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Numerical integration
In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral. The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for "numerical integration", especially as applied to one-dimensional integrals. Some authors refer to numerical integration over more than one dimension as cubature; others take "quadrature" to include higher-dimensional integration.
The basic problem in numerical integration is to compute an approximate solution to a definite integral
to a given degree of accuracy. If f(x) is a smooth function integrated over a small number of dimensions, and the domain of integration is bounded, there are many methods for approximating the integral to the desired precision.
Numerical integration has roots in the geometrical problem of finding a square with the same area as a given plane figure (quadrature or squaring), as in the quadrature of the circle. The term is also sometimes used to describe the numerical solution of differential equations.
There are several reasons for carrying out numerical integration, as opposed to analytical integration by finding the antiderivative:
The term "numerical integration" first appears in 1915 in the publication A Course in Interpolation and Numeric Integration for the Mathematical Laboratory by David Gibb.
"Quadrature" is a historical mathematical term that means calculating area. Quadrature problems have served as one of the main sources of mathematical analysis. Mathematicians of Ancient Greece, according to the Pythagorean doctrine, understood calculation of area as the process of constructing geometrically a square having the same area (squaring) — that is why the process was named "quadrature". Examples include quadrature of the circle, Lune of Hippocrates, and the treatise Quadrature of the Parabola. This construction must be performed only by means of compass and straightedge.
The ancient Babylonians used the trapezoidal rule to integrate the motion of Jupiter along the ecliptic.
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Numerical integration
In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral. The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for "numerical integration", especially as applied to one-dimensional integrals. Some authors refer to numerical integration over more than one dimension as cubature; others take "quadrature" to include higher-dimensional integration.
The basic problem in numerical integration is to compute an approximate solution to a definite integral
to a given degree of accuracy. If f(x) is a smooth function integrated over a small number of dimensions, and the domain of integration is bounded, there are many methods for approximating the integral to the desired precision.
Numerical integration has roots in the geometrical problem of finding a square with the same area as a given plane figure (quadrature or squaring), as in the quadrature of the circle. The term is also sometimes used to describe the numerical solution of differential equations.
There are several reasons for carrying out numerical integration, as opposed to analytical integration by finding the antiderivative:
The term "numerical integration" first appears in 1915 in the publication A Course in Interpolation and Numeric Integration for the Mathematical Laboratory by David Gibb.
"Quadrature" is a historical mathematical term that means calculating area. Quadrature problems have served as one of the main sources of mathematical analysis. Mathematicians of Ancient Greece, according to the Pythagorean doctrine, understood calculation of area as the process of constructing geometrically a square having the same area (squaring) — that is why the process was named "quadrature". Examples include quadrature of the circle, Lune of Hippocrates, and the treatise Quadrature of the Parabola. This construction must be performed only by means of compass and straightedge.
The ancient Babylonians used the trapezoidal rule to integrate the motion of Jupiter along the ecliptic.
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