NAG FL Interface
d01tdf (dim1_​gauss_​wrec)

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1 Purpose

d01tdf computes the weights and abscissae of a Gaussian quadrature rule using the method of Golub and Welsch.

2 Specification

Fortran Interface
Subroutine d01tdf ( n, a, b, c, muzero, weight, abscis, ifail)
Integer, Intent (In) :: n
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (In) :: a(n), muzero
Real (Kind=nag_wp), Intent (Inout) :: b(n), c(n)
Real (Kind=nag_wp), Intent (Out) :: weight(n), abscis(n)
C Header Interface
#include <nag.h>
void  d01tdf_ (const Integer *n, const double a[], double b[], double c[], const double *muzero, double weight[], double abscis[], Integer *ifail)
The routine may be called by the names d01tdf or nagf_quad_dim1_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 routines 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) Real (Kind=nag_wp) array Input
On entry: a contains the coefficients a(j).
3: b(n) Real (Kind=nag_wp) array 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) Real (Kind=nag_wp) array 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 Real (Kind=nag_wp) Input
On entry: muzero contains the definite integral of the weight function for the interval of interest.
6: weight(n) Real (Kind=nag_wp) array Output
On exit: weight(j) contains the weight corresponding to the jth abscissa.
7: abscis(n) Real (Kind=nag_wp) array Output
On exit: abscis(j) the jth abscissa.
8: ifail Integer Input/Output
On entry: ifail must be set to 0, −1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of −1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value −1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

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:
The number of weights and abscissae requested (n) is less than 1: n=value.
Unexpected failure in eigenvalue computation. Please contact NAG.
The algorithm failed to converge. The ith diagonal was not zero: i=value.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL 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 FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL 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.
d01tdf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
d01tdf 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 routine. 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 routine 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 (d01tdfe.f90)

10.2 Program Data


10.3 Program Results

Program Results (d01tdfe.r)