NAG FL Interface
d02ubf (bvp_​ps_​lin_​cgl_​vals)

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1 Purpose

d02ubf 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 d02uaf or d02uef.

2 Specification

Fortran Interface
Subroutine d02ubf ( n, a, b, q, c, f, ifail)
Integer, Intent (In) :: n, q
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (In) :: a, b, c(n+1)
Real (Kind=nag_wp), Intent (Out) :: f(n+1)
C Header Interface
#include <nag.h>
void  d02ubf_ (const Integer *n, const double *a, const double *b, const Integer *q, const double c[], double f[], Integer *ifail)
The routine may be called by the names d02ubf or nagf_ode_bvp_ps_lin_cgl_vals.

3 Description

d02ubf 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 Tj(x¯) 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(π(r-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).

4 References

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

5 Arguments

1: n Integer Input
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 Real (Kind=nag_wp) Input
On entry: a, the lower bound of domain [a,b].
Constraint: a<b.
3: b Real (Kind=nag_wp) Input
On entry: b, the upper bound of domain [a,b].
Constraint: b>a.
4: q Integer Input
On entry: the order, q, of the derivative to evaluate.
Constraint: 0q4.
5: c(n+1) Real (Kind=nag_wp) array Input
On entry: the Chebyshev coefficients, ci, for i=1,2,,n+1.
6: f(n+1) Real (Kind=nag_wp) array Output
On exit: the derivatives S(q) xi , for i=1,2,,n+1, of the Chebyshev series, S.
7: ifail Integer Input/Output
On entry: ifail must be set to 0, −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 −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 −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 -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry 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:
On entry, n=value.
Constraint: n>0.
On entry, n=value.
Constraint: n is even.
On entry, a=value and b=value.
Constraint: a<b.
On entry, q=value.
Constraint: 0q4.
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.
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.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

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.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
d02ubf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
d02ubf 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.

9 Further Comments

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

10 Example

See d02uef.