NAG CL Interface
d01tcc (dim1_​gauss_​wgen)

1 Purpose

d01tcc 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

#include <nag.h>
void  d01tcc (Nag_QuadType quad_type, double a, double b, double c, double d, Integer n, double weight[], double abscis[], NagError *fail)
The function may be called by the names: d01tcc, nag_quad_dim1_gauss_wgen or nag_quad_1d_gauss_wgen.

3 Description

d01tcc returns the weights wi and abscissae xi for use in the summation
which approximates a definite integral (see Davis and Rabinowitz (1975) or Stroud and Secrest (1966)). The following types are provided:
  1. (a)Gauss–Legendre
    Sab fxdx,   exact for ​fx=P2n- 1x.  
    Constraint: b>a.
  2. (b)Gauss–Jacobi
    normal weights:
    Sabb-xcx-adfxdx,   exact for ​fx=P2n-1x,  
    adjusted weights:
    Sab fxdx,   exact for ​fx=b-xcx-ad P2n- 1x.  
    Constraint: c>-1, d>-1, b>a.
  3. (c)Exponential Gauss
    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.
  4. (d)Gauss–Laguerre
    normal weights:
    S ax-ace-bxfxdxb>0, -ax-ace-bxfxdxb<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, b0.
  5. (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.
  6. (f)Rational Gauss
    normal weights:
    S ax-acx+bdfxdxa+b>0, -ax-acx+bdfxdxa+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+b0.
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 d01tcc may be passed to d01fbc, 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 Arguments

1: quad_type Nag_QuadType Input
On entry: 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.
Constraint: quad_type=Nag_Quad_Gauss_Legendre, Nag_Quad_Gauss_Jacobi, Nag_Quad_Gauss_Jacobi_Adjusted, Nag_Quad_Gauss_Exponential, Nag_Quad_Gauss_Exponential_Adjusted, Nag_Quad_Gauss_Laguerre, Nag_Quad_Gauss_Laguerre_Adjusted, Nag_Quad_Gauss_Hermite, Nag_Quad_Gauss_Hermite_Adjusted, Nag_Quad_Gauss_Rational or Nag_Quad_Gauss_Rational_Adjusted.
2: a double Input
3: b double Input
4: c double Input
5: d double Input
On entry: the parameters a, b, c and d which occur in the quadrature formulae. c is not used if quad_type=Nag_Quad_Gauss_Legendre; d is not used unless quad_type=Nag_Quad_Gauss_Jacobi, Nag_Quad_Gauss_Jacobi_Adjusted, Nag_Quad_Gauss_Rational or Nag_Quad_Gauss_Rational_Adjusted. For some rules c and d must not be too large (see Section 6).
  • if quad_type=Nag_Quad_Gauss_Legendre, a<b;
  • if quad_type=Nag_Quad_Gauss_Jacobi or Nag_Quad_Gauss_Jacobi_Adjusted, a<b and c>-1.0 and d>-1.0;
  • if quad_type=Nag_Quad_Gauss_Exponential or Nag_Quad_Gauss_Exponential_Adjusted, a<b and c>-1.0;
  • if quad_type=Nag_Quad_Gauss_Laguerre or Nag_Quad_Gauss_Laguerre_Adjusted, b0.0 and c>-1.0;
  • if quad_type=Nag_Quad_Gauss_Hermite or Nag_Quad_Gauss_Hermite_Adjusted, b>0.0 and c>-1.0;
  • if quad_type=Nag_Quad_Gauss_Rational or Nag_Quad_Gauss_Rational_Adjusted, a+b0.0 and c>-1.0 and d>c+1.0.
6: n Integer Input
On entry: n, the number of weights and abscissae to be returned. If quad_type=Nag_Quad_Gauss_Exponential_Adjusted or Nag_Quad_Gauss_Hermite_Adjusted and c0.0, an odd value of n may raise problems (see fail.code= NE_INDETERMINATE).
Constraint: n>0.
7: weight[n] double Output
On exit: the n weights.
8: abscis[n] double Output
On exit: the n abscissae.
9: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument value had an illegal value.
On entry, a, b, c, or d is not in the allowed range: a=value, b=value c=value, d=value and quad_type=value.
The algorithm for computing eigenvalues of a tridiagonal matrix has failed to converge.
The contribution of the central abscissa to the summation is indeterminate.
On entry, n=value.
Constraint: n>0.
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
One or more of the weights are larger than rmax, the largest floating point number on this computer (see X02ALC): rmax=value.
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.
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.

7 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 n20, and three or four decimal digits may be lost for n100.

8 Parallelism and Performance

d01tcc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 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 n3.

10 Example

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

10.1 Program Text

Program Text (d01tcce.c)

10.2 Program Data

Program Data (d01tcce.d)

10.3 Program Results

Program Results (d01tcce.r)
GnuplotProduced by GNUPLOT 4.6 patchlevel 3 0 1 2 3 4 5 6 7 8 9 −5 0 5 10 15 20 25 Weights at Abscissae x Example Program Abscissae and Weights for the 7-point Gauss-Laguerre Formula (a=0, b=1) gnuplot_plot_1