The method employed is based on the three-term recurrence relation due to Clenshaw (1955), with modifications to give greater numerical stability due to Reinsch and Gentleman (see Gentleman (1969)).

For further details of the algorithm and its use see Cox (1974) and Cox and Hayes (1973).

4 References

Clenshaw C W (1955) A note on the summation of Chebyshev series Math. Tables Aids Comput. 9 118–120

Cox M G (1974) A data-fitting package for the non-specialist user Software for Numerical Mathematics (ed D J Evans) Academic Press

Cox M G and Hayes J G (1973) Curve fitting: a guide and suite of algorithms for the non-specialist user NPL Report NAC26 National Physical Laboratory

Gentleman W M (1969) An error analysis of Goertzel's (Watt's) method for computing Fourier coefficients Comput. J. 12 160–165

5 Arguments

1: $nplus1$ – Integer Input: On entry: the number $n + 1$ of terms in the series (i.e., one greater than the degree of the polynomial).

Constraint: $nplus1 \geq 1$ .
2: $a (nplus1)$ – Real (Kind=nag_wp) array Input: On entry: $a (i)$ must be set to the value of the $i$ th coefficient in the series, for $i = 1, 2, \dots, n + 1$ .
3: $xcap$ – Real (Kind=nag_wp) Input: On entry: $\bar{x}$ , the argument at which the polynomial is to be evaluated. It should lie in the range $- 1$ to $+ 1$ , but a value just outside this range is permitted (see Section 6) to allow for possible rounding errors committed in the transformation from $x$ to $\bar{x}$ discussed in Section 3. Provided the recommended form of the transformation is used, a successful exit is thus assured whenever the value of $x$ lies in the range $x_{\min}$ to $x_{\max}$ .
4: $p$ – Real (Kind=nag_wp) Output: On exit: the value of the polynomial.
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, $xcap = ⟨ value ⟩$ and $EPS = ⟨ value ⟩$ .
Constraint: $| xcap | \leq 1 + 4 \times EPS$ , where $EPS$ is machine precision.

$ifail = 2$: On entry, $nplus1 = ⟨ value ⟩$ .
Constraint: $nplus1 \geq 1$ .

$ifail = - 99$: 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 rounding errors committed are such that the computed value of the polynomial is exact for a slightly perturbed set of coefficients

a_{i} + δ a_{i}

. The ratio of the sum of the absolute values of the

δ a_{i}

to the sum of the absolute values of the

a_{i}

is less than a small multiple of

(n + 1) \times machine precision

8 Parallelism and Performance

e02aef is not threaded in any implementation.

9 Further Comments

The time taken is approximately proportional to

n + 1

It is expected that a common use of e02aef will be the evaluation of the polynomial approximations produced by e02adf and e02aff.

10 Example

Evaluate at

11

equally-spaced points in the interval

- 1 \leq \bar{x} \leq 1

the polynomial of degree

4

with Chebyshev coefficients,

2.0

0.5

0.25

0.125

0.0625

The example program is written in a general form that will enable a polynomial of degree

n

in its Chebyshev series form to be evaluated at

m

equally-spaced points in the interval

- 1 \leq \bar{x} \leq 1

. The program is self-starting in that any number of datasets can be supplied.

e02ae: FL CL CPP AD

NAG FL Interfacee02aef (dim1_​cheb_​eval)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
e02aef (dim1_cheb_eval)