NAG Library Function Document
nag_ode_bvp_ps_lin_cgl_deriv (d02udc)
1 Purpose
nag_ode_bvp_ps_lin_cgl_deriv (d02udc) differentiates a function discretized on Chebyshev Gauss–Lobatto points. The grid points on which the function values are to be provided are normally returned by a previous call to
nag_ode_bvp_ps_lin_cgl_grid (d02ucc).
2 Specification
#include <nag.h> 
#include <nagd02.h> 
void 
nag_ode_bvp_ps_lin_cgl_deriv (Integer n,
const double f[],
double fd[],
NagError *fail) 

3 Description
nag_ode_bvp_ps_lin_cgl_deriv (d02udc) differentiates a function discretized on Chebyshev Gauss–Lobatto points on $\left[1,1\right]$. The polynomial interpolation on Chebyshev points is equivalent to trigonometric interpolation on equally spaced points. Hence the differentiation on the Chebyshev points can be implemented by the Fast Fourier transform (FFT).
Given the function values $f\left({x}_{i}\right)$ on Chebyshev Gauss–Lobatto points
${x}_{\mathit{i}}=\mathrm{cos}\left(\left(\mathit{i}1\right)\pi /n\right)$, for $\mathit{i}=1,2,\dots ,n+1$, $f$ is differentiated with respect to $x$ by means of forward and backward FFTs on the function values $f\left({x}_{i}\right)$. nag_ode_bvp_ps_lin_cgl_deriv (d02udc) returns the computed derivative values
${f}^{\prime}\left({x}_{\mathit{i}}\right)$, for $\mathit{i}=1,2,\dots ,n+1$. The derivatives are computed with respect to the standard Chebyshev Gauss–Lobatto points on $\left[1,1\right]$; for derivatives of a function on $\left[a,b\right]$ the returned values have to be scaled by a factor $2/\left(ba\right)$.
4 References
Canuto C, Hussaini M Y, Quarteroni A and Zang T A (2006) Spectral Methods: Fundamentals in Single Domains Springer
Greengard L (1991) Spectral integration and twopoint boundary value problems SIAM J. Numer. Anal. 28(4) 1071–80
Trefethen L N (2000) Spectral Methods in MATLAB SIAM
5 Arguments
 1:
$\mathbf{n}$ – IntegerInput

On entry: $n$, where the number of grid points is $n+1$.
Constraint:
${\mathbf{n}}>0$ and
n is even.
 2:
$\mathbf{f}\left[{\mathbf{n}}+1\right]$ – const doubleInput

On entry: the function values
$f\left({x}_{\mathit{i}}\right)$, for $\mathit{i}=1,2,\dots ,n+1$
 3:
$\mathbf{fd}\left[{\mathbf{n}}+1\right]$ – doubleOutput

On exit: the approximations to the derivatives of the function evaluated at the Chebyshev Gauss–Lobatto points. For functions defined on $\left[a,b\right]$, the returned derivative values (corresponding to the domain $\left[1,1\right]$) must be multiplied by the factor $2/\left(ba\right)$ to obtain the correct values on $\left[a,b\right]$.
 4:
$\mathbf{fail}$ – NagError *Input/Output

The NAG error argument (see
Section 2.7 in How to Use the NAG Library and its Documentation).
6 Error Indicators and Warnings
 NE_ALLOC_FAIL

Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
 NE_BAD_PARAM

On entry, argument $\u2329\mathit{\text{value}}\u232a$ had an illegal value.
 NE_INT

On entry, ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{n}}>0$.
On entry,
${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint:
n is even.
 NE_INTERNAL_ERROR

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact
NAG for assistance.
An unexpected error has been triggered by this function. Please contact
NAG.
See
Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
 NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
7 Accuracy
The accuracy is close to
machine precision for small numbers of grid points, typically less than 100. For larger numbers of grid points, the error in differentiation grows with the number of grid points. See
Greengard (1991) for more details.
8 Parallelism and Performance
nag_ode_bvp_ps_lin_cgl_deriv (d02udc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_ode_bvp_ps_lin_cgl_deriv (d02udc) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the
x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the
Users' Note for your implementation for any additional implementationspecific information.
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 $n$).
10 Example
The function $2x+\mathrm{exp}\left(x\right)$, defined on $\left[0,1.5\right]$, is supplied and then differentiated on a grid.
10.1 Program Text
Program Text (d02udce.c)
10.2 Program Data
Program Data (d02udce.d)
10.3 Program Results
Program Results (d02udce.r)