In the above formulae, stands for any polynomial of degree or less in .
The method used to calculate the abscissae involves finding the eigenvalues of the appropriate tridiagonal matrix (see Golub and Welsch (1969)). The weights are then determined by the formula
where is the th orthogonal polynomial with respect to the weight function over the appropriate interval.
The weights and abscissae produced by d01tcc may be passed to d01fbc, which will evaluate the summations in one or more dimensions.
Davis P J and Rabinowitz P (1975) Methods of Numerical Integration Academic Press
Golub G H and Welsch J H (1969) Calculation of Gauss quadrature rules Math. Comput.23 221–230
Stroud A H and Secrest D (1966) Gaussian Quadrature Formulas Prentice–Hall
1: – Nag_QuadTypeInput
On entry: indicates the type of quadrature rule.
Gauss–Legendre, with normal weights.
Gauss–Jacobi, with normal weights.
Gauss–Jacobi, with adjusted weights.
Exponential Gauss, with normal weights.
Exponential Gauss, with adjusted weights.
Gauss–Laguerre, with normal weights.
Gauss–Laguerre, with adjusted weights.
Gauss–Hermite, with normal weights.
Gauss–Hermite, with adjusted weights.
Rational Gauss, with normal weights.
Rational Gauss, with adjusted weights.
, , , , , , , , , or .
2: – doubleInput
3: – doubleInput
4: – doubleInput
5: – doubleInput
On entry: the parameters , , and which occur in the quadrature formulae described in Section 3. c is not used if ; d is not used unless , , or . For some rules c and d must not be too large (see Section 6).
if , ;
if or , and and ;
if or , and ;
if or , and ;
if or , and ;
if or , and and .
6: – IntegerInput
On entry: , the number of weights and abscissae to be returned. If or and , an odd value of n may raise problems (see NE_INDETERMINATE).
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument had an illegal value.
On entry, a, b, c, or d is not in the allowed range: , , and .
The algorithm for computing eigenvalues of a tridiagonal matrix has failed to converge.
The contribution of the central abscissa to the summation is indeterminate.
On entry, .
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 7.5 in the Introduction to the NAG Library CL 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 CL Interface for further information.
One or more of the weights are larger than , the largest floating point number on this computer (see X02ALC): . Possible solutions are to use a smaller value of ; or, if using adjusted weights to change to normal weights.
One or more of the weights are too small to be distinguished from zero on this machine. The underflowing weights are returned as zero, which may be a usable approximation. Possible solutions are to use a smaller value of ; or, if using normal weights, to change to adjusted weights.
The accuracy depends mainly on , with increasing loss of accuracy for larger values of . Typically, one or two decimal digits may be lost from machine accuracy with , and three or four decimal digits may be lost for .
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
d01tcc 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 major portion of the time is taken up during the calculation of the eigenvalues of the appropriate tridiagonal matrix, where the time is roughly proportional to .
This example returns the abscissae and (adjusted) weights for the seven-point Gauss–Laguerre formula.