NAG Library Routine Document
D02UDF
1 Purpose
D02UDF 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
D02UCF.
2 Specification
INTEGER 
N, IFAIL 
REAL (KIND=nag_wp) 
F(N+1), FD(N+1) 

3 Description
D02UDF 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 $f\left({x}_{i}\right)$. D02UDF 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: $\mathrm{N}$ – INTEGERInput

On entry: $n$, where the number of grid points is $n+1$.
Constraint:
${\mathbf{N}}>0$ and
N is even.
 2: $\mathrm{F}\left({\mathbf{N}}+1\right)$ – REAL (KIND=nag_wp) arrayInput

On entry: the function values
$f\left({x}_{\mathit{i}}\right)$, for $\mathit{i}=1,2,\dots ,n+1$
 3: $\mathrm{FD}\left({\mathbf{N}}+1\right)$ – REAL (KIND=nag_wp) arrayOutput

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: $\mathrm{IFAIL}$ – INTEGERInput/Output

On entry:
IFAIL must be set to
$0$,
$1\text{ or}1$. If you are unfamiliar with this argument you should refer to
Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
$1\text{ or}1$ is recommended. If the output of error messages is undesirable, then the value
$1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is
$0$.
When the value $\mathbf{1}\text{ or}\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit:
${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see
Section 6).
6 Error Indicators and Warnings
If on entry
${\mathbf{IFAIL}}=0$ or
$1$, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Errors or warnings detected by the routine:
 ${\mathbf{IFAIL}}=1$

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.
 ${\mathbf{IFAIL}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
See
Section 3.9 in How to Use the NAG Library and its Documentation for further information.
 ${\mathbf{IFAIL}}=399$
Your licence key may have expired or may not have been installed correctly.
See
Section 3.8 in How to Use the NAG Library and its Documentation for further information.
 ${\mathbf{IFAIL}}=999$
Dynamic memory allocation failed.
See
Section 3.7 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
D02UDF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
D02UDF 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 routine. 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 (d02udfe.f90)
10.2 Program Data
Program Data (d02udfe.d)
10.3 Program Results
Program Results (d02udfe.r)