d01tef 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: $mu (0 : 2 * n)$ – Real (Kind=nag_wp) array 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: $a (n)$ – Real (Kind=nag_wp) array Output: On exit: values helping define the three term recurrence used by d01tdf.
4: $b (n)$ – Real (Kind=nag_wp) array Output: On exit: values helping define the three term recurrence used by d01tdf.
5: $c (n)$ – Real (Kind=nag_wp) array Output: On exit: values helping define the three term recurrence used by d01tdf.
6: $ifail$ – Integer Input/Output: On entry: ifail must be set to $0$ , $−1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $−1$ means that an error message is printed while a value of $1$ means that it is not.

If halting is not appropriate, the value $−1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit: $ifail = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

−1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: The number of weights and abscissae requested (n) is less than $1$ : $n = ⟨ value ⟩$ .

$ifail = 2$: The problem is too ill conditioned, it breaks down at row $⟨ value ⟩$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

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

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

d01tef is not threaded in any implementation.

9 Further Comments

Because the routine 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 d01tef and subsequently d01tdf, to compare answers.

10 Example

This example program uses d01tef 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 d01tdf.