Numerical integration

Numerical integration is the term used for a number of methods to find an approximation for an integral.^[1] Numerical integration has also been called quadrature. Very often, it is not possible to solve integration analytically, for example when the data consists of a number of distinct measurements, or when the antiderivative is not known, and it is difficult, impractical or impossible to find it. In such cases, the integral can be written as a mathematical function defined over the interval in question, plus a function giving the error.

One way to find a numerical integral is using interpolation. Very often these interpolating functions are polynomials.

Various formulas have been studied for many years and become famous. For example, there is the Gaussian quadrature^[2] (named after Gauss), the Newton-Cotes formula^[3] (named after Isaac Newton), and the Euler-Maclaurin formula^[4] (named after Leonhard Euler).

Numerical errors[change | change source]

Numerical errors can occur in any kind of numerical computation including numerical integration. Errors in numerical integration are considered in another area called "validated numerics".^[5]

People who studied about numerical integration[change | change source]

References[change | change source]

↑ Davis, P. J., & Rabinowitz, P. (2007). Methods of numerical integration. Courier Corporation.
↑ Weisstein, Eric W. "Gaussian Quadrature." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GaussianQuadrature.html
↑ Weisstein, Eric W. "Newton-Cotes Formulas." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Newton-CotesFormulas.html
↑ Weisstein, Eric W. "Euler-Maclaurin Integration Formulas." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Euler-MaclaurinIntegrationFormulas.html
↑ Rump, S. M. (2010). Verification methods: Rigorous results using floating-point arithmetic. Acta Numerica, 19, 287-449.

Numerical integration software[change | change source]

MATLAB, INTLAB
- GNU Octave, Scilab - open source versions of MATLAB
Wolfram Mathematica - made by Wolfram Research

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[1] Davis, P. J., & Rabinowitz, P. (2007). Methods of numerical integration. Courier Corporation.

[2] Weisstein, Eric W. "Gaussian Quadrature." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GaussianQuadrature.html

[3] Weisstein, Eric W. "Newton-Cotes Formulas." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Newton-CotesFormulas.html

[4] Weisstein, Eric W. "Euler-Maclaurin Integration Formulas." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Euler-MaclaurinIntegrationFormulas.html

[5] Rump, S. M. (2010). Verification methods: Rigorous results using floating-point arithmetic. Acta Numerica, 19, 287-449.

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