NAG Library Function Document
nag_fft_multiple_qtr_cosine (c06hdc) computes the discrete quarter-wave Fourier cosine transforms of sequences of real data values.
||nag_fft_multiple_qtr_cosine (Nag_TransformDirection direct,
const double trig,
real data values
, this function simultaneously calculates the quarter-wave Fourier cosine transforms of all the sequences defined by
or its inverse
(Note the scale factor in this definition.)
A call of the function with
followed by a call with
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 derivative of the solution is specified at the left boundary, and the solution is specified at the right boundary (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.
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
direct – Nag_TransformDirectionInput
: if the forward transform as defined in Section 3
is to be computed, then direct
must be set equal to
. If the backward transform is to be computed, that is the inverse, then direct
must be set equal to
m – IntegerInput
On entry: the number of sequences to be transformed, .
n – IntegerInput
On entry: the number of real values in each sequence, .
x – doubleInput/Output
data sequences stored in x
consecutively. If the data values of the
th sequence to be transformed are denoted by
, then the first
elements of the array x
must contain the values
On exit: the quarter-wave cosine transforms stored consecutively overwriting the corresponding original sequence.
trig – const doubleInput
: trigonometric coefficients as returned by a call of nag_fft_init_trig (c06gzc)
. nag_fft_multiple_qtr_cosine (c06hdc) makes a simple check to ensure that trig
has been initialized and that the initialization is compatible with the value of n
fail – NagError *Input/Output
The NAG error argument (see Section 3.6
in the Essential Introduction).
6 Error Indicators and Warnings
Dynamic memory allocation failed.
On entry, argument direct
had an illegal value.
Value of n
array are incompatible or trig
array not initialized.
On entry, .
On entry, .
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
The time taken is approximately proportional to , but also depends on the factors of . The function is fastest if the only prime factors of are 2, 3 and 5, and is particularly slow if is a large prime, or has large prime factors.
This program reads in sequences of real data values and prints their quarter-wave cosine transforms as computed by nag_fft_multiple_qtr_cosine (c06hdc) with . It then calls nag_fft_multiple_qtr_cosine (c06hdc) again with and prints the results which may be compared with the original data.
10.1 Program Text
Program Text (c06hdce.c)
10.2 Program Data
Program Data (c06hdce.d)
10.3 Program Results
Program Results (c06hdce.r)