NAG CL Interface
d01tdc (dim1_​gauss_​wrec)

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1 Purpose

d01tdc computes the weights and abscissae of a Gaussian quadrature rule using the method of Golub and Welsch.

2 Specification

#include <nag.h>
void  d01tdc (Integer n, const double a[], double b[], double c[], double muzero, double weight[], double abscis[], NagError *fail)
The function may be called by the names: d01tdc, nag_quad_dim1_gauss_wrec or nag_quad_1d_gauss_wrec.

3 Description

A tri-diagonal system of equations is formed from the coefficients of an underlying three-term recurrence formula:
for a set of othogonal polynomials p(j) induced by the quadrature. This is described in greater detail in the D01 Chapter Introduction. The user is required to specify the three-term recurrence and the value of the integral of the chosen weight function over the chosen interval.
As described in Golub and Welsch (1969) the abscissae are computed from the eigenvalues of this matrix and the weights from the first component of the eigenvectors.
LAPACK functions are used for the linear algebra to speed up computation.

4 References

Golub G H and Welsch J H (1969) Calculation of Gauss quadrature rules Math. Comput. 23 221–230

5 Arguments

1: n Integer Input
On entry: n, the number of Gauss points required. The resulting quadrature rule will be exact for all polynomials of degree 2n-1.
Constraint: n>0.
2: a[n] const double Input
On entry: a contains the coefficients a(j).
3: b[n] double Input/Output
On entry: b contains the coefficients b(j).
On exit: elements of b are altered to make the underlying eigenvalue problem symmetric.
4: c[n] double Input/Output
On entry: c contains the coefficients c(j).
On exit: elements of c are altered to make the underlying eigenvalue problem symmetric.
5: muzero double Input
On entry: muzero contains the definite integral of the weight function for the interval of interest.
6: weight[n] double Output
On exit: weight[j-1] contains the weight corresponding to the jth abscissa.
7: abscis[n] double Output
On exit: abscis[j-1] the jth abscissa.
8: 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, n=value.
Constraint: n1.
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.

7 Accuracy

In general the computed weights and abscissae are accurate to a reasonable multiple of machine precision.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
d01tdc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
d01tdc 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 weight function must be non-negative to obtain sensible results. This and the validity of muzero are not something that the function can check, so please be particularly careful. If possible check the computed weights and abscissae by integrating a function with a function for which you already know the integral.

10 Example

This example program generates the weights and abscissae for the 4-point Gauss rules: Legendre, Chebyshev1, Chebyshev2, Jacobi, Laguerre and Hermite.

10.1 Program Text

Program Text (d01tdce.c)

10.2 Program Data

Program Data (d01tdce.d)

10.3 Program Results

Program Results (d01tdce.r)