The routine may be called by the names c06faf or nagf_sum_fft_real_1d_rfmt.
Given a sequence of real data values , for , c06faf calculates their discrete Fourier transform defined by
(Note the scale factor of in this definition.) 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 (see also the C06 Chapter Introduction).
To compute the inverse discrete Fourier transform defined by
this routine should be followed by forming the complex conjugates of the ; that is, , for .
c06faf uses the fast Fourier transform (FFT) algorithm (see Brigham (1974)).
Brigham E O (1974) The Fast Fourier Transform Prentice–Hall
1: – Real (Kind=nag_wp) arrayInput/Output
On entry: if x is declared with bounds in the subroutine from which c06faf is called,
must contain , for .
On exit: the discrete Fourier transform stored in Hermitian form. If the components of the transform are written as , and if x is declared with bounds in the subroutine from which c06faf is called, then for , is contained in , and for , is contained in . (See also Section 2.1.2 in the C06 Chapter Introduction.)
2: – IntegerInput
On entry: , the number of data values.
3: – Real (Kind=nag_wp) arrayWorkspace
4: – 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, .
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.
c06faf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c06faf 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 . c06faf is faster if the only prime factors of are , or ; and fastest of all if is a power of .
This example reads in a sequence of real data values and prints their discrete Fourier transform (as computed by c06faf), after expanding it from Hermitian form into a full complex sequence. It then performs an inverse transform using c06fbf and conjugation, and prints the sequence so obtained alongside the original data values.