NAG C Library Function Document

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.

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References

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

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Arguments

1: $n$ – IntegerInput: 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: $a [n]$ – const doubleInput: On entry: a contains the coefficients $a (j)$ .
3: $b [n]$ – doubleInput/Output: On entry: b contains the coefficients $b (j)$ .

On exit: elements of b are altered to make the underlying eigenvalue problem symmetric.
4: $c [n]$ – doubleInput/Output: On entry: c contains the coefficients $c (j)$ .

On exit: elements of c are altered to make the underlying eigenvalue problem symmetric.
5: $muzero$ – doubleInput: On entry: muzero contains the definite integral of the weight function for the interval of interest.
6: $weight [n]$ – doubleOutput: On exit: $weight (j)$ contains the weight corresponding to the $j$ th abscissa.
7: $abscis [n]$ – doubleOutput: On exit: $abscis (j)$ the $j$ th abscissa.
8: $fail$ – NagError *Input/Output: The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

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Error 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.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $n = 〈value〉$ .
Constraint: $n \geq 1$ .
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.

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Accuracy

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

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Parallelism and Performance

nag_quad_1d_gauss_wrec (d01tdc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

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.

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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.