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:

p (j) (x) = (a (j) x + b (j)) p (j - 1) (x) - c (j) p (j - 2) (x)

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 $2 n - 1$ .

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 $j$ th abscissa.
7: $abscis (n)$ – Real (Kind=nag_wp) array Output: On exit: $abscis (j)$ the $j$ th 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

−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 number of weights and abscissae requested (n) is less than $1$ : $n = ⟨ value ⟩$ .

$ifail = 4$: Unexpected failure in eigenvalue computation. Please contact NAG.

$ifail = 5$: The algorithm failed to converge. The $i$ th diagonal was not zero: $i = ⟨ value ⟩$ .

$ifail = - 99$: 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.

$ifail = - 399$: 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.

$ifail = - 999$: 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.