The NAG Fortran SMP Library

Fast Fourier Transform (FFT) Routines

The NAG Fortran SMP Library contains a set of high performance FFT routines, for performing Discrete Fourier Transforms (DFTs), within its C06 chapter. The main differences between the C06 chapter routines of the standard NAG Fortran Library and those of the NAG Fortran SMP Library are:

A number of new complex interface routines have been added
The new interfaces contain some features which differ significantly from the standard interfaces
On some implementations, many of the original C06 routines have been specially recoded for the Fortran SMP Library
Some affected routines make one or more calls to recoded original routines resulting in enhanced performance

New NAG Fortran SMP Library FFT Routines

The following routines have been specially coded to provide a more convenient interface for the calculation of Discrete Fourier Transforms (DFTs) on sequences of real and complex data values. The new routines include the following features:

Sequences of complex data are stored in complex arrays.
Hermitian sequences are stored, in a compact complex storage format, in real arrays: real and imaginary parts are stored in consecutive elements of an array.
The routines, where required, contain a direction parameter determining the direction of the transform. A call to a routine in one direction followed by a call to the same routine in the other direction restores the original data.
There are no INIT or TRIG parameters. This results in a much simple user interface at the expense of a small reduction in performance whenever a routine is called more than once, on data arrays of fixed dimensions, in the same calling (sub)program.
There is now a routine for performing convolutions and correlations on periodic complex sequences.
Where a given vendor performance library FFT routine is found to be particularly efficient on a given platform, that routine will be called in preference to the NAG Library equivalent.

The new routine names and their function are:

Routine	Function
C06PAF	Single DFT of a Hermitian or real sequence.
C06PCF	Single DFT of a complex sequence.
C06PFF	DFT in one-dimension of a multi-dimensional complex sequence.
C06PJF	DFT of a multi-dimensional complex sequence.
C06PKF	Convolution or correlation of two periodic complex sequences.
C06PPF	Multiple DFTs of a Hermitian or real sequence.
C06PRF	Multiple DFTs of a complex sequence.
C06PUF	Two-dimensional DFT of a complex sequence.
C06PXF	Three-dimensional DFT of a complex sequence.
C06RAF	Multiple Fourier sine transforms on a real sequence.
C06RBF	Multiple Fourier cosine transforms on a real sequence.
C06RCF	Multiple quarter-wave sine transforms on a real sequence.
C06RDF	Multiple quarter-wave cosine transforms on a real sequence.

NAG Fortran Library Performance Improved FFT Routines (some implementations only)

On some implementations of the Fortran SMP Library a number of the routines (real storage format), as documented in the C06 chapter of the NAG Fortran Library manual, have been recoded. These routines are:

C06EAF	C06EBF	C06ECF	C06EKF
C06FAF	C06FBF	C06FCF	C06FKF
C06FPF	C06FQF	C06FRF	C06FUF
C06FXF
C06HAF	C06HBF	C06HCF	C06HDF

Routines affected by replacement routines

The following routines make calls to one or more of the above listed routines.

Routine	Function
C06FFF	1D complex discrete Fourier transform of multi-dimensional data, real storage format.
C06FJF	Multi-dimensional complex discrete Fourier Transform of multi-dimensional data, real storage format.
D03FAF	Helmholtz equation in Cartesian co-ordinates in three dimensions
D05BDF	Nonlinear convolution Volterra-Abel equation of the second kind, weakly singular
D05BEF	Nonlinear convolution Volterra-Abel integral equation of the first kind, weakly singular
D05BYF	Generate weights for use in solving weakly singular Abel type equations
G10BAF	Kernel density estimation using a Gaussian kernel
G13CAF	Univariate time series, smoothed sample spectrum using rectangular, Bartlett, Tukey or Parzen lag window
G13CBF	Univariate time series, smoothed sample spectrum using spectral smoothing by the trapezium frequency (Daniell) window.
G13CCF	Multivariate time series, smoothed sample cross spectrum using rectangular, Bartlett, Tukey or Parzen lag window
G13CDF	Multivariate time series, smoothed sample cross spectrum using spectral smoothing by the trapezium frequency (Daniell) window.
G13CGF	Multivariate time series: noise spectrum, bounds, impulse response function and its standard error

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