d01bc 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.

Syntax

C#
public static void d01bc( int itype, double a, double b, double c, double d, int n, double[] weight, double[] abscis, out int ifail )

Visual Basic
Public Shared Sub d01bc ( _ itype As Integer, _ a As Double, _ b As Double, _ c As Double, _ d As Double, _ n As Integer, _ weight As Double(), _ abscis As Double(), _ <OutAttribute> ByRef ifail As Integer _ )

Visual Basic

Public Shared Sub d01bc ( _
	itype As Integer, _
	a As Double, _
	b As Double, _
	c As Double, _
	d As Double, _
	n As Integer, _
	weight As Double(), _
	abscis As Double(), _
	<OutAttribute> ByRef ifail As Integer _
)

Visual C++
public: static void d01bc( int itype, double a, double b, double c, double d, int n, array<double>^ weight, array<double>^ abscis, [OutAttribute] int% ifail )

F#
static member d01bc : itype : int * a : float * b : float * c : float * d : float * n : int * weight : float[] * abscis : float[] * ifail : int byref -> unit

Parameters

itype

Type: System..::..Int32

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

a: Type: System..::..Double
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 $itype = 1$ , $-1$ , $5$ or $-5$ . For some rules c and d must not be too large (see [Error Indicators and Warnings]).

b: Type: System..::..Double
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 $itype = 1$ , $-1$ , $5$ or $-5$ . For some rules c and d must not be too large (see [Error Indicators and Warnings]).

c: Type: System..::..Double
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 $itype = 1$ , $-1$ , $5$ or $-5$ . For some rules c and d must not be too large (see [Error Indicators and Warnings]).

d: Type: System..::..Double
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 $itype = 1$ , $-1$ , $5$ or $-5$ . For some rules c and d must not be too large (see [Error Indicators and Warnings]).

n: Type: System..::..Int32
On entry: $n$ , the number of weights and abscissae to be returned. If $itype = -2$ or $-4$ and $c \neq 0.0$ , an odd value of n may raise problems (see $ifail = 6$ ).

Constraint: $n > 0$ .

weight: Type: array<System..::..Double>[]()[][]
An array of size [n]
On exit: the n weights.

abscis: Type: array<System..::..Double>[]()[][]
An array of size [n]
On exit: the n abscissae.

ifail: Type: System..::..Int32%
On exit: $ifail = 0$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

Description

d01bc 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 d01bc may be passed to (D01FBF not in this release), 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

Error Indicators and Warnings

Errors or warnings detected by the method:

$ifail = 1$: 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.

$ifail = 2$

On entry,	$n < 1$ ,
or	$itype < - 5$ ,
or	$itype > 5$ .

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

$ifail = 3$

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 s14aa).

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

$ifail = 4$: One or more of the weights are larger than $rmax$ , the largest floating-point number on this machine. $rmax$ is given by the function x02al. 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.

$ifail = 5$: 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.

$ifail = 6$

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.

$ifail = -9000$: An error occured, see message report.
$ifail = -8000$: Negative dimension for array $〈value〉$
$ifail = -6000$: Invalid Parameters $〈value〉$

Accuracy

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

Parallelism and Performance

None.

Further Comments

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}

Example

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

Example program (C#): d01bce.cs

Example program results: d01bce.r

Syntax

Parameters

Description

References

Error Indicators and Warnings

Accuracy

Parallelism and Performance

Further Comments

Example

See Also