NAG C Library Function Document

nag_ode_bvp_ps_lin_cgl_vals (d02ubc)


nag_ode_bvp_ps_lin_cgl_vals (d02ubc) evaluates a function, or one of its lower order derivatives, from its Chebyshev series representation at Chebyshev Gauss–Lobatto points on a,b. The coefficients of the Chebyshev series representation required are usually derived from those returned by nag_ode_bvp_ps_lin_coeffs (d02uac) or nag_ode_bvp_ps_lin_solve (d02uec).


#include <nag.h>
#include <nagd02.h>
void  nag_ode_bvp_ps_lin_cgl_vals (Integer n, double a, double b, Integer q, const double c[], double f[], NagError *fail)


nag_ode_bvp_ps_lin_cgl_vals (d02ubc) evaluates the Chebyshev series
S x- = 12 c1 T0 x- + c2 T1 x- + c3T2 x- ++ cn+1 Tn x- ,  
or its derivative (up to fourth order) at the Chebyshev Gauss–Lobatto points on a,b. Here Tjx- denotes the Chebyshev polynomial of the first kind of degree j with argument x- defined on -1,1. In terms of your original variable, x say, the input values at which the function values are to be provided are
xr = - 12 b - a cos πr-1 /n + 1 2 b + a ,   r=1,2,,n+1 , ​  
where b and a are respectively the upper and lower ends of the range of x over which the function is required.
The calculation is implemented by a forward one-dimensional discrete Fast Fourier Transform (DFT).


Canuto C (1988) Spectral Methods in Fluid Dynamics 502 Springer
Canuto C, Hussaini M Y, Quarteroni A and Zang T A (2006) Spectral Methods: Fundamentals in Single Domains Springer
Trefethen L N (2000) Spectral Methods in MATLAB SIAM


1:     n IntegerInput
On entry: n, where the number of grid points is n+1. This is also the largest order of Chebyshev polynomial in the Chebyshev series to be computed.
Constraint: n>0 and n is even.
2:     a doubleInput
On entry: a, the lower bound of domain a,b.
Constraint: a<b.
3:     b doubleInput
On entry: b, the upper bound of domain a,b.
Constraint: b>a.
4:     q IntegerInput
On entry: the order, q, of the derivative to evaluate.
Constraint: 0q4.
5:     c[n+1] const doubleInput
On entry: the Chebyshev coefficients, ci, for i=1,2,,n+1.
6:     f[n+1] doubleOutput
On exit: the derivatives Sq xi , for i=1,2,,n+1, of the Chebyshev series, S.
7:     fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

Error Indicators and Warnings

Dynamic memory allocation failed.
See Section in How to Use the NAG Library and its Documentation for further information.
On entry, argument value had an illegal value.
On entry, n=value.
Constraint: n>0.
On entry, n=value.
Constraint: n is even.
On entry, q=value.
Constraint: 0q4.
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
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.
On entry, a=value and b=value.
Constraint: a<b.


Evaluations of DFT to obtain function or derivative values should be an order n multiple of machine precision assuming full accuracy to machine precision in the given Chebyshev series representation.

Parallelism and Performance

nag_ode_bvp_ps_lin_cgl_vals (d02ubc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_ode_bvp_ps_lin_cgl_vals (d02ubc) 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.

Further Comments

The number of operations is of the order n logn  and the memory requirements are On; thus the computation remains efficient and practical for very fine discretizations (very large values of n).


See Section 10 in nag_ode_bvp_ps_lin_solve (d02uec).