nag_ode_bvp_ps_lin_cgl_vals (d02ub) evaluates a function, or one of its lower order derivatives, from its Chebyshev series representation at Chebyshev Gauss–Lobatto points on

[a, b]

. The coefficients of the Chebyshev series representation required are usually derived from those returned by nag_ode_bvp_ps_lin_coeffs (d02ua) or nag_ode_bvp_ps_lin_solve (d02ue).

Syntax

[f, ifail] = d02ub(n, a, b, q, c)

[f, ifail] = nag_ode_bvp_ps_lin_cgl_vals(n, a, b, q, c)

Description

nag_ode_bvp_ps_lin_cgl_vals (d02ub) evaluates the Chebyshev series

S (\bar{x}) = \frac{1}{2} c_{1} T_{0} (\bar{x}) + c_{2} T_{1} (\bar{x}) + c_{3} T_{2} (\bar{x}) + \dots + c_{n + 1} T_{n} (\bar{x}),

or its derivative (up to fourth order) at the Chebyshev Gauss–Lobatto points on

[a, b]

. Here

T_{j} (\bar{x})

denotes the Chebyshev polynomial of the first kind of degree

j

with argument

\bar{x}

defined on

[- 1, 1]

. In terms of your original variable,

x

say, the input values at which the function values are to be provided are

x_{r} = - \frac{1}{2} (b - a) \cos (π (r - 1) / n) + \frac{1}{2} (b + a), r = 1, 2, \dots, n + 1, ​

where

b

and

a

are respectively the upper and lower ends of the range of

x

over which the function is required.

The calculation is implemented by a forward one-dimensional discrete Fast Fourier Transform (DFT).

References

Canuto C (1988) Spectral Methods in Fluid Dynamics 502 Springer

Canuto C, Hussaini M Y, Quarteroni A and Zang T A (2006) Spectral Methods: Fundamentals in Single Domains Springer

Trefethen L N (2000) Spectral Methods in MATLAB SIAM

Parameters

Compulsory Input Parameters

1: $n$ – int64int32nag_int scalar: $n$ , where the number of grid points is $n + 1$ . This is also the largest order of Chebyshev polynomial in the Chebyshev series to be computed.

Constraint: $n > 0$ and n is even.
2: $a$ – double scalar: $a$ , the lower bound of domain $[a, b]$ .

Constraint: $a < b$ .
3: $b$ – double scalar: $b$ , the upper bound of domain $[a, b]$ .

Constraint: $b > a$ .
4: $q$ – int64int32nag_int scalar: The order, $q$ , of the derivative to evaluate.

Constraint: $0 \leq q \leq 4$ .
5: $c (n + 1)$ – double array: The Chebyshev coefficients, $c_{i}$ , for $i = 1, 2, \dots, n + 1$ .

Optional Input Parameters

None.

Output Parameters

1: $f (n + 1)$ – double array: The derivatives $S^{(q)} x_{i}$ , for $i = 1, 2, \dots, n + 1$ , of the Chebyshev series, $S$ .
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: $n > 0$ .

Constraint: n is even.

$ifail = 2$: Constraint: $a < b$ .

$ifail = 3$: Constraint: $0 \leq q \leq 4$ .

$ifail = - 99$: 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

Evaluations of DFT to obtain function or derivative values should be an order

n

multiple of machine precision assuming full accuracy to machine precision in the given Chebyshev series representation.

Further Comments

The number of operations is of the order

n \log (n)

and the memory requirements are

O (n)

; thus the computation remains efficient and practical for very fine discretizations (very large values of

n

Example

See Example in nag_ode_bvp_ps_lin_solve (d02ue).

Open in the MATLAB editor: d02ub_example

function d02ub_example


fprintf('d02ub example results\n\n');

n = int64(16);
a = -pi/2;
b =  pi/2;



% Set up boundary condition on left side of domain
y = [a, a, b];
% Set up Dirichlet condition using exact solution at x=a.
bmat = zeros(3, 4);
bmat(1, 1) = 1;
bmat(2, 1:3) = [1, 2, 3];
bmat(3, 1:3) = [1, 2, 3];
bvec = [0, 2, -2];

% Set up problem definition
f = [1, 2, 3, 4];

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

% Set up problem right hand sides for grid and transform
f0 = 2*sin(x) - 2*cos(x);
[c, ifail] = d02ua(n, f0);

% Solve in coefficient space
[bmat, f, uc, resid, ifail] = d02ue(n, a, b, c, bmat, y, bvec, f);

% Transform solution and derivative back to real space.
u = zeros(17, 4);
for q=0:3
  [u(:, q+1),  ifail] = d02ub(n, a, b, int64(q), uc(:, q+1));
end

% Print Solution
fprintf('\nNumerical solution U and its first three derivatives\n');
fprintf('      x          U          Ux         Uxx       Uxxx\n');
for i=1:17
  fprintf('%10.4f %10.4f %10.4f %10.4f %10.4f\n', x(i), u(i, :));
end

d02ub example results


Numerical solution U and its first three derivatives
      x          U          Ux         Uxx       Uxxx
   -1.5708    -0.0000     1.0000     0.0000    -1.0000
   -1.5406     0.0302     0.9995    -0.0302    -0.9995
   -1.4512     0.1193     0.9929    -0.1193    -0.9929
   -1.3061     0.2616     0.9652    -0.2616    -0.9652
   -1.1107     0.4440     0.8960    -0.4440    -0.8960
   -0.8727     0.6428     0.7661    -0.6428    -0.7661
   -0.6011     0.8247     0.5656    -0.8247    -0.5656
   -0.3064     0.9534     0.3017    -0.9534    -0.3017
   -0.0000     1.0000     0.0000    -1.0000    -0.0000
    0.3064     0.9534    -0.3017    -0.9534     0.3017
    0.6011     0.8247    -0.5656    -0.8247     0.5656
    0.8727     0.6428    -0.7661    -0.6428     0.7661
    1.1107     0.4440    -0.8960    -0.4440     0.8960
    1.3061     0.2616    -0.9652    -0.2616     0.9652
    1.4512     0.1193    -0.9929    -0.1193     0.9929
    1.5406     0.0302    -0.9995    -0.0302     0.9995
    1.5708    -0.0000    -1.0000     0.0000     1.0000

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

Chapter Contents

Chapter Introduction

NAG Toolbox