ for $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \xi \in (x_0, x_2)$ Both of these are “closed”, meaning that they include the endpoints of the interval in their approximation. There are also “open” formulas which dont, such as the midpoint rule: $$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \int_{x_{-1}}^{x_1} f(x) \id x = 2 h f(x_0) + \frac{h^3}3 f''(\xi)$$ this uses the single node $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } x_0$ in its approximation. Accuracy of an approximation method is measured by its degree of precision (or degree of accuracy), which is the highest $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } k$ for which the approximation is exact when applied to $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } f(x) = x^k$. (It is a theorem that, given some niceness assumptions, this is equivalent to asking for exactness on all degree-$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } k$ polynomials)

If you have some way to approximate an integral, you can apply it $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } n$ times to $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } n$ subintervals to produce a better approximation. For the following two we have $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } h = (b - a)/n$ and $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } x_i = a + i h$. Composite Simpson’s rule ($\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } n$ must be even) $$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \int_a^b f(x) \id x = \frac h 3 \tbrak{ f(a) + 2 \sum_{j=1}^{(n/2)-1} f(x_{2j}) + 4 \sum_{j=1}^{n/2} f(x_{2j-1}) + f(b) } - \frac{b-a}{180}h^4 f^{(4)}(\mu) \qquad \exists \mu \in (a,b)$$ Composite Trapezoidal rule $$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \int_a^b f(x) \id x = \frac h 2 \tbrak{ f(a) + 2 \sum_{j=1}^{n-1} f(x_j) + f(b) } - \frac{b-a}{12} h^2 f''(\mu) \qquad \exists\mu \in (a,b)$$ For the last one we have $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } h = (b - a)/(n+2)$ and $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } x_i = a + (i+1) h$. Composite Midpoint rule ($\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } n$ must be even) $$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \int_a^b f(x) \id x = 2 h \sum_{j=0}^{n/2} f(x_{2j}) + \frac{b-a}6 h^2 f''(\mu) \qquad \exists\mu \in (a,b)$$

Is just extrapolation applied to the composite trapezoidal rule

Using equally-spaced nodes for computing integrals doesn’t make sense for a function with non-constant variabiliy. We do some magic and find a way to, in some way, detect which parts of the function are more/less variable. We make some vaguely-strong guarantees about error bounds.

The Trapezoidal rule, Simpson’s rule, midpoint rule, etc, all have the form $$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \int_a^b f(x) \id x \approx \sum_{i=1}^n c_i f(x_i)$$ for some choice of $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \{ c_i \}$ and $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \{ x_i \}$. Gaussian Quadrature aims to find the optimal approximation of that form. It does so by maximizing the approximation degree of accuracy. For $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } n=2$ and integrating over $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } [{-1}, 1]$ the resultant approximation is $$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \int_{-1}^1 f(x) \id x \approx f (-\sqrt 3 / 3) + f(\sqrt 3 / 3)$$ with degree of precision three. In general the result for $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \renewcommand{\dd}{{\ ..\ }}n$ nodes and integrating over $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \renewcommand{\dd}{{\ ..\ }}[{-1}, 1]$ is given as follows. $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \renewcommand{\dd}{{\ ..\ }}\{ x_i \}$ are the roots of the $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \renewcommand{\dd}{{\ ..\ }}n^\text{th}$ Legendre polynomial (the first 5 are given by $$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \renewcommand{\dd}{{\ ..\ }}\begin{align*} P_0(x) &= 1 \\ P_1(x) &= x \\ P_2(x) &= x^2 - (1/3) \\ P_3(x) &= x^3 - (3/5)x \\ P_4(x) &= x^4 - (6/7)x^2 + (3/35) \end{align*}$$ ) and $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \renewcommand{\dd}{{\ ..\ }}\{ c_i \}$ are $$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \renewcommand{\dd}{{\ ..\ }}c_i = \int_{-1}^1 \prod_{j=1,\ j \neq i}^n \frac{x-x_j}{x_i-x_j} \id x$$ This approximation has degree of precision $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \renewcommand{\dd}{{\ ..\ }}2n{-1}$. The fact that the result approximation is over $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \renewcommand{\dd}{{\ ..\ }}[{-1},1]$ is superficial; we can refit it to any integral with $$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \renewcommand{\dd}{{\ ..\ }}\int_a^b f(x) \id x = \int_{-1}^1 f \tpar{ \frac{ (b-a)t + (b+a) }2 } \frac {b-a}2 \id t$$ The values $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \renewcommand{\dd}{{\ ..\ }}\{x_i\},\{c_i\}$ can also be derived (without restricting to $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \renewcommand{\dd}{{\ ..\ }}[{-1}, 1]$!) by solving for the $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \renewcommand{\dd}{{\ ..\ }}2n$ unknowns $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \renewcommand{\dd}{{\ ..\ }}\{ x_i \}, \{ c_i \}$ by constraining the solution to have degree of precision $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \renewcommand{\dd}{{\ ..\ }}2n - 1$.

This is just Gaussian Quadrature applied in multiple dimensions The technique is nothing beyond using $$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \iint_R f(x, y) \id A = \int_a^b \int_{c(x)}^{d(x)} f(x,y) \id y \id x$$ on e.g. composite Simpson’s. The results are hairy though.