# NAG Library Routine Document

## 1Purpose

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

## 2Specification

Fortran Interface
 Subroutine d01tdf ( n, a, b, c,
 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)
#include nagmk26.h
 void d01tdf_ (const Integer *n, const double a[], double b[], double c[], const double *muzero, double weight[], double abscis[], Integer *ifail)

## 3Description

A tri-diagonal system of equations is formed from the coefficients of an underlying three-term recurrence formula:
 $pjx=ajx+bjpj-1x-cjpj-2x$
for a set of othogonal polynomials $p\left(j\right)$ 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.

## 4References

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

## 5Arguments

1:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of Gauss points required. The resulting quadrature rule will be exact for all polynomials of degree $2n-1$.
Constraint: ${\mathbf{n}}>0$.
2:     $\mathbf{a}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: a contains the coefficients $a\left(j\right)$.
3:     $\mathbf{b}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: b contains the coefficients $b\left(j\right)$.
On exit: elements of b are altered to make the underlying eigenvalue problem symmetric.
4:     $\mathbf{c}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: c contains the coefficients $c\left(j\right)$.
On exit: elements of c are altered to make the underlying eigenvalue problem symmetric.
5:     $\mathbf{muzero}$ – Real (Kind=nag_wp)Input
On entry: muzero contains the definite integral of the weight function for the interval of interest.
6:     $\mathbf{weight}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{weight}}\left(j\right)$ contains the weight corresponding to the $j$th abscissa.
7:     $\mathbf{abscis}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{abscis}}\left(j\right)$ the $j$th abscissa.
8:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
The number of weights and abscissae requested (n) is less than $1$: ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=4$
${\mathbf{ifail}}=5$
The algorithm failed to converge. The $i$th diagonal was not zero: $i=〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

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

## 8Parallelism and Performance

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.

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.

## 10Example

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

### 10.1Program Text

Program Text (d01tdfe.f90)

None.

### 10.3Program Results

Program Results (d01tdfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017