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Chapter Contents
Chapter Introduction
NAG Toolbox

# 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 ${c}_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,n+1$, of the interpolating Chebyshev series
 $12 c1 T0 x- + c2 T1 x- + c3T2 x- +⋯+ cn+1 Tn x- ,$
which interpolates the function $f\left(x\right)$ evaluated at the Chebyshev Gauss–Lobatto points
 $x-r = - cos r-1 π/n , r=1,2,…,n+1 .$
Here ${T}_{j}\left(\stackrel{-}{x}\right)$ denotes the Chebyshev polynomial of the first kind of degree $j$ with argument $\stackrel{-}{x}$ defined on $\left[-1,1\right]$. In terms of your original variable, $x$ say, the input values at which the function values are to be provided are
 $xr = - 12 b - a cos πr-1 /n + 1 2 b + a , r=1,2,…,n+1 , ​$
where $b$ and $a$ are respectively the upper and lower ends of the range of $x$ 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:     $\mathrm{n}$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: ${\mathbf{n}}>0$ and n is even.
2:     $\mathrm{f}\left({\mathbf{n}}+1\right)$ – double array
The function values $f\left({x}_{\mathit{r}}\right)$, for $\mathit{r}=1,2,\dots ,n+1$.

None.

### Output Parameters

1:     $\mathrm{c}\left({\mathbf{n}}+1\right)$ – double array
The Chebyshev coefficients, ${c}_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,n+1$.
2:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:
${\mathbf{ifail}}=1$
Constraint: ${\mathbf{n}}>1$.
Constraint: n is even.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

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

The number of operations is of the order $n\mathrm{log}\left(n\right)$ and the memory requirements are $\mathit{O}\left(n\right)$; 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).
```function d02ua_example

fprintf('d02ua example results\n\n');

% On [0,4], Solve u + ux = f0(x); u(0) = 0
% where f) is such that u(x) = sin(10xcos^2x).
% Set up Chebyshev grid on [a,b]
a = 0;
b = 4;
n = int64(128);
[x, ifail] = d02uc(n, a, b);

% Get Chebyshev coeficients on grid for f0(x) = 1.
z = 10*x.*cos(x).^2;
f0 = sin(z) - 10*cos(x).^2.*cos(10*x.*cos(x).^2).*(2*x.*tan(x)-1);
[f0_c, ifail] = d02ua(n, f0);

% Set up problem definition for  u + ux = f0 [(1 1).(u ux) = f0]
f = [1, 1];
% subject to u(a) = 0  [(1 0).(u ux)(a) = 0]
y = [a];
B = [1, 0];
beta = 0;

% Solve in coefficient space using f0_c for rhs.
[B, f, uc, resid, ifail] = d02ue(...
n, a, b, f0_c, B, y, beta, 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));

maxerr = max(abs(u - sin(z)));
fprintf('With n = %4d, maximum error in solution = %13.2e\n',n,maxerr);

% Plot solution
fig1 = figure;
plot(x,u,x,ux);
title('Solution of u + u_x = f_0 on [0,4]');
xlabel('x');
ylabel('u(x) and u_x(x)');
legend('u','u_x','Location','Northwest');

```
```d02ua example results

With n =  128, maximum error in solution =      5.33e-09
```