Method of the trapezoids

In numerical Analysis, the method of the trapezoids is a method making it possible to carry out the numerical Calcul of an integral

$\backslash \; int\_\; \{has\}\; ^\; \{B\}\; F\; (X)\; dx$

Single interval

The principle is to approach the area under the curve representative of the function $f$ by a Trapèze and to calculate the surface of it:

$T=\; (Ba)\; \backslash \; frac\; \{F\; (a)\; +\; F\; (b)\}\; \{2\}.$

For a function with actual values, twice continuously differentiable on the segment $$, the made error is form

$\backslash \; int\_a^b\; F\; (X)\; \backslash ,\; dx\; -\; (Ba)\; \backslash \; frac\; \{F\; (a)\; +\; F\; (b)\}\; \{2\}\; =\; -\; \backslash \; frac\; \{(Ba)\; ^3\}\; \{12\}\; F$ (\ xi)

for some $\backslash \; xi\; \backslash \; in\; \backslash ,$.

In particular, in the case of a convex Function, the surface of the trapezoid is a value approached by excess of the integral.

Multiple intervals

To obtain better results, one cuts out the interval $b$ in N smaller intervals and one applies the method to each one of them:

$\backslash \; int\_a^b\; F\; (X)\; \backslash ,\; dx\; =\; \backslash \; frac\; \{Ba\}\; \{N\}\; \backslash \; left\; (\{F\; (a)\; +\; F\; (b)\; \backslash \; over\; 2\}\; +\; \backslash \; sum\_\; \{k=1\}\; ^\; \{n-1\}\; F\; \backslash \; left\; (a+k\; \backslash \; frac\; \{Ba\}\; \{N\}\; \backslash \; right)\; \backslash \; right)\; +\; R\_n\; (F)$

$R\_n\; (F)\; \backslash ,$ is the error of squaring and is worth: $-\; \backslash \; frac\; \{(Ba)\; ^3\}\; \{12n^2\}\; F$ (\ xi) for a $\backslash \; xi\; \backslash \; in\; \backslash ,$

The method of the trapezoids thus consists in replacing the function by a function continuous and closely connected per pieces (operation of linear Interpolation) and regarding the surface of the latter as approximate value of the surface of the initial function.

Example of approximation of a function by trapezoids

Here the cutting of a function F which one wants to integrate on the interval $$ $f\; (X)\; =1.1\; +\; \backslash \; ln\; (E\; \backslash \; frac\; \{X\}\; \{100\}\; +\; \backslash \; frac\; \{3\}\; \{5\}\; \backslash \; operatorname\; \{tanh\}\; (\backslash \; ln\; (x+10^\; \{-\; 7\})\; +1))\backslash \; cos\; \backslash ,\; X\; +\; \backslash \; frac\; \{2\}\; \{5\}\; (X\; \backslash \; frac\; \{\backslash \; cos\; (3x)\}\{5\})\; ^2\; \backslash ,\; \backslash !$ $\backslash \; cdot\; \backslash \; cdot\; \backslash \; cdot\; +\; \backslash \; frac\; \{11\}\; \{100\}\; \backslash \; sqrt\; \{2+2x\}\; \backslash \; sin\; (\backslash \; frac\; \{44\}\; \{25\}\; (4+3\; \backslash \; sqrt\; \{X\})\; X\; \backslash \; frac\; \{19\}\; \{20\}\; x^5)\; -\; e^\; \{\backslash \; frac\; \{X\}\; \{3\}\}\; \backslash ,\; \backslash !$

Découpage for various values of N (2,8 and 16).

Various theorems

Theorem : If F is 2 times continuously differentiable on $$, the method of the trapezoids is convergent on $C^2\; ()$.
Theorem : The method of the trapezoids is stable.

Bond with the other methods of integration

The method of the trapezoids is an application of the Formules of Newton-Dimensions, the Méthode of Simpson in is another, the more precise.

The Méthode of Romberg is a process of acceleration of the convergence of the method of the trapezoids.

See too

• Integral calculus (elementary mathematics)
• Formula of Euler-Maclaurin
• Formulas of Newton-Cotessur Math-Linux.com

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