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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_fit_1dcheb_eval2 (e02ak)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_fit_1dcheb_eval2 (e02ak) evaluates a polynomial from its Chebyshev series representation, allowing an arbitrary index increment for accessing the array of coefficients.


[result, ifail] = e02ak(n, xmin, xmax, a, ia1, x)
[result, ifail] = nag_fit_1dcheb_eval2(n, xmin, xmax, a, ia1, x)


If supplied with the coefficients ai, for i=0,1,,n, of a polynomial px- of degree n, where
nag_fit_1dcheb_eval2 (e02ak) returns the value of px- at a user-specified value of the variable x. Here Tjx- denotes the Chebyshev polynomial of the first kind of degree j with argument x-. It is assumed that the independent variable x- in the interval -1,+1 was obtained from your original variable x in the interval xmin,xmax by the linear transformation
x-=2x-xmax+xmin xmax-xmin .  
The coefficients ai may be supplied in the array a, with any increment between the indices of array elements which contain successive coefficients. This enables the function to be used in surface fitting and other applications, in which the array might have two or more dimensions.
The method employed is based on the three-term recurrence relation due to Clenshaw (see Clenshaw (1955)), with modifications due to Reinsch and Gentleman (see Gentleman (1969)). For further details of the algorithm and its use see Cox (1973) 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 (1973) A data-fitting package for the non-specialist user NPL Report NAC 40 National Physical Laboratory
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:     n int64int32nag_int scalar
n, the degree of the given polynomial px-.
Constraint: n0.
2:     xmin – double scalar
3:     xmax – double scalar
The lower and upper end points respectively of the interval xmin,xmax. The Chebyshev series representation is in terms of the normalized variable x-, where
x-=2x-xmax+xmin xmax-xmin .  
Constraint: xmin<xmax.
4:     ala – double array
la, the dimension of the array, must satisfy the constraint lanp1-1 ×ia1+1.
The Chebyshev coefficients of the polynomial px-. Specifically, element i×ia1+1 must contain the coefficient ai, for i=0,1,,n. Only these n+1 elements will be accessed.
5:     ia1 int64int32nag_int scalar
The index increment of a. Most frequently, the Chebyshev coefficients are stored in adjacent elements of a, and ia1 must be set to 1. However, if, for example, they are stored in a1,a4,a7,, then the value of ia1 must be 3.
Constraint: ia11.
6:     x – double scalar
The argument x at which the polynomial is to be evaluated.
Constraint: xminxxmax.

Optional Input Parameters


Output Parameters

1:     result – double scalar
The value of the polynomial px-.
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:
On entry,np1<1,
x does not satisfy the restriction xminxxmax.
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 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.


Suppose a polynomial has been computed in Chebyshev series form to fit data over the interval -0.5,2.5. The following program evaluates the polynomial at 4 equally spaced points over the interval. (For the purposes of this example, xmin, xmax and the Chebyshev coefficients are supplied . Normally a program would first read in or generate data and compute the fitted polynomial.)
function e02ak_example

fprintf('e02ak example results\n\n');

xmin = -0.5; xmax = 2.5;
a = [2.53213  1.13032  0.2715  0.04434   0.00547   0.00054  4e-05];

n = int64(6);
ia1 = int64(1);

% Evaluate polynomial from coefficients a at x
m = 21;
dx =  (xmax-xmin)/(m-1);
x = [xmin:dx:xmax];

for i = 1:m
  [fit(i), ifail] = e02ak( ...
                             n, xmin, xmax, a, ia1, x(i));

sol = [x; fit];
fprintf('     x           p(x)\n');
fprintf('%9.4f    %9.4f\n',sol(1:2,1:5:21));

fig1 = figure;
title('Evaluation of Chebyshev Polynomial');

e02ak example results

     x           p(x)
  -0.5000       0.3679
   0.2500       0.6065
   1.0000       1.0000
   1.7500       1.6487
   2.5000       2.7183

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Chapter Contents
Chapter Introduction
NAG Toolbox

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