NAG Library Function Document
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)
||nag_ode_bvp_ps_lin_cgl_deriv (Integer n,
const double f,
nag_ode_bvp_ps_lin_cgl_deriv (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 . nag_ode_bvp_ps_lin_cgl_deriv (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 .
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
n – IntegerInput
On entry: , where the number of grid points is . The fast Fourier transform requires that the prime factorization of contain no more than prime factors.
f – const doubleInput
On entry: the function values
fd – doubleOutput
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 .
fail – NagError *Input/Output
The NAG error argument (see Section 3.6
in the Essential Introduction).
6 Error Indicators and Warnings
Dynamic memory allocation failed.
On entry, argument had an illegal value.
On entry, .
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
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.
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 ).
The function , defined on , is supplied and then differentiated on a grid.
9.1 Program Text
Program Text (d02udce.c)
9.2 Program Data
Program Data (d02udce.d)
9.3 Program Results
Program Results (d02udce.r)