The routine may be called by the names c06fxf or nagf_sum_fft_complex_3d_sep.
c06fxf computes the three-dimensional discrete Fourier transform of a trivariate sequence of complex data values
, for , and .
The discrete Fourier transform is here defined by
where , , .
(Note the scale factor of in this definition.)
To compute the inverse discrete Fourier transform, defined with in the above formula instead of , this routine should be preceded and followed by forming the complex conjugates of the data values and the transform.
This routine performs, for each dimension, multiple one-dimensional discrete Fourier transforms by the fast Fourier transform (FFT) algorithm (see Brigham (1974)). It is designed to be particularly efficient on vector processors.
Brigham E O (1974) The Fast Fourier Transform Prentice–Hall
Temperton C (1983) Self-sorting mixed-radix fast Fourier transforms J. Comput. Phys.52 1–23
1: – IntegerInput
On entry: , the first dimension of the transform.
2: – IntegerInput
On entry: , the second dimension of the transform.
3: – IntegerInput
On entry: , the third dimension of the transform.
4: – Real (Kind=nag_wp) arrayInput/Output
5: – Real (Kind=nag_wp) arrayInput/Output
On entry: the real and imaginary parts of the complex data values must be stored in arrays x and y respectively. If x and y are regarded as three-dimensional arrays of dimension , and must contain the real and imaginary parts of .
On exit: the real and imaginary parts respectively of the corresponding elements of the computed transform.
6: – Character(1)Input
7: – Real (Kind=nag_wp) arrayInput/Output
8: – Real (Kind=nag_wp) arrayInput/Output
9: – Real (Kind=nag_wp) arrayInput/Output
10: – Real (Kind=nag_wp) arrayOutput
These arguments are no longer accessed by c06fxf.
11: – 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, .
On entry, .
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
Background information to multithreading can be found in the Multithreading documentation.
c06fxf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c06fxf 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 is approximately proportional to , but also depends on the factorization of the individual dimensions , and . c06fxf is faster if the only prime factors are , or ; and fastest of all if they are powers of .
This example reads in a trivariate sequence of complex data values and prints the three-dimensional Fourier transform. It then performs an inverse transform and prints the sequence so obtained, which may be compared to the original data values.