The routine may be called by the names c06raf or nagf_sum_fft_real_sine_simple.
Given sequences of real data values , for and , c06raf simultaneously calculates the Fourier sine transforms of all the sequences defined by
(Note the scale factor in this definition.)
Since the Fourier sine transform defined above is its own inverse, two consecutive calls of this routine will restore the original data.
The transform calculated by this routine can be used to solve Poisson's equation when the solution is specified at both left and right boundaries (see Swarztrauber (1977)).
The routine uses a variant of the fast Fourier transform (FFT) algorithm (see 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 , , and .
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
1: – IntegerInput
On entry: , the number of sequences to be transformed.
2: – IntegerInput
On entry: one more than the number of real values in each sequence, i.e., the number of values in each sequence is .
3: – Real (Kind=nag_wp) arrayInput/Output
On entry: the data must be stored in x as if in a two-dimensional array of dimension ; each of the sequences is stored in a row of the array.
In other words, if the data values of the th sequence to be transformed are denoted by , for and , the first elements of the array x must contain the values
The th to th elements of each row , for , are required as workspace. These elements may contain arbitrary values as they are set to zero by the routine.
On exit: the Fourier sine transforms stored as if in a two-dimensional array of dimension . Each of the transforms is stored in a row of the array, overwriting the corresponding original sequence. If the components of the th Fourier sine transform are denoted by
, for and , the elements of the array x contain the values
4: – Real (Kind=nag_wp) arrayWorkspace
Note: the dimension of the array work
must be at least
The workspace requirements as documented for c06raf may be an overestimate in some implementations.
On exit: contains the minimum workspace required for the current values of m and n with this implementation.
5: – IntegerInput/Output
On entry: ifail must be set to , or to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of means that an error message is printed while a value of means that it is not.
If halting is not appropriate, the value or is recommended. If message printing is undesirable, then the value is recommended. Otherwise, the value is recommended. When the value or is used it is essential to test the value of ifail on exit.
On exit: unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry or , explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, .
On entry, .
An internal error has occurred in this routine.
Check the routine call and any array sizes.
If the call is correct then please contact NAG for assistance.
An unexpected error has been triggered by this routine. Please
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
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.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
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
c06raf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c06raf 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.
The time taken by c06raf is approximately proportional to , but also depends on the factors of . c06raf is fastest if the only prime factors of are , and , and is particularly slow if is a large prime, or has large prime factors.
This example reads in sequences of real data values and prints their Fourier sine transforms (as computed by c06raf). It then calls c06raf again and prints the results which may be compared with the original sequence.