1: $k$ – int64int32nag_int scalar: The order of the Chebyshev polynomial.

Constraint: $k \geq 0$ .
2: $x$ – double scalar: The point at which to evaluate the polynomial.

Constraint: $- 1.0 \leq x \leq 1.0$ .

Optional Input Parameters

None.

Output Parameters

1: $t$ – double scalar: The value, $T$ , of the Chebyshev polynomial order $k$ evaluated at $x$ .
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:

$ifail = 1$: Constraint: $k \geq 0$ .

$ifail = 2$: Constraint: $- 1.0 \leq x \leq 1.0$ .

$ifail = - 99$: 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 accuracy should be close to machine precision.

Further Comments

None.

Example

A set of Chebyshev coefficients is obtained for the function

x + \exp (- x)

defined on

[- 0.24 \times π, 0.5 \times π]

using nag_ode_bvp_ps_lin_cgl_grid (d02uc). At each of a set of new grid points in the domain of the function nag_ode_bvp_ps_lin_cheb_eval (d02uz) is used to evaluate each Chebshev polynomial in the series representation. The values obtained are multiplied to the Chebyshev coefficients and summed to obtain approximations to the given function at the new grid points.

Open in the MATLAB editor: d02uz_example

function d02uz_example


fprintf('d02uz example results\n\n');

n = int64(16);
m = 9;
a = -0.24*pi;
b =  pi/2;


% Set up Chebyshev grid
[x, ifail] = d02uc(n, a, b);

% Evaluate function on grid.and get interpolating Chebyshev coefficients
f = x + exp(-x);
[c, ifail] = d02ua(n, f);

% Evaluate Chebyshev series manually by evaluating each Chebyshev
% polynomial in turn at new equispaced (m+1) grid points.
% Chebyshev series on [-1,1] map of [a,b].
dmap  = 2/(m-1);
xmap  = [-1:dmap:1];
deven = (b-a)/(m-1);
xeven = [a:deven:b];

fprintf('    x_even     x_map      Sum(f)\n');
for i=1:m
  for k=int64(0:n)
    [t(k+1), ifail] = d02uz(k, xmap(i));
  end
  fseries(i) = dot(c,t);
end
fprintf('%10.4f %10.4f %10.4f\n', [xmap xeven fseries]');

d02uz example results

    x_even     x_map      Sum(f)
   -1.0000    -0.7500    -0.5000
   -0.2500     0.0000     0.2500
    0.5000     0.7500     1.0000
   -0.7540    -0.4634    -0.1728
    0.1178     0.4084     0.6990
    0.9896     1.2802     1.5708
    1.3715     1.1261     1.0158
    1.0067     1.0731     1.1961
    1.3613     1.5582     1.7787

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

Chapter Contents

Chapter Introduction

NAG Toolbox