NAG Library Function Document

nag_fft_multiple_cosine (c06hbc)

The Fourier cosine transform defined above is its own inverse, and two consecutive calls of this function with the same data will restore the original data (but see Section 9).

The transform calculated by this function can be used to solve Poisson's equation when the solution is specified at both left and right boundaries (Swarztrauber (1977)).

The function uses a variant of the fast Fourier transform (FFT) algorithm (Brigham (1974)) known as the Stockham self-sorting algorithm, described in Temperton (1983), together with pre- and post-processing stages described in Swarztrauber (1982). Special coding is provided for the factors 2, 3, 4, 5 and 6.

4 References

Brigham E O (1974) The Fast Fourier Transform Prentice–Hall

Swarztrauber P N (1977) The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson's equation on a rectangle SIAM Rev. 19(3) 490–501

Swarztrauber P N (1982) Vectorizing the FFT's Parallel Computation (ed G Rodrique) 51–83 Academic Press

Temperton C (1983) Fast mixed-radix real Fourier transforms J. Comput. Phys. 52 340–350

5 Arguments

1: m – IntegerInput

On entry: the number of sequences to be transformed,

m

Constraint:

m \geq 1

2: n – IntegerInput

On entry: one less than the number of real values in each sequence, i.e., the number of values in each sequence is

n + 1

Constraint:

n \geq 1

3: x[ $m \times (n + 1)$ ] – doubleInput/Output

On entry: the

m

data sequences stored in x consecutively. If the

n + 1

data values of the

p

th sequence to be transformed are denoted by

x_{j}^{p}

, for

j = 0, 1, \dots, n

and

p = 1, 2, \dots, m

, then the first

m (n + 1)

elements of the array x must contain the values

x_{0}^{1}, x_{1}^{1}, \dots, x_{n}^{1}, x_{0}^{2}, x_{1}^{2}, \dots, x_{n}^{2}, \dots, x_{0}^{m}, x_{1}^{m}, \dots, x_{n}^{m} .

On exit: the

m

Fourier cosine transforms stored consecutively, overwriting the corresponding original sequence.

4: trig[ $2 \times n$ ] – const doubleInput

On entry: trigonometric coefficients as returned by a call of nag_fft_init_trig (c06gzc). nag_fft_multiple_cosine (c06hbc) makes a simple check to ensure that trig has been initialized and that the initialization is compatible with the value of n.

5: 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_C06_NOT_TRIG: Value of n and trig array are incompatible or trig array not initialized.
NE_INT_ARG_LT: On entry, $m = ⟨value⟩$ .
Constraint: $m \geq 1$ .
On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 1$ .

7 Accuracy

Some indication of accuracy can be obtained by performing a subsequent inverse transform and comparing the results with the original sequence (in exact arithmetic they would be identical).

8 Parallelism and Performance

Not applicable.

9 Further Comments

The time taken is approximately proportional to

n m \log (n)

, but also depends on the factors of

n

. The function is fastest if the only prime factors of

n

are 2, 3 and 5, and is particularly slow if

n

is a large prime, or has large prime factors.

10 Example

This program reads in sequences of real data values and prints their Fourier cosine transforms (as computed by nag_fft_multiple_cosine (c06hbc)). It then calls nag_fft_multiple_cosine (c06hbc) again and prints the results which may be compared with the original sequence.

NAG Library Function Documentnag_fft_multiple_cosine (c06hbc)

+− Contents

1 Purpose

2 Specification

3 Description