naginterfaces.library.quad.dim1_​gauss_​wgen

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

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_29.3/flhtml/d01/d01tcf.html

Parameters

itypeint

Indicates the type of quadrature rule.

$itype=0$

Gauss–Legendre, with normal weights.

$itype=1$

Gauss–Jacobi, with normal weights.

$itype=-1$

Gauss–Jacobi, with adjusted weights.

$itype=2$

Exponential Gauss, with normal weights.

$itype=-2$

Exponential Gauss, with adjusted weights.

$itype=3$

Gauss–Laguerre, with normal weights.

$itype=-3$

Gauss–Laguerre, with adjusted weights.

$itype=4$

Gauss–Hermite, with normal weights.

$itype=-4$

Gauss–Hermite, with adjusted weights.

$itype=5$

Rational Gauss, with normal weights.

$itype=-5$

Rational Gauss, with adjusted weights.

afloat

The parameter $a$ which occurs in the quadrature formula

bfloat

The parameter $b$ which occurs in the quadrature formula

cfloat

The parameter $c$ which occurs in the quadrature formula

dfloat

The parameter $d$ which occurs in the quadrature formula

nint

$n$, the number of weights and abscissae to be returned. If $itype=-2$ or $-4$ and $c\ne 0.0$, an odd value of $n$ may raise problems (see $errno$ = 6).

Returns

weightfloat, ndarray, shape $\left(n\right)$

The $n$ weights.

abscisfloat, ndarray, shape $\left(n\right)$

The $n$ abscissae.

Raises

NagValueError

(errno $2$)

On entry, $itype=⟨value⟩$.

Constraint: $\left|itype\right|<6$.

(errno $2$)

On entry, $n=⟨value⟩$.

Constraint: $n>0$.

(errno $3$)

On entry, $a$, $b$, $c$, or $d$ is not in the allowed range: $a=⟨value⟩$, $b=⟨value⟩$ $c=⟨value⟩$, $d=⟨value⟩$ and $itype=⟨value⟩$.

Warns

NagAlgorithmicWarning

(errno $1$)

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

(errno $4$)

One or more of the weights are larger than $\text{rmax}$, the largest floating point number on this computer (see machine.real_largest): $\text{rmax}=⟨value⟩$.

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

(errno $5$)

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 $n$; or, if using normal weights, to change to adjusted weights.

(errno $6$)

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

Notes

dim1_gauss_wgen returns the weights $w_i$ and abscissae $x_i$ for use in the summation

$$S=\sum _{i=1}^n{w}_{i}f\left(x_i\right)\text{,}$$

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

1. Gauss–Legendre

   $$S\simeq {\int }_a^{b}f\left(x\right)dx\text{, exact for}f\left(x\right)=P_{2n-1}\left(x\right)\text{.}$$

   Constraint: $b>a$.

2. Gauss–Jacobi

   normal weights:

   $$S\simeq {\int }_a^b{(b-x)}^c{(x-a)}^{d}f\left(x\right)dx\text{, exact for}f\left(x\right)=P_{2n-1}\left(x\right)\text{,}$$

   adjusted weights:

   $$S\simeq {\int }_a^{b}f\left(x\right)dx\text{, exact for}f\left(x\right)={(b-x)}^c{(x-a)}^d{P}_{2n-1}\left(x\right)\text{.}$$

   Constraint: $c>-1$, $d>-1$, $b>a$.

3. Exponential Gauss

   normal weights:

   $$S\simeq {\int }_a^b{|x-\frac{a+b}{2}|}^{c}f\left(x\right)dx\text{, exact for}f\left(x\right)=P_{2n-1}\left(x\right)\text{,}$$

   adjusted weights:

   $$S\simeq {\int }_a^{b}f\left(x\right)dx\text{, exact for}f\left(x\right)={|x-\frac{a+b}{2}|}^c{P}_{2n-1}\left(x\right)\text{.}$$

   Constraint: $c>-1$, $b>a$.

4. Gauss–Laguerre

   normal weights:

   $$\begin{array}{c}\begin{array}{cc}S& \simeq {\int }_a^{\infty }{|x-a|}^c{e}^{-bx}f\left(x\right)dx(b>0)\text{,}\\ & \\ & \simeq {\int }_{-\infty }^a{|x-a|}^c{e}^{-bx}f\left(x\right)dx(b<0)\text{, exact for}f\left(x\right)=P_{2n-1}\left(x\right)\text{,}\end{array}\end{array}$$

   adjusted weights:

   $$\begin{array}{c}\begin{array}{cc}S& \simeq {\int }_a^{\infty }f\left(x\right)dx(b>0)\text{,}\\ & \\ & \simeq {\int }_{-\infty }^{a}f\left(x\right)dx(b<0)\text{, exact for}f\left(x\right)={|x-a|}^c{e}^{-bx}{P}_{2n-1}\left(x\right)\text{.}\end{array}\end{array}$$

   Constraint: $c>-1$, $b\ne 0$.

5. Gauss–Hermite

   normal weights:

   $$S\simeq {\int }_{-\infty }^{+\infty }{|x-a|}^c{e}^{-b{(x-a)}^2}f\left(x\right)dx\text{, exact for}f\left(x\right)=P_{2n-1}\left(x\right)\text{,}$$

   adjusted weights:

   $$S\simeq {\int }_{-\infty }^{+\infty }f\left(x\right)dx\text{, exact for}f\left(x\right)={|x-a|}^c{e}^{-b{(x-a)}^2}{P}_{2n-1}\left(x\right)\text{.}$$

   Constraint: $c>-1$, $b>0$.

6. Rational Gauss

   normal weights:

   $$\begin{array}{c}\begin{array}{cc}S& \simeq {\int }_a^{\infty }\frac{{|x-a|}^c}{{|x+b|}^d}f\left(x\right)dx(a+b>0)\text{,}\\ & \\ & \simeq {\int }_{-\infty }^a\frac{{|x-a|}^c}{{|x+b|}^d}f\left(x\right)dx(a+b<0)\text{, exact for}f\left(x\right)=P_{2n-1}\left(\frac{1}{x+b}\right)\text{,}\end{array}\end{array}$$

   adjusted weights:

   $$\begin{array}{c}\begin{array}{cc}S& \simeq {\int }_a^{\infty }f\left(x\right)dx(a+b>0)\text{,}\\ & \\ & \simeq {\int }_{-\infty }^{a}f\left(x\right)dx(a+b<0)\text{, exact for}f\left(x\right)=\frac{{|x-a|}^c}{{|x+b|}^d}{P}_{2n-1}\left(\frac{1}{x+b}\right)\text{.}\end{array}\end{array}$$

   Constraint: $c>-1$, $d>c+1$, $a+b\ne 0$.

In the above formulae, $P_{2n-1}\left(x\right)$ stands for any polynomial of degree $2n-1$ or less in $x$.

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

$$w_i={\left\{\sum _{j=0}^{n-1}{P}_j^{\ast }{\left(x_i\right)}^2\right\}}^{-1}\text{,}$$

where $P_j^{\ast }\left(x\right)$ is the $j$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