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A formula for calculating an integral over a finite interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080120/r0801201.png" />:
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{{TEX|done}}
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A formula for calculating an integral over a finite interval $[a,b]$:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080120/r0801202.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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\begin{equation}\label{eq:1}
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\int\limits_a^bf(x)dx\cong h\sum_{k=1}^Nf(\alpha+(k-1)h),
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\end{equation}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080120/r0801203.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080120/r0801204.png" />. Its algebraic degree of accuracy is 1 if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080120/r0801205.png" /> and 0 otherwise.
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where $h=(b-a)/N$ and $\alpha\in[a,a+h]$. Its algebraic degree of accuracy is 1 if $\alpha=a+h/2$ and 0 otherwise.
  
The [[Quadrature formula|quadrature formula]] (*) is exact for the trigonometric functions
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The [[quadrature formula]] \eqref{eq:1} is exact for the trigonometric functions
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080120/r0801206.png" /></td> </tr></table>
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$$\cos\frac{2\pi}{b-a}kx,\quad\sin\frac{2\pi}{b-a}kx,\quad k=0,\dots,N-1.$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080120/r0801207.png" />, then (*) is exact for all trigonometric polynomials of order at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080120/r0801208.png" />; moreover, its trigonometric degree of accuracy is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080120/r0801209.png" />. No other quadrature formula with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080120/r08012010.png" /> real nodes can have trigonometric degree of accuracy larger than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080120/r08012011.png" />, so that the rectangle rule with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080120/r08012012.png" /> has the highest trigonometric degree of accuracy.
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If $b-a=2\pi$, then \eqref{eq:1} is exact for all trigonometric polynomials of order at most $N-1$; moreover, its trigonometric degree of accuracy is $N-1$. No other quadrature formula with $N$ real nodes can have trigonometric degree of accuracy larger than $N-1$, so that the rectangle rule with $b-a=2\pi$ has the highest trigonometric degree of accuracy.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080120/r08012013.png" /> be the error of the rectangle rule, i.e. the difference between the left- and right-hand sides of (*). If the integrand <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080120/r08012014.png" /> is twice continuously differentiable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080120/r08012015.png" />, then for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080120/r08012016.png" /> one has
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Let $R(f,\alpha)$ be the error of the rectangle rule, i.e. the difference between the left- and right-hand sides of \eqref{eq:1}. If the integrand $f$ is twice continuously differentiable on $[a,b]$, then for $\alpha=a+h/2$ one has
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080120/r08012017.png" /></td> </tr></table>
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$$R\left(f,a+\frac h2\right)=\frac{b-a}{24}h^2f''(\xi),$$
  
for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080120/r08012018.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080120/r08012019.png" /> is a periodic function with period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080120/r08012020.png" /> and has a continuous derivative of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080120/r08012021.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080120/r08012022.png" /> is a natural number) on the entire real axis, then for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080120/r08012023.png" />,
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for some $\xi\in[a,b]$. If $f$ is a periodic function with period $b-a$ and has a continuous derivative of order $2k$ (where $k$ is a natural number) on the entire real axis, then for any $\alpha\in[a,a+h]$,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080120/r08012024.png" /></td> </tr></table>
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$$R(f,\alpha)=-(b-a)B_{2k}\frac{h^{2k}}{(2k)!}f^{(2k)}(\eta),$$
  
for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080120/r08012025.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080120/r08012026.png" /> is the Bernoulli number (cf. [[Bernoulli numbers|Bernoulli numbers]]).
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for some $\eta\in[a,b]$, where $B_{2k}$ is the Bernoulli number (cf. [[Bernoulli numbers]]).
  
  

Latest revision as of 17:35, 24 March 2018

A formula for calculating an integral over a finite interval $[a,b]$:

\begin{equation}\label{eq:1} \int\limits_a^bf(x)dx\cong h\sum_{k=1}^Nf(\alpha+(k-1)h), \end{equation}

where $h=(b-a)/N$ and $\alpha\in[a,a+h]$. Its algebraic degree of accuracy is 1 if $\alpha=a+h/2$ and 0 otherwise.

The quadrature formula \eqref{eq:1} is exact for the trigonometric functions

$$\cos\frac{2\pi}{b-a}kx,\quad\sin\frac{2\pi}{b-a}kx,\quad k=0,\dots,N-1.$$

If $b-a=2\pi$, then \eqref{eq:1} is exact for all trigonometric polynomials of order at most $N-1$; moreover, its trigonometric degree of accuracy is $N-1$. No other quadrature formula with $N$ real nodes can have trigonometric degree of accuracy larger than $N-1$, so that the rectangle rule with $b-a=2\pi$ has the highest trigonometric degree of accuracy.

Let $R(f,\alpha)$ be the error of the rectangle rule, i.e. the difference between the left- and right-hand sides of \eqref{eq:1}. If the integrand $f$ is twice continuously differentiable on $[a,b]$, then for $\alpha=a+h/2$ one has

$$R\left(f,a+\frac h2\right)=\frac{b-a}{24}h^2f''(\xi),$$

for some $\xi\in[a,b]$. If $f$ is a periodic function with period $b-a$ and has a continuous derivative of order $2k$ (where $k$ is a natural number) on the entire real axis, then for any $\alpha\in[a,a+h]$,

$$R(f,\alpha)=-(b-a)B_{2k}\frac{h^{2k}}{(2k)!}f^{(2k)}(\eta),$$

for some $\eta\in[a,b]$, where $B_{2k}$ is the Bernoulli number (cf. Bernoulli numbers).


Comments

References

[a1] D.M. Young, R.T. Gregory, "A survey of numerical mathematics" , Dover, reprint (1988) pp. 362ff
How to Cite This Entry:
Rectangle rule. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rectangle_rule&oldid=13753
This article was adapted from an original article by I.P. Mysovskikh (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article