Numerical calculation of an integral

In numerical Analysis, there exists a whole family of algorithms making it possible to approach the numerical value of a Intégrale. All consist in approaching the integral $I\; =\; \backslash \; int\; F\; (X)\; \backslash ,\; dx$ by a formula known as of squaring , type $I\; (F)\; =\; \backslash \; sum\_\; \{i=0\}\; ^p\; \backslash \; omega\_i\; F\; (x\_i)$. The choice of p , weightings $\backslash \; omega\_i$ and the nodes $x\_i$ depends on the method employed. It will also be advisable to be interested in the precision of the formulas used.

A first indication of the effectiveness of a method is given by its order . By definition, a method of squaring is of order N when it gives the exact value of the integral for all Polynôme of degree lower or equal to N , and a false result for at least a polynomial of degree n+1 .

Method of calculating of integral to a dimension

These methods use the interpolation functions to be integrated. Generally, the functions are interpolated by Polynôme S which one knows the primitive easily.

Simple formulas

Formulas of the rectangle and the point medium

It is the simplest method which consists in interpolating the function $f$ to be integrated by a constant function (polynomial of degree 0). That is to say $\backslash \; xi$ the point of interpolation; the formula becomes then:

$I\; (F)\; =\; (Ba)\; F\; (\backslash \; xi)\; \backslash ,$

The choice of the point has importance for the determination of the term of error $E\; (F)\; =\; I\; -\; I\; (F)\; \backslash ,$:

• If $\backslash \; xi\; =\; has\; \backslash ,$ or $\backslash \; xi\; =\; B\; \backslash ,$, the error is $E\; (F)\; =\; \backslash \; frac\; \{(Ba)\; ^2\}\; \{2\}\; f\text{'}\; (\backslash \; eta),\; \backslash \; quad\; \backslash \; eta\; \backslash \; in$. It is the method of the rectangle ;
• If $\backslash \; xi\; =\; (a+b)/2\; \backslash ,$, then the error becomes $E\; (F)\; =\; \backslash \; frac\; \{(Ba)\; ^3\}\; \{24\}\; F$ (\ eta), \ quad \ eta \ in \, . It is about the method of the point medium.

Thus, the choice of the point medium improves the order of the method. The method of the rectangle is exact (i.e. $E\; (F)\; =\; 0$) for the constant functions and that of the point medium is exact for the polynomials of degree lower or equal to 1. This is explained by the fact that for the integration of $x$, the method of the point medium gives place to two errors of evaluation, equal in value absolute and opposite in sign.

Formulate trapezoid

If one interpolates F by a polynomial of degree one (Fonction refines), one needs two points of interpolation, namely ( has , F ( has )) and ( B , F ( B )). The integral is then approached by the surface of the polynomial interpolater, in fact a trapezoid. This justifies the name of Méthode of the trapezoids:

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

The made error is

$E\; (F)\; =\; -\; \backslash \; frac\; \{(Ba)\; ^3\}\; \{12\}\; F$ (\ eta), \ quad \ eta \ in

The error is cancelled for any polynomial of degree lower or equal to one. According to this criterion, the method of the trapezoids is thus less powerful than that of the point medium, since the degrees of exactitude are the same ones and that the number of evaluations is larger for the method of the trapezoids than for that of the point medium.

Formulate of Simpson

The function F is now replaced by a parabola, which requires three points of interpolation. The ends has , B , and their medium m are selected. The Méthode of Simpson then consists in replacing the integral by

$I\; (F)\; =\; \backslash \; frac\; \{(Ba)\}\{6\}\; \backslash \; left\; F\; (a)\; +\; 4\; F\; (m)\; +\; F\; (b)\; \backslash \; right$

The error is

$E\; (F)\; =\; -\; \backslash \; frac\; \{(Ba)\; ^5\}\; \{2880\}\; f^\; \{(4)\}(\backslash \; eta),\; \backslash \; quad\; \backslash \; eta\; \backslash \; in$

The degree of exactitude is of 3 for this method, for 3 evaluations of F .

The following table summarizes the performances of each method

The Formules of Newton-Dimensions make it possible to generalize these results on constant intervals, where the function F is interpolated by polynomials of increasingly high degree. For questions of numerical stability, it is preferable to limit the degree of the polynomial of interpolation by subdividing the interval in subintervals, for which a linear interpolation is sufficient.

Composite formulas

For each preceding method, the term of error depends on Ba . If this amplitude is too high, one can reduce the error simply by cutting out the interval '' B '' in N subintervals, on which one will calculate the approximate value of the integral. One then speaks about composite formula . The value on the interval '' B '' will be the sum of the value on each subinterval.

For the method of the point medium, the formula becomes

$I\; (F)\; =\; \backslash \; frac\; \{(Ba)\}\{N\}\; \backslash \; sum\_\; \{k=0\}\; ^\; \{n-1\}\; F\; (m\_k)$

where $m\_k$ is the medium of the K - ième subinterval. Since N subintervals are identical, they are form + '' K '' '' H '', '' has '' + ('' K '' +1) '' H '', with H = ( B - has )/ N and K = 0,1,2,…, N -1. This involves finally that $m\_k\; =\; has\; +\; K\; H\; +\; h/2$. The term of error is written $E\; (H)\; =\; h^2\; \backslash \; frac\; \{(Ba)\}\{24\}\; F$ (\ eta), \ quad \ eta \ in . The composite formula has an order 1, like previously. Aggregation cause a drop in by a power the term in (a-b) .

For the method of the trapezoids, the composite formula is

$I\; (F)\; =\; \backslash \; frac\; \{(Ba)\}\{N\}\; \backslash \; left\; (\{F\; (a)\; +\; F\; (b)\; \backslash \; over\; 2\}\; +\; \backslash \; sum\_\; \{k=1\}\; ^\; \{n-1\}\; F\; \backslash \; left\; (+\; K\; H\; \backslash \; right\; has)\; \backslash \; right)$

The term of error is written $E\; (F)\; =-h^2\; \backslash \; frac\; \{(Ba)\}\{12\}\; F$ (\ eta), \ quad \ eta \ in .

The composite formula of Simpson takes the form

$I\; (F)\; =\; \backslash \; frac\; \{H\}\; \{6\}\; \backslash \; left\; +\; 2\; \backslash \; sum\_\; \{i=1\}\; ^\; \{n-1\}\; F\; (x\_i)\; +\; 4\; \backslash \; sum\_\; \{i=0\}\; ^\; \{n-1\}\; F\; (x\_i\; +\; h/2)\; \backslash \; right$

and the error becomes

$E\; (F)\; =\; -\; h^4\; \backslash \; frac\; \{(Ba)\}\{2880\}\; f^\; \{(4)\}(\backslash \; eta),\; \backslash \; quad\; \backslash \; eta\; \backslash \; in$

Other methods of numerical squaring

• Method of Romberg

• Methods of squaring of Gauss

Method of calculating of integral to several dimensions

• Method of Monte Carlo

Method of calculating of integral of particular form

Method of Laplace ($\backslash \; int\_a^b\; \backslash !\; e^\; \{M\; F\; (X)\}\backslash ,\; dx\; \backslash ,$), Method of the point collar ($I\; (\backslash \; lambda)\; =\; \backslash \; int\_\; \backslash \; mathcal\; \{C\}\; F\; (Z)\; e^\; \{\backslash \; lambda\; G\; (Z)\}\; \backslash ,\; dz\; \backslash ,$)

See too

• Rules of Bioche

Random links:

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