NAG Library Function Document

nag_ode_bvp_ps_lin_cgl_vals (d02ubc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

1 Purpose

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).

2 Specification

#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)

3 Description

nag_ode_bvp_ps_lin_cgl_vals (d02ubc) evaluates the Chebyshev series

S (\bar{x}) = \frac{1}{2} c_{1} T_{0} (\bar{x}) + c_{2} T_{1} (\bar{x}) + c_{3} T_{2} (\bar{x}) + \dots + c_{n + 1} T_{n} (\bar{x}),

or its derivative (up to fourth order) at the Chebyshev Gauss–Lobatto points on

[a, b]

. Here

T_{j} (\bar{x})

denotes the Chebyshev polynomial of the first kind of degree

j

with argument

\bar{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

x_{r} = - \frac{1}{2} (b - a) \cos (π (r - 1) / n) + \frac{1}{2} (b + a), r = 1, 2, \dots, 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 – 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: $0 \leq q \leq 4$ .
5: c[ $n + 1$ ] – const doubleInput: On entry: the Chebyshev coefficients, $c_{i}$ , for $i = 1, 2, \dots, n + 1$ .
6: f[ $n + 1$ ] – doubleOutput: On exit: the derivatives $S^{(q)} x_{i}$ , for $i = 1, 2, \dots, n + 1$ , of the Chebyshev series, $S$ .
7: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $n = 〈value〉$ .
Constraint: $n > 0$ .; On entry, $n = 〈value〉$ .
Constraint: n is even.; On entry, $q = 〈value〉$ .
Constraint: $0 \leq q \leq 4$ .
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.
NE_REAL_2: On entry, $a = 〈value〉$ and $b = 〈value〉$ .
Constraint: $a < b$ .

7 Accuracy

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