d02uaf 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 d02ucf.

2 Specification

Fortran Interface

Subroutine d02uaf (

n, f, c, ifail)

Integer, Intent (In)	::	n
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	f(n+1)
Real (Kind=nag_wp), Intent (Out)	::	c(n+1)

C Header Interface

#include <nag.h>

void	d02uaf_ (const Integer n, const double f[], double c[], Integer ifail)

The routine may be called by the names d02uaf or nagf_ode_bvp_ps_lin_coeffs.

3 Description

d02uaf 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: $n$ – 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: $f (n + 1)$ – Real (Kind=nag_wp) array Input: On entry: the function values $f (x_{r})$ , for $r = 1, 2, \dots, n + 1$ .
3: $c (n + 1)$ – Real (Kind=nag_wp) array Output: On exit: the Chebyshev coefficients, $c_{j}$ , for $j = 1, 2, \dots, n + 1$ .
4: $ifail$ – 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

−1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $n = ⟨ value ⟩$ .
Constraint: $n > 1$ .

On entry, $n = ⟨ value ⟩$ .
Constraint: n is even.

$ifail = - 99$: 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.

$ifail = - 399$: 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.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

The Chebyshev coefficients computed should be accurate to within a small multiple of machine precision.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

d02uaf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

d02uaf 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.