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).

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

Parameters

Compulsory Input Parameters

1: $a (nplus1)$ – double array: $a (i)$ must be set to the value of the $i$ th coefficient in the series, for $i = 1, 2, \dots, n + 1$ .
2: $xcap$ – double scalar: $\bar{x}$ , 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 Error Indicators and Warnings) to allow for possible rounding errors committed in the transformation from $x$ to $\bar{x}$ discussed in Description. 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 $x_{\max}$ .

Optional Input Parameters

1: $nplus1$ – int64int32nag_int scalar: Default: the dimension of the array a.
The number $n + 1$ of terms in the series (i.e., one greater than the degree of the polynomial).

Constraint: $nplus1 \geq 1$ .

Output Parameters

1: $p$ – double scalar: The value of the polynomial.
2: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$: $ABS (xcap) > 1.0 + 4 ε$ , where $ε$ is the machine precision. In this case the value of p is set arbitrarily to zero.

$ifail = 2$

On entry,

nplus1 < 1

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

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

Further Comments

The time taken is approximately proportional to

n + 1

It is expected that a common use of nag_fit_1dcheb_eval (e02ae) will be the evaluation of the polynomial approximations produced by nag_fit_1dcheb_arb (e02ad) and nag_fit_1dcheb_glp (e02af).

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.

Open in the MATLAB editor: e02ae_example

function e02ae_example


fprintf('e02ae example results\n\n');

a = [2     0.5     0.25     0.125     0.0625];

xcap = [-1:0.2:1]';
for i=1:11
  [p(i), ifail] = e02ae(a, xcap(i));
end
disp('      x        p(x)');
disp([xcap p']);

e02ae example results

      x        p(x)
   -1.0000    0.6875
   -0.8000    0.6613
   -0.6000    0.6943
   -0.4000    0.7433
   -0.2000    0.7843
         0    0.8125
    0.2000    0.8423
    0.4000    0.9073
    0.6000    1.0603
    0.8000    1.3733
    1.0000    1.9375

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox