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PreGLQ

Calculate the weights and nodes used in integrating a function by Gauss-Legendre quadrature.

Usage

zero, w = pyshtools.PreGLQ (lower, upper, n)

Returns

zero : float, dimension (n)
The zeros used in the Gauss-Legendre quadrature.
w : float, dimension (n)
The weights used in the Gauss-Legendre quadrature.

Parameters

lower : float
The lower bound of the integration.
upper : float
The upper bound of the integration.
n : integer
The number of integration points to use. This will integrate exactly a polynomial of degree 2n-1.

Description

PreGLQ will calculate the weights and zeros used to integrate a function using Gauss-Legendre quadrature. For n quadrature points, the integration will be exact if the function is a polynomial of degree 2n-1, or less. The quadrature nodes correspond to the zeros of the Legendre polynomial of degree n. The number of quadrature points required to integrate a polynomial of degree L is ceiling((L+1)/2).

To integrate a function between the bounds lower and upper it is only necessary to calculate the sum of the function evaluated at the nodes zero multiplied by the weights.

This is a slightly modified version of the algorithm that was published in NUMERICAL RECIPES.

References

Press, W.H., S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed., Cambridge Univ. Press, Cambridge, UK, 1992.

See also

shglq

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