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.
The routine may be called by the names d02udf or nagf_ode_bvp_ps_lin_cgl_deriv.
3Description
d02udf differentiates a function discretized on Chebyshev Gauss–Lobatto points on $[\mathrm{-1},1]$. 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
${x}_{\mathit{i}}=-\mathrm{cos}((\mathit{i}-1)\pi /n)$, 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 $[\mathrm{-1},1]$; for derivatives of a function on $[a,b]$ the returned values have to be scaled by a factor $2/(b-a)$.
4References
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 two-point boundary value problems SIAM J. Numer. Anal.28(4) 1071–80
Trefethen L N (2000) Spectral Methods in MATLAB SIAM
5Arguments
1: $\mathbf{n}$ – IntegerInput
On entry: $n$, where the number of grid points is $n+1$.
2: $\mathbf{f}\left({\mathbf{n}}+1\right)$ – 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: $\mathbf{fd}\left({\mathbf{n}}+1\right)$ – 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 $[a,b]$, the returned derivative values (corresponding to the domain $[\mathrm{-1},1]$) must be multiplied by the factor $2/(b-a)$ to obtain the correct values on $[a,b]$.
4: $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $\mathrm{-1}$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $\mathrm{-1}$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $\mathrm{-1}$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ 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).
6Error Indicators and Warnings
If on entry ${\mathbf{ifail}}=0$ or $\mathrm{-1}$, 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, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}>0$.
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: n is even.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please
contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
7Accuracy
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.
8Parallelism 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 implementation-specific information.
9Further Comments
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 $n$).
10Example
The function $2x+\mathrm{exp}(-x)$, defined on $[0,1.5]$, is supplied and then differentiated on a grid.