NAG Library Function Document

nag_ode_bvp_ps_lin_cgl_deriv (d02udc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

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).

2 Specification

#include <nag.h>

#include <nagd02.h>

void	nag_ode_bvp_ps_lin_cgl_deriv (Integer n, const double f[], double fd[], NagError *fail)

3 Description

nag_ode_bvp_ps_lin_cgl_deriv (d02udc) differentiates a function discretized on Chebyshev Gauss–Lobatto points on

[- 1, 1]

. 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

f (x_{i})

on Chebyshev Gauss–Lobatto points

x_{i} = - \cos ((i - 1) π / n)

, for

i = 1, 2, \dots, n + 1

f

is differentiated with respect to

x

by means of forward and backward FFTs on the function values

f (x_{i})

. nag_ode_bvp_ps_lin_cgl_deriv (d02udc) returns the computed derivative values

f^{'} (x_{i})

, for

i = 1, 2, \dots, n + 1

. The derivatives are computed with respect to the standard Chebyshev Gauss–Lobatto points on

[- 1, 1]

; for derivatives of a function on

[a, b]

the returned values have to be scaled by a factor

2 / (b - a)

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: n – IntegerInput: On entry: $n$ , where the number of grid points is $n + 1$ . The fast Fourier transform requires that the prime factorization of $n$ contain no more than $30$ prime factors.
Constraint: $n > 0$ and n is even.
2: f[ $n + 1$ ] – const doubleInput: On entry: the function values $f (x_{i})$ , for $i = 1, 2, \dots, n + 1$
3: fd[ $n + 1$ ] – doubleOutput: On exit: the approximations to the derivatives of the function evaluated at the Chebyshev Gauss–Lobatto points. For functions defined on $[a, b]$ , the returned derivative values (corresponding to the domain $[- 1, 1]$ ) must be multiplied by the factor $2 / (b - a)$ to obtain the correct values on $[a, b]$ .
4: 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.
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.

7 Accuracy

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.

8 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

9 Example

The function

2 x + \exp (- x)

, defined on

[0, 1.5]

, is supplied and then differentiated on a grid.

NAG Library Function Documentnag_ode_bvp_ps_lin_cgl_deriv (d02udc)