The function may be called by the names: c06fpc, nag_sum_withdraw_fft_real_1d_multi_rfmt or nag_fft_multiple_real.
Given sequences of real data values , for and , this function simultaneously calculates the Fourier transforms of all the sequences defined by
(Note the scale factor in this definition.)
The transformed values are complex, but for each value of the form a Hermitian sequence (i.e., is the complex conjugate of ), so they are completely determined by real numbers. The first call of c06fpc must be preceded by a call to c06gzc to initialize the array trig with trigonometric coefficients according to the value of n.
The discrete Fourier transform is sometimes defined using a positive sign in the exponential term
To compute this form, this function should be followed by a call to c06gqc to form the complex conjugates of the .
The function uses a variant of the fast Fourier transform algorithm (Brigham (1974)) known as the Stockham self-sorting algorithm, which is described in Temperton (1983). Special coding is provided for the factors , , , 5 and 6.
Brigham E O (1974) The Fast Fourier Transform Prentice–Hall
Temperton C (1983) Fast mixed-radix real Fourier transforms J. Comput. Phys.52 340–350
1: – IntegerInput
On entry: the number of sequences to be transformed, .
2: – IntegerInput
On entry: the number of real values in each sequence, .
3: – doubleInput/Output
On entry: the data sequences must be stored in x consecutively. If the data values of the th sequence to be transformed are denoted by , for , then the elements of the array x must contain the values
On exit: the discrete Fourier transforms in Hermitian form, stored consecutively, overwriting the corresponding original sequences. If the components of the discrete Fourier transform are written as , then for , is in array element and for , is in array element .
4: – const doubleInput
On entry: trigonometric coefficients as returned by a call of c06gzc. c06fpc makes a simple check to ensure that trig has been initialized and that the initialization is compatible with the value of n
5: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
Dynamic memory allocation failed.
Value of n and trig 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).
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
c06fpc is not threaded in any implementation.
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 , 3 and , 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 discrete Fourier transforms (as computed by c06fpc). The Fourier transforms are expanded into full complex form using c06gsc and printed. Inverse transforms are then calculated by calling c06gqc followed by c06fqc showing that the original sequences are restored.