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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_fit_1dcheb_eval (e02ae)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_fit_1dcheb_eval (e02ae) evaluates a polynomial from its Chebyshev series representation.


[p, ifail] = e02ae(a, xcap, 'nplus1', nplus1)
[p, ifail] = nag_fit_1dcheb_eval(a, xcap, 'nplus1', nplus1)


nag_fit_1dcheb_eval (e02ae) evaluates the polynomial
for any value of x- satisfying -1x-1. Here Tjx- denotes the Chebyshev polynomial of the first kind of degree j with argument x-. The value of n is prescribed by you.
In practice, the variable x- will usually have been obtained from an original variable x, where xminxxmax and
x-=x-xmin-xmax-x xmax-xmin  
Note that this form of the transformation should be used computationally rather than the mathematical equivalent
x-= 2x-xmin-xmax xmax-xmin  
since the former guarantees that the computed value of x- differs from its true value by at most 4ε, where ε is the machine precision, whereas the latter has no such guarantee.
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).


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


Compulsory Input Parameters

1:     anplus1 – double array
ai must be set to the value of the ith coefficient in the series, for i=1,2,,n+1.
2:     xcap – double scalar
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 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 xmin to xmax.

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: nplus11.

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:
ABSxcap>1.0+4ε, where ε is the machine precision. In this case the value of p is set arbitrarily to zero.
On entry,nplus1<1.
An unexpected error has been triggered by this routine. Please contact NAG.
Your licence key may have expired or may not have been installed correctly.
Dynamic memory allocation failed.


The rounding errors committed are such that the computed value of the polynomial is exact for a slightly perturbed set of coefficients ai+δai. The ratio of the sum of the absolute values of the δai to the sum of the absolute values of the ai is less than a small multiple of n+1×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).


Evaluate at 11 equally-spaced points in the interval -1x-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 -1x-1. The program is self-starting in that any number of datasets can be supplied.
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));
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

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