d01tec should only be used if the three-term recurrence cannot be determined analytically. A system of equations are formed, using the moments provided. This set of equations becomes ill-conditioned for moderate values of

n

, the number of abscissae and weights required. In most implementations quadruple precision calculation is used to maintain as much accuracy as possible.

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 weights and abscissae required.

Constraint: $n > 0$ .
2: $mu [0 : 2 * n]$ – const double Input: On entry: $mu [i]$ must contain the value of the moment with respect to $x^{i}$ i.e., $\int w (x) x^{i} d x$ , for $i = 0, 1, \dots, 2 n$ .
3: $a [n]$ – double Output: On exit: values helping define the three term recurrence used by d01tdc.
4: $b [n]$ – double Output: On exit: values helping define the three term recurrence used by d01tdc.
5: $c [n]$ – double Output: On exit: values helping define the three term recurrence used by d01tdc.
6: $fail$ – NagError * Input/Output: The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_DATA_ILL_CONDITIONED: The problem is too ill conditioned, it breaks down at row $〈value〉$ .
NE_INT: The number of weights and abscissae requested (n) is less than $1$ : $n = 〈value〉$ .
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 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: 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.

7 Accuracy

Internally quadruple precision is used to minimize loss of accuracy as much as possible.

8 Parallelism and Performance

d01tec is not threaded in any implementation.

9 Further Comments

Because the function cannot check the validity of all the data presented, you are advised to independently check the result, perhaps by integrating a function whose integral is known, using d01tec and subsequently d01tdc, to compare answers.

10 Example

This example program uses d01tec and moments to calculate a three-term recurrence relationship appropriate for Gauss–Legendre quadrature. It then uses the recurrence relationship to derive the weights and abscissae by calling d01tdc.