D01BCF : NAG Library, Mark 24

Note: before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

D01BCF 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.

D01BCF returns the weights

w_{i}

and abscissae

x_{i}

for use in the summation

S = \sum_{i = 1}^{n} w_{i} f (x_{i})

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

(a)

Gauss–Legendre

S ≃ \int_{a}^{b} f (x) d x,   exact for ​ f (x) = P_{2 n - 1} (x) .

Constraint:

b > a

(b)

Gauss–Jacobi

normal weights:

S ≃ \int_{a}^{b} {(b - x)}^{c} {(x - a)}^{d} f (x) d x,   exact for ​ f (x) = P_{2 n - 1} (x),

adjusted weights:

S ≃ \int_{a}^{b} f (x) d x,   exact for ​ f (x) = {(b - x)}^{c} {(x - a)}^{d} P_{2 n - 1} (x) .

Constraint:

c > - 1

d > - 1

b > a

(c)

Exponential Gauss

normal weights:

S ≃ \int_{a}^{b} {|x - \frac{a + b}{2}|}^{c} f (x) d x,   exact for ​ f (x) = P_{2 n - 1} (x),

adjusted weights:

S ≃ \int_{a}^{b} f (x) d x,   exact for ​ f (x) = {|x - \frac{a + b}{2}|}^{c} P_{2 n - 1} (x) .

Constraint:

c > - 1

b > a

(d)

Gauss–Laguerre

normal weights:

\begin{array}{l} S & ≃ \int_{a}^{\infty} {|x - a|}^{c} e^{- b x} f (x) d x (b > 0), \\ ≃ \int_{- \infty}^{a} {|x - a|}^{c} e^{- b x} f (x) d x (b < 0),   exact for ​ f (x) = P_{2 n - 1} (x), \end{array}

adjusted weights:

\begin{array}{l} S & ≃ \int_{a}^{\infty} f (x) d x (b > 0), \\ ≃ \int_{- \infty}^{a} f (x) d x (b < 0),   exact for ​ f (x) = {|x - a|}^{c} e^{- b x} P_{2 n - 1} (x) . \end{array}

Constraint:

c > - 1

b \neq 0

(e)

Gauss–Hermite

normal weights:

S ≃ \int_{- \infty}^{+ \infty} {|x - a|}^{c} e^{- b {(x - a)}^{2}} f (x) d x,   exact for ​ f (x) = P_{2 n - 1} (x),

adjusted weights:

S ≃ \int_{- \infty}^{+ \infty} f (x) d x,   exact for ​ f (x) = {|x - a|}^{c} e^{- b {(x - a)}^{2}} P_{2 n - 1} (x) .

Constraint:

c > - 1

b > 0

(f)

Rational Gauss

normal weights:

\begin{array}{l} S & ≃ \int_{a}^{\infty} \frac{{|x - a|}^{c}}{{|x + b|}^{d}} f (x) d x (a + b > 0), \\ ≃ \int_{- \infty}^{a} \frac{{|x - a|}^{c}}{{|x + b|}^{d}} f (x) d x (a + b < 0),   exact for ​ f (x) = P_{2 n - 1} (\frac{1}{x + b}), \end{array}

adjusted weights:

\begin{array}{l} S & ≃ \int_{a}^{\infty} f (x) d x (a + b > 0), \\ ≃ \int_{- \infty}^{a} f (x) d x (a + b < 0),   exact for ​ f (x) = \frac{{|x - a|}^{c}}{{|x + b|}^{d}} P_{2 n - 1} (\frac{1}{x + b}) . \end{array}

Constraint:

c > - 1

d > c + 1

a + b \neq 0

In the above formulae,

P_{2 n - 1} (x)

stands for any polynomial of degree

2 n - 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} = {\{\sum_{j = 0}^{n - 1} P_{j}^{*} {(x_{i})}^{2}\}}^{- 1}

where

P_{j}^{*} (x)

is the

j

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

The weights and abscissae produced by D01BCF may be passed to D01FBF, which will evaluate the summations in one or more dimensions.

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

On entry: 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.

Constraint:

ITYPE = 0

1

-1

2

-2

3

-3

4

-4

5

-5

On entry: the parameters

a

b

c

and

d

which occur in the quadrature formulae. C is not used if

ITYPE = 0

; D is not used unless

ITYPE = 1

-1

5

-5

. For some rules C and D must not be too large (see Section 6).

On entry:

n

, the number of weights and abscissae to be returned. If

ITYPE = -2

-4

and

C \neq 0.0

, an odd value of N may raise problems (see

IFAIL = 6

Constraint:

N > 0

On exit: the N weights.

On exit: the N abscissae.

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

The algorithm for computing eigenvalues of a tridiagonal matrix has failed to obtain convergence. If the soft fail option is used, the values of the weights and abscissae on return are indeterminate.

If the soft fail option is used, weights and abscissae are returned as zero.

A, B, C or D is not in the allowed range:

if $ITYPE = 0$ , $A \geq B$ ;
if $ITYPE = \pm 1$ , $A \geq B$ or $C \leq - 1.0$ or $D \leq - 1.0$ or $C + D + 2.0 > gmax$ ;
if $ITYPE = \pm 2$ , $A \geq B$ or $C \leq - 1.0$ ;
if $ITYPE = \pm 3$ , $B = 0.0$ or $C \leq - 1.0$ or $C + 1.0 > gmax$ ;
if $ITYPE = \pm 4$ , $B \leq 0.0$ or $C \leq - 1.0$ or $(C + 1.0 / 2.0) > gmax$ ;
if $ITYPE = \pm 5$ , $A + B = 0.0$ or $C \leq - 1.0$ or $D \leq C + 1.0$ .

Here

gmax

is the (machine-dependent) largest integer value such that

Γ (gmax)

can be computed without overflow (see the Users' Note for your implementation for S14AAF).

If the soft fail option is used, weights and abscissae are returned as zero.

One or more of the weights are larger than

rmax

, the largest floating point number on this machine.

rmax

is given by the function X02ALF. If the soft fail option is used, the overflowing weights are returned as

rmax

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

One or more of the weights are too small to be distinguished from zero on this machine. If the soft fail option is used, 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.

Exponential Gauss or Gauss–Hermite adjusted weights with N odd and

C \neq 0.0

. Theoretically, in these cases:

for $C > 0.0$ , the central adjusted weight is infinite, and the exact function $f (x)$ is zero at the central abscissa.
for $C < 0.0$ , the central adjusted weight is zero, and the exact function $f (x)$ is infinite at the central abscissa.

In either case, the contribution of the central abscissa to the summation is indeterminate.

In practice, the central weight may not have overflowed or underflowed, if there is sufficient rounding error in the value of the central abscissa.

If the soft fail option is used, the weights and abscissa returned may be usable; you must be particularly careful not to ‘round’ the central abscissa to its true value without simultaneously ‘rounding’ the central weight to zero or

\infty

as appropriate, or the summation will suffer. It would be preferable to use normal weights, if possible.

Note: remember that, when switching from normal weights to adjusted weights or vice versa, redefinition of

f (x)

is involved.

The accuracy depends mainly on

n

, with increasing loss of accuracy for larger values of

n

. Typically, one or two decimal digits may be lost from machine accuracy with

n ≃ 20

, and three or four decimal digits may be lost for

n ≃ 100

The major portion of the time is taken up during the calculation of the eigenvalues of the appropriate tridiagonal matrix, where the time is roughly proportional to

n^{3}

This example returns the abscissae and (adjusted) weights for the seven-point Gauss–Laguerre formula.

On entry,	$N < 1$ ,
or	$ITYPE < - 5$ ,
or	$ITYPE > 5$ .

NAG Library Routine Document

D01BCF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine DocumentD01BCF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

D01BCF