NAG Library Function Document

1Purpose

nag_quad_1d_gauss_wrec (d01tdc) computes the weights and abscissae of a Gaussian quadrature rule using the method of Golub and Welsch.

2Specification

 #include #include
 void nag_quad_1d_gauss_wrec (Integer n, const double a[], double b[], double c[], double muzero, double weight[], double abscis[], NagError *fail)

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 functions are used for the linear algebra to speed up computation.
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:    $\mathbf{a}\left[{\mathbf{n}}\right]$const doubleInput
On entry: a contains the coefficients $a\left(j\right)$.
3:    $\mathbf{b}\left[{\mathbf{n}}\right]$doubleInput/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:    $\mathbf{c}\left[{\mathbf{n}}\right]$doubleInput/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:    $\mathbf{muzero}$doubleInput
On entry: muzero contains the definite integral of the weight function for the interval of interest.
6:    $\mathbf{weight}\left[{\mathbf{n}}\right]$doubleOutput
On exit: ${\mathbf{weight}}\left(j\right)$ contains the weight corresponding to the $j$th abscissa.
7:    $\mathbf{abscis}\left[{\mathbf{n}}\right]$doubleOutput
On exit: ${\mathbf{abscis}}\left(j\right)$ the $j$th abscissa.
8:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.
NE_INTERNAL_ERROR
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 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

nag_quad_1d_gauss_wrec (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.

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.

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 (d01tdce.c)

None.

10.3Program Results

Program Results (d01tdce.r)

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