The method employed is based upon 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), 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: $a [nplus1]$ – const double Input: On entry: $a [i - 1]$ must be set to the value of the $i$ th coefficient in the series, for $i = 1, 2, \dots, n + 1$ .
3: $xcap$ – double 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 9) 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: $p$ – double * Output: On exit: the value of the polynomial.
5: $fail$ – 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_INT_ARG_LT: On entry, nplus1 must not be less than 1: $nplus1 = ⟨ value ⟩$ .
NE_INVALID_XCAP: On entry, $abs (xcap) > 1.0 + 4 ε$ , where $ε$ is the machine precision.
In this case the value of p is set arbitrarily to zero.

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

e02aec is not threaded in any implementation.

9 Further Comments

The time taken by e02aec is approximately proportional to

n + 1

It is expected that a common use of e02aec will be the evaluation of the polynomial approximations produced by e02adc and e02afc.

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 CL Interfacee02aec (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 CL Interface
e02aec (dim1_cheb_eval)