NAG Library Function Document

nag_ode_bvp_ps_lin_coeffs (d02uac)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

1 Purpose

nag_ode_bvp_ps_lin_coeffs (d02uac) obtains the Chebyshev coefficients of a function discretized on Chebyshev Gauss–Lobatto points. The set of discretization points on which the function is evaluated is usually obtained 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_coeffs (Integer n, const double f[], double c[], NagError *fail)

3 Description

nag_ode_bvp_ps_lin_coeffs (d02uac) computes the coefficients

c_{j}

, for

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

, of the interpolating Chebyshev series

\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}),

which interpolates the function

f (x)

evaluated at the Chebyshev Gauss–Lobatto points

{\bar{x}}_{r} = - \cos ((r - 1) π / n), r = 1, 2, \dots, n + 1 .

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.

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: f[ $n + 1$ ] – const doubleInput: On entry: the function values $f (x_{r})$ , for $r = 1, 2, \dots, n + 1$ .
3: c[ $n + 1$ ] – doubleOutput: On exit: the Chebyshev coefficients, $c_{j}$ , for $j = 1, 2, \dots, n + 1$ .
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: $n > 1$ .
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 Chebyshev coefficients computed should be accurate to within a small multiple of machine precision.

8 Parallelism and Performance

nag_ode_bvp_ps_lin_coeffs (d02uac) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_ode_bvp_ps_lin_coeffs (d02uac) 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 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 Section 10 in nag_ode_bvp_ps_lin_solve (d02uec).

nag_ode_bvp_ps_lin_coeffs (d02uac) (PDF version)

d02 Chapter Contents

d02 Chapter Introduction

NAG Library Manual

NAG Library Function Documentnag_ode_bvp_ps_lin_coeffs (d02uac)