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.

Numerical errors

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"^[2].

People who studied about numerical integration

Famous numerical integration formulas

There are various formulas which were studied for many years. For example, there is the Gaussian quadrature^[3] (named after Gauss), the Newton-Cotes formula^[4] (named after Isaac Newton), and the Euler-Maclaurin formula^[5] (named after Leonhard Euler).

References

↑ Davis, P. J., & Rabinowitz, P. (2007). Methods of numerical integration. Courier Corporation.
↑ Rump, S. M. (2010). Verification methods: Rigorous results using floating-point arithmetic. Acta Numerica, 19, 287-449.
↑ 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

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

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

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

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

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

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