d02udf 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 d02ucf.

2 Specification

Fortran Interface

Subroutine d02udf (

n, f, fd, ifail)

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

C Header Interface

#include <nag.h>

void	d02udf_ (const Integer n, const double f[], double fd[], Integer ifail)

The routine may be called by the names d02udf or nagf_ode_bvp_ps_lin_cgl_deriv.

3 Description

d02udf 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})

. d02udf 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: $n$ – Integer Input: On entry: $n$ , where the number of grid points is $n + 1$ .

Constraint: $n > 0$ and n is even.
2: $f (n + 1)$ – Real (Kind=nag_wp) array Input: On entry: the function values $f (x_{i})$ , for $i = 1, 2, \dots, n + 1$
3: $fd (n + 1)$ – Real (Kind=nag_wp) array Output: 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: $ifail$ – 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: $n > 0$ .

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 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 Parallelism and Performance

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

d02udf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

d02udf 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

The function

2 x + \exp (- x)

, defined on

[0, 1.5]

, is supplied and then differentiated on a grid.

d02ud: FL CL CPP AD PY MB

NAG FL Interfaced02udf (bvp_​ps_​lin_​cgl_​deriv)

▸▿ 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

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
d02udf (bvp_ps_lin_cgl_deriv)