Quadrature Calculator Precision 81

Quadrature Calculator Precision 81

Quadrature Calculator Precision 81

Developer’s Description

Quadrature Calculator Precision 81 calculates definite integrals of large variety of functions by tanh-sinh quadrature scheme. Numerical values are calculated with precision 81 digits. All calculation results are reflected in History rich-text-box, which can be saved in file or printedQuadrature Calculator Accuracy 81 calculates precise integrals of large variety of functions by tanh-sinh quadrature scheme. Numerical values ​​are calculated with an accuracy of 81 digits. All calculation results can also be seen in the History in rich-text-box, which can be saved to file or printed

Limitations :.

3-days already

To calculate definite integral by the rectangle method, trapezoid method, Simpson method or other Newton-Cotes quadrature methods.

Numerical methods can be used for definite integral value approximation. Numerical integration is used in case of impossibility to evaluate antiderivative analytically and then calculate definite integral using Newton–Leibniz axiom.

Numerical integration of a single argument function can be represented as the area (or quadrature) calculation of a curvilinear trapezoid bounded by the graph of a given function, the x-axis, and vertical lines bounding given limits.

The integrand function is replaced by a simpler one (which has antiderivative) approximating the integrand with a given accuracy. Replacing the integrand with Lagrange polynomials evaluated at equally spaced points in given limits yields the Newton-Cotes integration formulas, such as:

The boundary element method requires accurate calculation of line and surface integrals in which a factor such as 1n(1/r), 1/r or 1/r2 is ill-behaved when r is nearly zero over a small portion of the domain. This paper shows that the underlying theory of Gaussian integration can be applied to get a quadrature formula for any particular distance from the element. An accurate integral can then be calculated with very few evaluation points by (a) computing the distance to the element; (b) interpolating in a table to obtain quadrature points and weights; and (c) summing the weighted values of the non-singular part of the integrand. This provides a fast, accurate value of the integral with no uncertainty as to the number of evaluation points that are required.

Line integrals with near-singular 1/r, 1/r2 and 1n(1/r) terms are used as examples. Accuracy of the formulas is verified experimentally for a wide range of distances. Interpolation in tabulated data is shown to intriduce only minor errors. The line integral formulas are applied to a triangular region, yielding exact formulas of any desired degree for the case where a 1/r singularity is at a vertex of the triangle.

The Jacobi matrix of the (2n+1)-point Gauss-Kronrod quadrature rule for a given measure is calculated eciently by a ve-term recurrence relation. The algorithm uses only rational operations and is therefore also useful for obtaining the Jacobi-Kronrod matrix analytically. The nodes and weights can then be computed directly by standard software for Gaussian quadrature formulas.

 

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