File:GaulegExample.png: Difference between revisions
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imported>Dmitrii Kouznetsov ({{Image notes |Description=Error of the approximation of the integral <math>\int_{-1}^1 f(x) {\rm d}x <math> with the <math>N</math>-point Legendre-Gaussian quadrature formula versus number <math>N</math> for the following functions: * <math>f(x)=\sqrt{1-x^2}</math> (red) * <math>f(x)=\frac{1}{1+x^2}</math> (green) * <math>f(x)=\frac{1}{3+x} </math> (blue) * <math>f(x)=x^16 </math> (black) In the last case, at <math> N>8 </math> the residual would be zero; practically, it is determined by the...) |
imported>Dmitrii Kouznetsov mNo edit summary |
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{{Image notes | {{Image notes | ||
|Description=Error of the approximation of the integral | |Description=Error of the approximation of the integral | ||
<math>\int_{-1}^1 f(x) {\rm d}x <math> with the <math>N</math>-point Legendre-Gaussian quadrature formula versus number <math>N</math> for the following functions: | <math>\int_{-1}^1 f(x) {\rm d}x </math> with the <math>N</math>-point Legendre-Gaussian quadrature formula versus number <math>N</math> for the following functions: | ||
* <math>f(x)=\sqrt{1-x^2}</math> (red) | * <math>f(x)=\sqrt{1-x^2}</math> (red) | ||
* <math>f(x)=\frac{1}{1+x^2}</math> (green) | * <math>f(x)=\frac{1}{1+x^2}</math> (green) | ||
* <math>f(x)=\frac{1}{3+x} </math> (blue) | * <math>f(x)=\frac{1}{3+x} </math> (blue) | ||
* <math>f(x)=x^16 </math> (black) | * <math>f(x)=x^{16} </math> (black) | ||
In the last case, at <math> N>8 </math> the residual would be zero; practically, it is determined by the precision of arithmetic used to perform the evaluation. In this example, long double variables were used. | In the last case, at <math> N>8 </math> the residual would be zero; practically, it is determined by the precision of arithmetic used to perform the evaluation. In this example, long double variables were used. | ||
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