NAG CL Interface
d02udc (bvp_ps_lin_cgl_deriv)
1
Purpose
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
d02ucc.
2
Specification
| void |
d02udc (Integer n,
const double f[],
double fd[],
NagError *fail) |
|
The function may be called by the names: d02udc or nag_ode_bvp_ps_lin_cgl_deriv.
3
Description
d02udc differentiates a function discretized on Chebyshev Gauss–Lobatto points on . 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 on Chebyshev Gauss–Lobatto points
, for , is differentiated with respect to by means of forward and backward FFTs on the function values . d02udc returns the computed derivative values
, for . The derivatives are computed with respect to the standard Chebyshev Gauss–Lobatto points on ; for derivatives of a function on the returned values have to be scaled by a factor .
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 two-point boundary value problems SIAM J. Numer. Anal. 28(4) 1071–80
Trefethen L N (2000) Spectral Methods in MATLAB SIAM
5
Arguments
-
1:
– Integer
Input
-
On entry: , where the number of grid points is .
Constraint:
and
n is even.
-
2:
– const double
Input
-
On entry: the function values
, for
-
3:
– double
Output
-
On exit: the approximations to the derivatives of the function evaluated at the Chebyshev Gauss–Lobatto points. For functions defined on , the returned derivative values (corresponding to the domain ) must be multiplied by the factor to obtain the correct values on .
-
4:
– NagError *
Input/Output
-
The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_INT
-
On entry, .
Constraint: .
On entry,
.
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.
See
Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library CL Interface for further information.
7
Accuracy
The accuracy is close to
machine precision for small numbers of grid points, typically less than
. 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
d02udc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
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 implementation-specific information.
The number of operations is of the order and the memory requirements are ; thus the computation remains efficient and practical for very fine discretizations (very large values of ).
10
Example
The function , defined on , is supplied and then differentiated on a grid.
10.1
Program Text
10.2
Program Data
10.3
Program Results