NAG CPP Interface
nagcpp::quad::dim1_gauss_wres (d01tb)

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1 Purpose

dim1_gauss_wres returns the weights and abscissae appropriate to a Gaussian quadrature formula with a specified number of abscissae. The formulae provided are for Gauss–Legendre, rational Gauss, Gauss–Laguerre and Gauss–Hermite.

2 Specification

#include "d01/nagcpp_d01tb.hpp"
template <typename WEIGHT, typename ABSCIS>

void function dim1_gauss_wres(const types::f77_integer key, const double a, const double b, const types::f77_integer n, WEIGHT &&weight, ABSCIS &&abscis, OptionalD01TB opt)
template <typename WEIGHT, typename ABSCIS>

void function dim1_gauss_wres(const types::f77_integer key, const double a, const double b, const types::f77_integer n, WEIGHT &&weight, ABSCIS &&abscis)

3 Description

dim1_gauss_wres returns the weights and abscissae for use in the Gaussian quadrature of a function f(x). The quadrature takes the form
S=i=1nwif(xi)  
where wi are the weights and xi are the abscissae (see Davis and Rabinowitz (1975), Fröberg (1970), Ralston (1965) or Stroud and Secrest (1966)).
Weights and abscissae are available for Gauss–Legendre, rational Gauss, Gauss–Laguerre and Gauss–Hermite quadrature, and for a selection of values of n (see Section 5).
  1. (a)Gauss–Legendre Quadrature:
    Sabf(x)dx  
    where a and b are finite and it will be exact for any function of the form
    f(x)=i=0 2n-1cixi.  
  2. (b)Rational Gauss quadrature, adjusted weights:
    Saf(x) dx (a+b>0)   or   S-a f(x) dx (a+b<0)  
    and will be exact for any function of the form
    f(x)=i=2 2n+1ci(x+b)i=i=0 2n-1c2n+1-i(x+b)i(x+b)2n+1.  
  3. (c)Gauss–Laguerre quadrature, adjusted weights:
    Saf(x) dx (b>0)   or   S-a f(x) dx (b<0)  
    and will be exact for any function of the form
    f(x)=e-bxi=0 2n-1cixi.  
  4. (d)Gauss–Hermite quadrature, adjusted weights:
    S- + f(x) dx  
    and will be exact for any function of the form
    f(x)=e-b (x-a) 2i=0 2n-1cixi(b>0).  
  5. (e)Gauss–Laguerre quadrature, normal weights:
    Sae-bxf(x) dx (b>0)   or   S-a e-bxf(x) dx (b<0)  
    and will be exact for any function of the form
    f(x)=i=0 2n-1cixi.  
  6. (f)Gauss–Hermite quadrature, normal weights:
    S- + e-b (x-a) 2f(x) dx 
    and will be exact for any function of the form
    f(x)=i=0 2n-1cixi.  
Note:  the Gauss–Legendre abscissae, with a=-1, b=+1, are the zeros of the Legendre polynomials; the Gauss–Laguerre abscissae, with a=0, b=1, are the zeros of the Laguerre polynomials; and the Gauss–Hermite abscissae, with a=0, b=1, are the zeros of the Hermite polynomials.

4 References

Davis P J and Rabinowitz P (1975) Methods of Numerical Integration Academic Press
Fröberg C E (1970) Introduction to Numerical Analysis Addison–Wesley
Ralston A (1965) A First Course in Numerical Analysis pp. 87–90 McGraw–Hill
Stroud A H and Secrest D (1966) Gaussian Quadrature Formulas Prentice–Hall

5 Arguments

1: key types::f77_integer Input
On entry: indicates the quadrature formula.
key=0
Gauss–Legendre quadrature on a finite interval, using normal weights.
key=3
Gauss–Laguerre quadrature on a semi-infinite interval, using normal weights.
key=-3
Gauss–Laguerre quadrature on a semi-infinite interval, using adjusted weights.
key=4
Gauss–Hermite quadrature on an infinite interval, using normal weights.
key=-4
Gauss–Hermite quadrature on an infinite interval, using adjusted weights.
key=-5
Rational Gauss quadrature on a semi-infinite interval, using adjusted weights.
Constraint: key=0, 3, -3, 4, -4 or -5.
2: a double Input
On entry: the parameters a and b which occur in the quadrature formulae described in Section 3.
Constraints:
  • if key=-5, a+b0.0;
  • if key=3 or -3, b0.0;
  • if key=4 or -4, b>0.0.
Constraints:
  • Rational Gauss: a+b0.0;
  • Gauss–Laguerre: b0.0;
  • Gauss–Hermite: b>0.
3: b double Input
On entry: the parameters a and b which occur in the quadrature formulae described in Section 3.
Constraints:
  • if key=-5, a+b0.0;
  • if key=3 or -3, b0.0;
  • if key=4 or -4, b>0.0.
Constraints:
  • Rational Gauss: a+b0.0;
  • Gauss–Laguerre: b0.0;
  • Gauss–Hermite: b>0.
4: n types::f77_integer Input
On entry: n, the number of weights and abscissae to be returned.
Constraint: n=1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 16, 20, 24, 32, 48 or 64.
Note: if n>0 and is not a member of the above list, the maxmium value of n stored below n will be used, and all subsequent elements of abscis and weight will be returned as zero.
5: weight(n) double array Output
On exit: the n weights.
6: abscis(n) double array Output
On exit: the n abscissae.
7: opt OptionalD01TB Input/Output
Optional parameter container, derived from Optional.

6 Exceptions and Warnings

Errors or warnings detected by the function:
All errors and warnings have an associated numeric error code field, errorid, stored either as a member of the thrown exception object (see errorid), or as a member of opt.ifail, depending on how errors and warnings are being handled (see Error Handling for more details).
Raises: WarningException
errorid=1
The n-point rule is not among those stored.
On entry: n = value.
n-rule used: n = value.
errorid=2
Underflow occurred in calculation of normal weights.
Reduce n or use adjusted weights: n = value.
errorid=3
No nonzero weights were generated for the provided parameters.
Raises: ErrorException
errorid=11
On entry, key = value.
Constraint: key = 0,3,-3,4,-4​ or ​-5.
errorid=12
The value of a and/or b is invalid for Gauss-Laguerre quadrature.
On entry, key = value.
On entry, a=value and b=value.
Constraint: |b|>0.0.
errorid=12
The value of a and/or b is invalid for Gauss-Hermite quadrature.
On entry, key = value.
On entry, a=value and b=value.
Constraint: b>0.0.
errorid=12
The value of a and/or b is invalid for rational Gauss quadrature.
On entry, key = value.
On entry, a=value and b=value.
Constraint: |a+b|>0.0.
errorid=14
On entry, n = value.
Constraint: n > 0.
errorid=10601
On entry, argument value must be a vector of size value array.
Supplied argument has value dimensions.
errorid=10601
On entry, argument value must be a vector of size value array.
Supplied argument was a vector of size value.
errorid=10601
On entry, argument value must be a vector of size value array.
The size for the supplied array could not be ascertained.
errorid=10602
On entry, the raw data component of value is null.
errorid=10603
On entry, unable to ascertain a value for value.
errorid=-99
An unexpected error has been triggered by this routine.
errorid=-399
Your licence key may have expired or may not have been installed correctly.
errorid=-999
Dynamic memory allocation failed.

7 Accuracy

The weights and abscissae are stored for standard values of a and b to full machine accuracy.

8 Parallelism and Performance

Please see the description for the underlying computational routine in this section of the FL Interface documentation.

9 Further Comments

Timing is negligible.

10 Example

Examples of the use of this method may be found in the examples for: md_​gauss.