naginterfaces.library.quad.dim1_​gauss_​wgen

naginterfaces.library.quad.dim1_gauss_wgen(itype, a, b, c, d, n)[source]

dim1_gauss_wgen returns the weights (normal or adjusted) and abscissae for a Gaussian integration rule with a specified number of abscissae. Six different types of Gauss rule are allowed.

For full information please refer to the NAG Library document for d01tc

https://www.nag.com/numeric/nl/nagdoc_28.5/flhtml/d01/d01tcf.html

Parameters
itypeint

Indicates the type of quadrature rule.

Gauss–Legendre, with normal weights.

Gauss–Jacobi, with normal weights.

Gauss–Jacobi, with adjusted weights.

Exponential Gauss, with normal weights.

Exponential Gauss, with adjusted weights.

Gauss–Laguerre, with normal weights.

Gauss–Laguerre, with adjusted weights.

Gauss–Hermite, with normal weights.

Gauss–Hermite, with adjusted weights.

Rational Gauss, with normal weights.

Rational Gauss, with adjusted weights.

afloat

The parameter which occurs in the quadrature formula

bfloat

The parameter which occurs in the quadrature formula

cfloat

The parameter which occurs in the quadrature formula

dfloat

The parameter which occurs in the quadrature formula

nint

, the number of weights and abscissae to be returned. If or and , an odd value of may raise problems (see = 6).

Returns
weightfloat, ndarray, shape

The weights.

abscisfloat, ndarray, shape

The abscissae.

Raises
NagValueError
(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .

(errno )

On entry, , , , or is not in the allowed range: , , and .

Warns
NagAlgorithmicWarning
(errno )

The algorithm for computing eigenvalues of a tridiagonal matrix has failed to converge.

(errno )

One or more of the weights are larger than , the largest floating point number on this computer (see machine.real_largest): .

Possible solutions are to use a smaller value of ; or, if using adjusted weights to change to normal weights.

(errno )

One or more of the weights are too small to be distinguished from zero on this machine.

The underflowing weights are returned as zero, which may be a usable approximation.

Possible solutions are to use a smaller value of ; or, if using normal weights, to change to adjusted weights.

(errno )

The contribution of the central abscissa to the summation is indeterminate.

Notes

dim1_gauss_wgen returns the weights and abscissae for use in the summation

which approximates a definite integral (see Davis and Rabinowitz (1975) and Stroud and Secrest (1966)). The following types are provided:

  1. Gauss–Legendre

    Constraint: .

  2. Gauss–Jacobi

    normal weights:

    adjusted weights:

    Constraint: , , .

  3. Exponential Gauss

    normal weights:

    adjusted weights:

    Constraint: , .

  4. Gauss–Laguerre

    normal weights:

    adjusted weights:

    Constraint: , .

  5. Gauss–Hermite

    normal weights:

    adjusted weights:

    Constraint: , .

  6. Rational Gauss

    normal weights:

    adjusted weights:

    Constraint: , , .

In the above formulae, stands for any polynomial of degree or less in .

The method used to calculate the abscissae involves finding the eigenvalues of the appropriate tridiagonal matrix (see Golub and Welsch (1969)). The weights are then determined by the formula

where is the th orthogonal polynomial with respect to the weight function over the appropriate interval.

The weights and abscissae produced by dim1_gauss_wgen may be passed to md_gauss(), which will evaluate the summations in one or more dimensions.

References

Davis, P J and Rabinowitz, P, 1975, Methods of Numerical Integration, Academic Press

Golub, G H and Welsch, J H, 1969, Calculation of Gauss quadrature rules, Math. Comput. (23), 221–230

Stroud, A H and Secrest, D, 1966, Gaussian Quadrature Formulas, Prentice–Hall