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

1 Purpose

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.

2 Specification

SUBROUTINE D01BCF (	ITYPE, A, B, C, D, N, WEIGHT, ABSCIS, IFAIL)
INTEGER	ITYPE, N, IFAIL
*double precision*	A, B, C, D, WEIGHT(N), ABSCIS(N)

3 Description

D01BCF returns the weights wi and abscissae xi for use in the summation

S=∑i=1nwifxi

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

(a)

Gauss–Legendre

S≃∫ab fxdx, exact for fx=P2n- 1x.

Constraint: b>a.

(b)

Gauss–Jacobi

normal weights:

S≃∫abb-xcx-adfxdx, exact for fx=P2n-1x,

adjusted weights:

S≃∫ab fxdx, exact for fx=b-xcx-ad P2n- 1x.

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

(c)

Gauss–Exponential

normal weights:

S≃ ∫ab x- a+b 2 c fxdx, exact for fx=P2n-1x,

adjusted weights:

S ≃ ∫ab fxdx, exact for fx = x-a+b2 c P2n- 1 x.

Constraint: c>-1, b>a.

(d)

Gauss–Laguerre

normal weights:

S ≃∫a∞x-ace-bxfxdx b>0, ≃∫-∞ax-ace-bxfxdx b<0, exact for fx=P2n-1x,

adjusted weights:

S ≃∫a∞ fx dx b> 0, ≃∫-∞a fx dx b< 0, exact for fx=x-ace-bxP2n- 1x.

Constraint: c>-1, b≠0.

(e)

Gauss–Hermite

normal weights:

S≃∫-∞ +∞x-ace-b x-a 2fxdx, exact for fx=P2n-1x,

adjusted weights:

S≃∫-∞ +∞ fxdx, exact for fx=x-ac e-b x-a 2 P2n- 1x.

Constraint: c>-1, b>0.

(f)

Gauss–Rational

normal weights:

S ≃∫a∞x-acx+bdfxdx a+b>0, ≃∫-∞ax-acx+bdfxdx a+b<0, exact for fx=P2n-1 1x+b ,

adjusted weights:

S ≃∫a∞ fx dx a+b> 0, ≃∫-∞a fx dx a+b< 0, exact for fx=x-acx+bd P2n- 1 1x+b .

Constraint: c>-1, d>c+1, a+b≠0.

In the above formulae, P2n-1x 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

wi= ∑j=0 n-1Pj* xi 2 -1

where Pj*x is the jth 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.

4 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

5 Parameters

1: ITYPE – INTEGERInput

On entry: indicates the type of quadrature rule.

ITYPE=0: Gauss–Legendre.
ITYPE=1: Gauss–Jacobi.
ITYPE=2: Gauss–Exponential.
ITYPE=3: Gauss–Laguerre.
ITYPE=4: Gauss–Hermite.
ITYPE=5: Gauss–Rational.

The above values give the normal weights; the adjusted weights are obtained if the value of ITYPE above is negated.

Constraint: -5≤ITYPE≤5.

2: A – double precisionInput

3: B – double precisionInput

4: C – double precisionInput

5: D – double precisionInput

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 or ±5. For some rules C and D must not be too large (See Section 6.)

Constraints:

if ITYPE=0, A<B;
if ITYPE=±1, A<B and C>-1 and D>-1;
if ITYPE=±2, A<B and C>-1;
if ITYPE=±3, B≠0 and C>-1;
if ITYPE=±4, B>0 and C>-1;
if ITYPE=±5, A+B≠0 and C>-1 and D>C+1.

6: N – INTEGERInput

On entry: n, the number of weights and abscissae to be returned. If ITYPE=-2 or -4 and C≠0.0, an odd value of N may raise problems (see IFAIL=6).

Constraint: N>0.

7: WEIGHT(N) – double precision arrayOutput

On exit: the N weights.

8: ABSCIS(N) – double precision arrayOutput

On exit: the N abscissae.

9: IFAIL – INTEGERInput/Output

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.

On exit: IFAIL=0 unless the routine detects an error (see Section 6).

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.

6 Error Indicators and Warnings

If on entry IFAIL=0 or -1, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

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≥B;
if ITYPE=±1, A≥B or C≤-1.0 or D≤-1.0 or C+D+2.0>gmax;
if ITYPE=±2, A≥B or C≤-1.0;
if ITYPE=±3, B=0.0 or C≤-1.0 or C+1.0>gmax;
if ITYPE=±4, B≤0.0 or C≤-1.0 or C+1.0/2.0>gmax;
if ITYPE=±5, A+B=0.0 or C≤-1.0 or D≤C+1.0.

Here gmax is the (machine-dependent) largest integer value such that Γgmax 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.

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 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.

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

Gauss–Exponential or Gauss–Hermite adjusted weights with N odd and C≠0.0. Theoretically, in these cases:

for C>0.0, the central adjusted weight is infinite, and the exact function fx is zero at the central abscissa.
for C<0.0, the central adjusted weight is zero, and the exact function fx 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 ∞ 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 fx is involved.