The function may be called by the names: c06pac or nag_sum_fft_realherm_1d.
Given a sequence of real data values , for , c06pac calculates their discrete Fourier transform (in the forward direction) defined by
The transformed values are complex, but they form a Hermitian sequence (i.e., is the complex conjugate of ), so they are completely determined by real numbers (since is real, as is for even).
Alternatively, given a Hermitian sequence of complex data values , this function calculates their inverse (backward) discrete Fourier transform defined by
The transformed values are real.
(Note the scale factor of in the above definitions.)
A call of c06pac with followed by a call with will restore the original data.
c06pac uses a variant of the fast Fourier transform (FFT) algorithm (see Brigham (1974)) known as the Stockham self-sorting algorithm, which is described in Temperton (1983).
The same functionality is available using the forward and backward transform function pair: c06pvcandc06pwc on setting . This pair use a different storage solution; real data is stored in a double array, while Hermitian data (the first unconjugated half) is stored in a Complex array.
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: – Nag_TransformDirectionInput
On entry: if the forward transform as defined in Section 3 is to be computed, direct must be set equal to .
If the backward transform is to be computed, direct must be set equal to .
2: – doubleInput/Output
must contain , for ;
if , and must contain the real and imaginary parts respectively of , for . (Note that for the sequence to be Hermitian, the imaginary part of , and of for even, must be zero.)
and will contain the real and imaginary parts respectively of , for ;
will contain , for .
3: – IntegerInput
On entry: , the number of data values.
4: – 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.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument had an illegal value.
On entry, .
An internal error has occurred in this function.
Check the function call and any array sizes.
If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL 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 CL 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.
c06pac is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c06pac 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 function. 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 . c06pac is faster if the only prime factors of are , or ; and fastest of all if is a power of . This function internally allocates a workspace of double values.
This example reads in a sequence of real data values and prints their discrete Fourier transform (as computed by c06pac with ), after expanding it from complex Hermitian form into a full complex sequence. It then performs an inverse transform using c06pac with , and prints the sequence so obtained alongside the original data values.