1: $n$ – Integer Input: On entry: $n$ , where the number of grid points is $n + 1$ . This is also the largest order of Chebyshev polynomial in the Chebyshev series to be computed.

Constraint: $n > 0$ and n is even.
2: $a$ – Real (Kind=nag_wp) Input: On entry: $a$ , the lower bound of domain $[a, b]$ .

Constraint: $a < b$ .
3: $b$ – Real (Kind=nag_wp) Input: On entry: $b$ , the upper bound of domain $[a, b]$ .

Constraint: $b > a$ .
4: $x (n + 1)$ – Real (Kind=nag_wp) array Output: On exit: the Chebyshev Gauss–Lobatto grid points, $x_{i}$ , for $i = 1, 2, \dots, n + 1$ , on $[a, b]$ .
5: $ifail$ – 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:

$ifail = 1$: On entry, $n = ⟨ value ⟩$ .
Constraint: $n > 0$ .

On entry, $n = ⟨ value ⟩$ .
Constraint: n is even.

$ifail = 2$: On entry, $a = ⟨ value ⟩$ and $b = ⟨ value ⟩$ .
Constraint: $a < b$ .

$ifail = - 99$: 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

The Chebyshev Gauss–Lobatto grid points computed should be accurate to within a small multiple of machine precision.

8 Parallelism and Performance

d02ucf is not threaded in any implementation.

9 Further Comments

The number of operations is of the order

n \log (n)

and there are no internal memory requirements; thus the computation remains efficient and practical for very fine discretizations (very large values of