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NAG Toolbox: nag_ode_bvp_ps_lin_coeffs (d02ua)

Purpose

nag_ode_bvp_ps_lin_coeffs (d02ua) obtains the Chebyshev coefficients of a function discretized on Chebyshev Gauss–Lobatto points. The set of discretization points on which the function is evaluated is usually obtained by a previous call to nag_ode_bvp_ps_lin_cgl_grid (d02uc).

Syntax

[c, ifail] = d02ua(n, f)
[c, ifail] = nag_ode_bvp_ps_lin_coeffs(n, f)

Description

nag_ode_bvp_ps_lin_coeffs (d02ua) computes the coefficients cjcj, for j = 1,2,,n + 1j=1,2,,n+1, of the interpolating Chebyshev series
(1/2) c1 T0 (x) + c2 T1 (x) + c3T2 (x) + + cn + 1 Tn (x) ,
12 c1 T0 (x-) + c2 T1 (x-) + c3T2 (x-) ++ cn+1 Tn (x-) ,
which interpolates the function f(x)f(x) evaluated at the Chebyshev Gauss–Lobatto points
xr = cos((r1)π / n) ,   r = 1,2,,n + 1 .
x-r = - cos( (r-1) π/n ) ,   r=1,2,,n+1 .
Here Tj(x)Tj(x-) denotes the Chebyshev polynomial of the first kind of degree jj with argument xx- defined on [1,1][-1,1]. In terms of your original variable, xx say, the input values at which the function values are to be provided are
xr = (1/2) (ba) cos(π(r1) / n) + (1/2) (b + a) ,   r = 1,2,,n + 1 , ​
xr = - 12 ( b - a ) cos( π(r-1) /n ) + 1 2 ( b + a ) ,   r=1,2,,n+1 , ​
where bb and aa are respectively the upper and lower ends of the range of xx over which the function is required.

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
nn, where the number of grid points is n + 1n+1. This is also the largest order of Chebyshev polynomial in the Chebyshev series to be computed.
Constraint: n > 0n>0 and n is even.
2:     f(n + 1n+1) – double array
The function values f(xr)f(xr), for r = 1,2,,n + 1r=1,2,,n+1.

Optional Input Parameters

None.

Input Parameters Omitted from the MATLAB Interface

None.

Output Parameters

1:     c(n + 1n+1) – double array
The Chebyshev coefficients, cjcj, for j = 1,2,,n + 1j=1,2,,n+1.
2:     ifail – int64int32nag_int scalar
ifail = 0ifail=0 unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Errors or warnings detected by the function:
  ifail = 1ifail=1
Constraint: n > 1n>1.
Constraint: n is even.
  ifail = 999ifail=-999
Dynamic memory allocation failed.

Accuracy

The Chebyshev coefficients computed should be accurate to within a small multiple of machine precision.

Further Comments

The number of operations is of the order n log(n) n log(n)  and the memory requirements are O(n) O(n) ; thus the computation remains efficient and practical for very fine discretizations (very large values of nn).

Example

function nag_ode_bvp_ps_lin_coeffs_example
n = int64(16);
a = 0;
b = 1.5;


% Set up boundary condition on left side of domain
y = [a];
% Set up Dirichlet condition using exact solution at x=a.
bmat = [1, 0];
bvec = exp(-a-1) + 1;

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

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

% Set up problem right hand sides for grid and transform
f0 = ones(17, 1);
[c, ifail] = nag_ode_bvp_ps_lin_coeffs(n, f0);

% Solve in coefficient space
[bmat, f, uc, resid, ifail] = nag_ode_bvp_ps_lin_solve(n, a, b, c, bmat, y, bvec, f);
% Transform solution and derivative back to real space.
[u,  ifail] = nag_ode_bvp_ps_lin_cgl_vals(n, a, b, int64(0), uc(:, 1));
[ux, ifail] = nag_ode_bvp_ps_lin_cgl_vals(n, a, b, int64(1), uc(:, 2));

% Print Solution
fprintf('\nNumerical solution U and derivative Ux\n');
fprintf('      x          U          Ux\n');
for i=1:17
  fprintf('%10.4f %10.4f %10.4f\n', x(i), u(i), ux(i));
end
 

Numerical solution U and derivative Ux
      x          U          Ux
    0.0000     1.3679    -0.3679
    0.0144     1.3626    -0.3626
    0.0571     1.3475    -0.3475
    0.1264     1.3242    -0.3242
    0.2197     1.2953    -0.2953
    0.3333     1.2636    -0.2636
    0.4630     1.2315    -0.2315
    0.6037     1.2012    -0.2012
    0.7500     1.1738    -0.1738
    0.8963     1.1501    -0.1501
    1.0370     1.1304    -0.1304
    1.1667     1.1146    -0.1146
    1.2803     1.1023    -0.1023
    1.3736     1.0931    -0.0931
    1.4429     1.0869    -0.0869
    1.4856     1.0833    -0.0833
    1.5000     1.0821    -0.0821

function d02ua_example
n = int64(16);
a = 0;
b = 1.5;


% Set up boundary condition on left side of domain
y = [a];
% Set up Dirichlet condition using exact solution at x=a.
bmat = [1, 0];
bvec = exp(-a-1) + 1;

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

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

% Set up problem right hand sides for grid and transform
f0 = ones(17, 1);
[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,  ifail] = d02ub(n, a, b, int64(0), uc(:, 1));
[ux, ifail] = d02ub(n, a, b, int64(1), uc(:, 2));

% Print Solution
fprintf('\nNumerical solution U and derivative Ux\n');
fprintf('      x          U          Ux\n');
for i=1:17
  fprintf('%10.4f %10.4f %10.4f\n', x(i), u(i), ux(i));
end
 

Numerical solution U and derivative Ux
      x          U          Ux
    0.0000     1.3679    -0.3679
    0.0144     1.3626    -0.3626
    0.0571     1.3475    -0.3475
    0.1264     1.3242    -0.3242
    0.2197     1.2953    -0.2953
    0.3333     1.2636    -0.2636
    0.4630     1.2315    -0.2315
    0.6037     1.2012    -0.2012
    0.7500     1.1738    -0.1738
    0.8963     1.1501    -0.1501
    1.0370     1.1304    -0.1304
    1.1667     1.1146    -0.1146
    1.2803     1.1023    -0.1023
    1.3736     1.0931    -0.0931
    1.4429     1.0869    -0.0869
    1.4856     1.0833    -0.0833
    1.5000     1.0821    -0.0821


PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

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