scipy.special._orthogonal

Members

Summary

A collection of functions to find the weights and abscissas for Gaussian Quadrature.

Extended Summary

These calculations are done by finding the eigenvalues of a tridiagonal matrix whose entries are dependent on the coefficients in the recursion formula for the orthogonal polynomials with the corresponding weighting function over the interval.

Many recursion relations for orthogonal polynomials are given:

a 1 n f_{n + 1} (x) = (a 2 n + a 3 n x) f_{n} (x) - a 4 n f_{n - 1} (x)

The recursion relation of interest is

P_{n + 1} (x) = (x - A_{n}) P_{n} (x) - B_{n} P_{n - 1} (x)

where $P$ has a different normalization than $f$ .

The coefficients can be found as:

A_{n} = - a 2 n / a 3 n B_{n} = (a 4 n / a 3 n h_{n} - 1/ h_{n})^{2}

where

h_{n} = \int_{a}^{b} w (x) f_{n} (x)^{2}

assume:

P_{0} (x) = 1 P_{- 1} (x) == 0

For the mathematical background, see [golub.welsch-1969-mathcomp] and [abramowitz.stegun-1965].

Additional content

A collection of functions to find the weights and abscissas for Gaussian Quadrature.

These calculations are done by finding the eigenvalues of a tridiagonal matrix whose entries are dependent on the coefficients in the recursion formula for the orthogonal polynomials with the corresponding weighting function over the interval.

Many recursion relations for orthogonal polynomials are given:

a 1 n f_{n + 1} (x) = (a 2 n + a 3 n x) f_{n} (x) - a 4 n f_{n - 1} (x)

The recursion relation of interest is

P_{n + 1} (x) = (x - A_{n}) P_{n} (x) - B_{n} P_{n - 1} (x)

where $P$ has a different normalization than $f$ .

The coefficients can be found as:

A_{n} = - a 2 n / a 3 n B_{n} = (a 4 n / a 3 n h_{n} - 1/ h_{n})^{2}

where

h_{n} = \int_{a}^{b} w (x) f_{n} (x)^{2}

assume:

P_{0} (x) = 1 P_{- 1} (x) == 0

For the mathematical background, see [golub.welsch-1969-mathcomp] and [abramowitz.stegun-1965].

References

[golub.welsch-1969-mathcomp]

Golub, Gene H, and John H Welsch. 1969. Calculation of Gauss Quadrature Rules. *Mathematics of Computation* 23, 221-230+s1--s10.

[abramowitz.stegun-1965]

Abramowitz, Milton, and Irene A Stegun. (1965) *Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables*. Gaithersburg, MD: National Bureau of Standards. http://www.math.sfu.ca/~cbm/aands/

[townsend.trogdon.olver-2014]

Townsend, A. and Trogdon, T. and Olver, S. (2014) *Fast computation of Gauss quadrature nodes and weights on the whole real line*. :arXiv:`1410.5286`.

[townsend.trogdon.olver-2015]

Townsend, A. and Trogdon, T. and Olver, S. (2015) *Fast computation of Gauss quadrature nodes and weights on the whole real line*. IMA Journal of Numerical Analysis :doi:`10.1093/imanum/drv002`.

`scipy.special._orthogonal`

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Extended Summary

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