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   This file documents the GNU Scientific Library.

   Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001 The GSL Team.

   Permission is granted to copy, distribute and/or modify this document
under the terms of the GNU Free Documentation License, Version 1.1 or
any later version published by the Free Software Foundation; with no
Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts.  A
copy of the license is included in the section entitled "GNU Free
Documentation License".


File: gsl-ref.info,  Node: Complex Hermitian Matrices,  Next: Sorting Eigenvalues and Eigenvectors,  Prev: Real Symmetric Matrices,  Up: Eigensystems

Complex Hermitian Matrices
==========================

 - Function: gsl_eigen_herm_workspace * gsl_eigen_herm_alloc (const
          size_t N)
     This function allocates a workspace for computing eigenvalues of
     N-by-N complex hermitian matrices.  The size of the workspace is
     O(3n).

 - Function: void gsl_eigen_herm_free (gsl_eigen_herm_workspace * W)
     This function frees the memory associated with the workspace W.

 - Function: int gsl_eigen_herm (gsl_matrix_complex * A, gsl_vector *
          EVAL, gsl_eigen_herm_workspace * W)
     This function computes the eigenvalues of the complex hermitian
     matrix A.  Additional workspace of the appropriate size must be
     provided in W.  The diagonal and lower triangular part of A are
     destroyed during the computation, but the strict upper triangular
     part is not referenced.  The imaginary parts of the diagonal are
     assumed to be zero and are not referenced. The eigenvalues are
     stored in the vector EVAL and are unordered.

 - Function: gsl_eigen_hermv_workspace * gsl_eigen_hermv_alloc (const
          size_t N)
     This function allocates a workspace for computing eigenvalues and
     eigenvectors of N-by-N complex hermitian matrices.  The size of
     the workspace is O(5n).

 - Function: void gsl_eigen_hermv_free (gsl_eigen_hermv_workspace * W)
     This function frees the memory associated with the workspace W.

 - Function: int gsl_eigen_hermv (gsl_matrix_complex * A, gsl_vector *
          EVAL, gsl_matrix_complex * EVEC, gsl_eigen_hermv_workspace *
          W)
     This function computes the eigenvalues and eigenvectors of the
     complex hermitian matrix A.  Additional workspace of the
     appropriate size must be provided in W.  The diagonal and lower
     triangular part of A are destroyed during the computation, but the
     strict upper triangular part is not referenced. The imaginary
     parts of the diagonal are assumed to be zero and are not
     referenced.  The eigenvalues are stored in the vector EVAL and are
     unordered.  The corresponding complex eigenvectors are stored in
     the columns of the matrix EVEC.  For example, the eigenvector in
     the first column corresponds to the first eigenvalue.  The
     eigenvectors are guaranteed to be mutually orthogonal and
     normalised to unit magnitude.


File: gsl-ref.info,  Node: Sorting Eigenvalues and Eigenvectors,  Next: Eigenvalue and Eigenvector Examples,  Prev: Complex Hermitian Matrices,  Up: Eigensystems

Sorting Eigenvalues and Eigenvectors
====================================

 - Function: int gsl_eigen_symmv_sort (gsl_vector * EVAL, gsl_matrix *
          EVEC, gsl_eigen_sort_t SORT_TYPE)
     This function simultaneously sorts the eigenvalues stored in the
     vector EVAL and the corresponding real eigenvectors stored in the
     columns of the matrix EVEC into ascending or descending order
     according to the value of the parameter SORT_TYPE,

    `GSL_EIGEN_SORT_VAL_ASC'
          ascending order in numerical value

    `GSL_EIGEN_SORT_VAL_DESC'
          descending order in numerical value

    `GSL_EIGEN_SORT_ABS_ASC'
          ascending order in magnitude

    `GSL_EIGEN_SORT_ABS_DESC'
          descending order in magnitude


 - Function: int gsl_eigen_hermv_sort (gsl_vector * EVAL,
          gsl_matrix_complex * EVEC, gsl_eigen_sort_t SORT_TYPE)
     This function simultaneously sorts the eigenvalues stored in the
     vector EVAL and the corresponding complex eigenvectors stored in
     the columns of the matrix EVEC into ascending or descending order
     according to the value of the parameter SORT_TYPE as shown above.


File: gsl-ref.info,  Node: Eigenvalue and Eigenvector Examples,  Next: Eigenvalue and Eigenvector References,  Prev: Sorting Eigenvalues and Eigenvectors,  Up: Eigensystems

Examples
========

   The following program computes the eigenvalues and eigenvectors of
the 4-th order Hilbert matrix, H(i,j) = 1/(i + j + 1).

     #include <stdio.h>
     #include <gsl/gsl_math.h>
     #include <gsl/gsl_eigen.h>
     
     int
     main (void)
     {
       double data[] = { 1.0  , 1/2.0, 1/3.0, 1/4.0,
                         1/2.0, 1/3.0, 1/4.0, 1/5.0,
                         1/3.0, 1/4.0, 1/5.0, 1/6.0,
                         1/4.0, 1/5.0, 1/6.0, 1/7.0 };
     
       gsl_matrix_view m
         = gsl_matrix_view_array(data, 4, 4);
     
       gsl_vector *eval = gsl_vector_alloc (4);
       gsl_matrix *evec = gsl_matrix_alloc (4, 4);
     
       gsl_eigen_symmv_workspace * w =
         gsl_eigen_symmv_alloc (4);
     
       gsl_eigen_symmv (&m.matrix, eval, evec, w);
     
       gsl_eigen_symmv_free(w);
     
       gsl_eigen_symmv_sort (eval, evec,
                             GSL_EIGEN_SORT_ABS_ASC);
     
       {
         int i;
     
         for (i = 0; i < 4; i++)
           {
             double eval_i
                = gsl_vector_get(eval, i);
             gsl_vector_view evec_i
                = gsl_matrix_column(evec, i);
     
             printf("eigenvalue = %g\n", eval_i);
             printf("eigenvector = \n");
             gsl_vector_fprintf(stdout,
                                &evec_i.vector, "%g");
           }
       }
     
       return 0;
     }

Here is the beginning of the output from the program,

     $ ./a.out
     eigenvalue = 9.67023e-05
     eigenvector =
     -0.0291933
     0.328712
     -0.791411
     0.514553
     ...

This can be compared with the corresponding output from GNU OCTAVE,

     octave> [v,d] = eig(hilb(4));
     octave> diag(d)
     ans =
     
        9.6702e-05
        6.7383e-03
        1.6914e-01
        1.5002e+00
     
     octave> v
     v =
     
        0.029193   0.179186  -0.582076   0.792608
       -0.328712  -0.741918   0.370502   0.451923
        0.791411   0.100228   0.509579   0.322416
       -0.514553   0.638283   0.514048   0.252161

Note that the eigenvectors can differ by a change of sign, since the
sign of an eigenvector is arbitrary.


File: gsl-ref.info,  Node: Eigenvalue and Eigenvector References,  Prev: Eigenvalue and Eigenvector Examples,  Up: Eigensystems

References and Further Reading
==============================

Further information on the algorithms described in this section can be
found in the following book,

     G. H. Golub, C. F. Van Loan, `Matrix Computations' (3rd Ed, 1996),
     Johns Hopkins University Press, ISBN 0-8018-5414-8.

The LAPACK library is described in,

     `LAPACK Users' Guide' (Third Edition, 1999), Published by SIAM,
     ISBN 0-89871-447-8.

     <http://www.netlib.org/lapack>

The LAPACK source code can be found at the website above along with an
online copy of the users guide.


File: gsl-ref.info,  Node: Fast Fourier Transforms,  Next: Numerical Integration,  Prev: Eigensystems,  Up: Top

Fast Fourier Transforms (FFTs)
******************************

   This chapter describes functions for performing Fast Fourier
Transforms (FFTs).  The library includes radix-2 routines (for lengths
which are a power of two) and mixed-radix routines (which work for any
length).  For efficiency there are separate versions of the routines
for real data and for complex data.  The mixed-radix routines are a
reimplementation of the FFTPACK library by Paul Swarztrauber.  Fortran
code for FFTPACK is available on Netlib (FFTPACK also includes some
routines for sine and cosine transforms but these are currently not
available in GSL).  For details and derivations of the underlying
algorithms consult the document `GSL FFT Algorithms' (*note FFT
References and Further Reading::)

* Menu:

* Mathematical Definitions::
* Overview of complex data FFTs::
* Radix-2 FFT routines for complex data::
* Mixed-radix FFT routines for complex data::
* Overview of real data FFTs::
* Radix-2 FFT routines for real data::
* Mixed-radix FFT routines for real data::
* FFT References and Further Reading::


File: gsl-ref.info,  Node: Mathematical Definitions,  Next: Overview of complex data FFTs,  Up: Fast Fourier Transforms

Mathematical Definitions
========================

   Fast Fourier Transforms are efficient algorithms for calculating the
discrete fourier transform (DFT),

     x_j = \sum_{k=0}^{N-1} z_k \exp(-2\pi i j k / N)

   The DFT usually arises as an approximation to the continuous fourier
transform when functions are sampled at discrete intervals in space or
time.  The naive evaluation of the discrete fourier transform is a
matrix-vector multiplication W\vec{z}. A general matrix-vector
multiplication takes O(N^2) operations for N data-points.  Fast fourier
transform algorithms use a divide-and-conquer strategy to factorize the
matrix W into smaller sub-matrices, corresponding to the integer
factors of the length N.  If N can be factorized into a product of
integers f_1 f_2 ... f_n then the DFT can be computed in O(N \sum f_i)
operations.  For a radix-2 FFT this gives an operation count of O(N
\log_2 N).

   All the FFT functions offer three types of transform: forwards,
inverse and backwards, based on the same mathematical definitions.  The
definition of the "forward fourier transform", x = FFT(z), is,

     x_j = \sum_{k=0}^{N-1} z_k \exp(-2\pi i j k / N)

and the definition of the "inverse fourier transform", x = IFFT(z), is,

     z_j = {1 \over N} \sum_{k=0}^{N-1} x_k \exp(2\pi i j k / N).

The factor of 1/N makes this a true inverse.  For example, a call to
`gsl_fft_complex_forward' followed by a call to
`gsl_fft_complex_inverse' should return the original data (within
numerical errors).

   In general there are two possible choices for the sign of the
exponential in the transform/ inverse-transform pair. GSL follows the
same convention as FFTPACK, using a negative exponential for the forward
transform.  The advantage of this convention is that the inverse
transform recreates the original function with simple fourier
synthesis.  Numerical Recipes uses the opposite convention, a positive
exponential in the forward transform.

   The "backwards FFT" is simply our terminology for an unscaled
version of the inverse FFT,

     z^{backwards}_j = \sum_{k=0}^{N-1} x_k \exp(2\pi i j k / N).

When the overall scale of the result is unimportant it is often
convenient to use the backwards FFT instead of the inverse to save
unnecessary divisions.


File: gsl-ref.info,  Node: Overview of complex data FFTs,  Next: Radix-2 FFT routines for complex data,  Prev: Mathematical Definitions,  Up: Fast Fourier Transforms

Overview of complex data FFTs
=============================

   The inputs and outputs for the complex FFT routines are "packed
arrays" of floating point numbers.  In a packed array the real and
imaginary parts of each complex number are placed in alternate
neighboring elements.  For example, the following definition of a packed
array of length 6,

     gsl_complex_packed_array data[6];

can be used to hold an array of three complex numbers, `z[3]', in the
following way,

     data[0] = Re(z[0])
     data[1] = Im(z[0])
     data[2] = Re(z[1])
     data[3] = Im(z[1])
     data[4] = Re(z[2])
     data[5] = Im(z[2])

A "stride" parameter allows the user to perform transforms on the
elements `z[stride*i]' instead of `z[i]'.  A stride greater than 1 can
be used to take an in-place FFT of the column of a matrix. A stride of
1 accesses the array without any additional spacing between elements.

   The array indices have the same ordering as those in the definition
of the DFT -- i.e. there are no index transformations or permutations of
the data.

   For physical applications it is important to remember that the index
appearing in the DFT does not correspond directly to a physical
frequency.  If the time-step of the DFT is \Delta then the
frequency-domain includes both positive and negative frequencies,
ranging from -1/(2\Delta) through 0 to +1/(2\Delta).  The positive
frequencies are stored from the beginning of the array up to the
middle, and the negative frequencies are stored backwards from the end
of the array.

   Here is a table which shows the layout of the array DATA, and the
correspondence between the time-domain data z, and the frequency-domain
data x.

     index    z               x = FFT(z)
     
     0        z(t = 0)        x(f = 0)
     1        z(t = 1)        x(f = 1/(N Delta))
     2        z(t = 2)        x(f = 2/(N Delta))
     .        ........        ..................
     N/2      z(t = N/2)      x(f = +1/(2 Delta),
                                    -1/(2 Delta))
     .        ........        ..................
     N-3      z(t = N-3)      x(f = -3/(N Delta))
     N-2      z(t = N-2)      x(f = -2/(N Delta))
     N-1      z(t = N-1)      x(f = -1/(N Delta))

When N is even the location N/2 contains the most positive and negative
frequencies +1/(2 \Delta), -1/(2 \Delta)) which are equivalent.  If N
is odd then general structure of the table above still applies, but N/2
does not appear.


File: gsl-ref.info,  Node: Radix-2 FFT routines for complex data,  Next: Mixed-radix FFT routines for complex data,  Prev: Overview of complex data FFTs,  Up: Fast Fourier Transforms

Radix-2 FFT routines for complex data
=====================================

   The radix-2 algorithms described in this section are simple and
compact, although not necessarily the most efficient.  They use the
Cooley-Tukey algorithm to compute in-place complex FFTs for lengths
which are a power of 2 -- no additional storage is required.  The
corresponding self-sorting mixed-radix routines offer better
performance at the expense of requiring additional working space.

   All these functions are declared in the header file
`gsl_fft_complex.h'.

 - Function: int gsl_fft_complex_radix2_forward
          (gsl_complex_packed_array DATA[], size_t STRIDE, size_t N)
 - Function: int gsl_fft_complex_radix2_transform
          (gsl_complex_packed_array DATA[], size_t STRIDE, size_t N)
 - Function: int gsl_fft_complex_radix2_backward
          (gsl_complex_packed_array DATA[], size_t STRIDE, size_t N)
 - Function: int gsl_fft_complex_radix2_inverse
          (gsl_complex_packed_array DATA[], size_t STRIDE, size_t N)
     These functions compute forward, backward and inverse FFTs of
     length N with stride STRIDE, on the packed complex array DATA
     using an in-place radix-2 decimation-in-time algorithm.  The
     length of the transform N is restricted to powers of two.

     The functions return a value of `GSL_SUCCESS' if no errors were
     detected, or `GSL_EDOM' if the length of the data N is not a power
     of two.

 - Function: int gsl_fft_complex_radix2_dif_forward
          (gsl_complex_packed_array DATA[], size_t STRIDE, size_t N)
 - Function: int gsl_fft_complex_radix2_dif_transform
          (gsl_complex_packed_array DATA[], size_t STRIDE, size_t N)
 - Function: int gsl_fft_complex_radix2_dif_backward
          (gsl_complex_packed_array DATA[], size_t STRIDE, size_t N)
 - Function: int gsl_fft_complex_radix2_dif_inverse
          (gsl_complex_packed_array DATA[], size_t STRIDE, size_t N)
     These are decimation-in-frequency versions of the radix-2 FFT
     functions.


   Here is an example program which computes the FFT of a short pulse
in a sample of length 128.  To make the resulting fourier transform
real the pulse is defined for equal positive and negative times (-10
... 10), where the negative times wrap around the end of the array.

     #include <stdio.h>
     #include <math.h>
     #include <gsl/gsl_errno.h>
     #include <gsl/gsl_fft_complex.h>
     
     #define REAL(z,i) ((z)[2*(i)])
     #define IMAG(z,i) ((z)[2*(i)+1])
     
     int
     main (void)
     {
       int i;
       double data[2*128];
     
       for (i = 0; i < 128; i++)
         {
            REAL(data,i) = 0.0;
            IMAG(data,i) = 0.0;
         }
     
       REAL(data,0) = 1.0;
     
       for (i = 1; i <= 10; i++)
         {
            REAL(data,i) = REAL(data,128-i) = 1.0;
         }
     
       for (i = 0; i < 128; i++)
         {
           printf ("%d %e %e\n", i,
                   REAL(data,i), IMAG(data,i));
         }
       printf ("\n");
     
       gsl_fft_complex_radix2_forward (data, 1, 128);
     
       for (i = 0; i < 128; i++)
         {
           printf ("%d %e %e\n", i,
                   REAL(data,i)/sqrt(128),
                   IMAG(data,i)/sqrt(128));
         }
     
       return 0;
     }

Note that we have assumed that the program is using the default error
handler (which calls `abort' for any errors).  If you are not using a
safe error handler you would need to check the return status of
`gsl_fft_complex_radix2_forward'.

   The transformed data is rescaled by 1/\sqrt N so that it fits on the
same plot as the input.  Only the real part is shown, by the choice of
the input data the imaginary part is zero.  Allowing for the
wrap-around of negative times at t=128, and working in units of k/N,
the DFT approximates the continuum fourier transform, giving a
modulated \sin function.


File: gsl-ref.info,  Node: Mixed-radix FFT routines for complex data,  Next: Overview of real data FFTs,  Prev: Radix-2 FFT routines for complex data,  Up: Fast Fourier Transforms

Mixed-radix FFT routines for complex data
=========================================

   This section describes mixed-radix FFT algorithms for complex data.
The mixed-radix functions work for FFTs of any length.  They are a
reimplementation of the Fortran FFTPACK library by Paul Swarztrauber.
The theory is explained in the review article `Self-sorting Mixed-radix
FFTs' by Clive Temperton.  The routines here use the same indexing
scheme and basic algorithms as FFTPACK.

   The mixed-radix algorithm is based on sub-transform modules - highly
optimized small length FFTs which are combined to create larger FFTs.
There are efficient modules for factors of 2, 3, 4, 5, 6 and 7.  The
modules for the composite factors of 4 and 6 are faster than combining
the modules for 2*2 and 2*3.

   For factors which are not implemented as modules there is a
fall-back to a general length-n module which uses Singleton's method for
efficiently computing a DFT. This module is O(n^2), and slower than a
dedicated module would be but works for any length n.  Of course,
lengths which use the general length-n module will still be factorized
as much as possible.  For example, a length of 143 will be factorized
into 11*13.  Large prime factors are the worst case scenario, e.g. as
found in n=2*3*99991, and should be avoided because their O(n^2)
scaling will dominate the run-time (consult the document `GSL FFT
Algorithms' included in the GSL distribution if you encounter this
problem).

   The mixed-radix initialization function
`gsl_fft_complex_wavetable_alloc' returns the list of factors chosen by
the library for a given length N.  It can be used to check how well the
length has been factorized, and estimate the run-time.  To a first
approximation the run-time scales as N \sum f_i, where the f_i are the
factors of N.  For programs under user control you may wish to issue a
warning that the transform will be slow when the length is poorly
factorized.  If you frequently encounter data lengths which cannot be
factorized using the existing small-prime modules consult `GSL FFT
Algorithms' for details on adding support for other factors.

   All these functions are declared in the header file
`gsl_fft_complex.h'.

 - Function: gsl_fft_complex_wavetable *
          gsl_fft_complex_wavetable_alloc (size_t N)
     This function prepares a trigonometric lookup table for a complex
     FFT of length N. The function returns a pointer to the newly
     allocated `gsl_fft_complex_wavetable' if no errors were detected,
     and a null pointer in the case of error.  The length N is
     factorized into a product of subtransforms, and the factors and
     their trigonometric coefficients are stored in the wavetable. The
     trigonometric coefficients are computed using direct calls to
     `sin' and `cos', for accuracy.  Recursion relations could be used
     to compute the lookup table faster, but if an application performs
     many FFTs of the same length then this computation is a one-off
     overhead which does not affect the final throughput.

     The wavetable structure can be used repeatedly for any transform
     of the same length.  The table is not modified by calls to any of
     the other FFT functions.  The same wavetable can be used for both
     forward and backward (or inverse) transforms of a given length.

 - Function: void gsl_fft_complex_wavetable_free
          (gsl_fft_complex_wavetable * WAVETABLE)
     This function frees the memory associated with the wavetable
     WAVETABLE.  The wavetable can be freed if no further FFTs of the
     same length will be needed.

These functions operate on a `gsl_fft_complex_wavetable' structure
which contains internal parameters for the FFT.  It is not necessary to
set any of the components directly but it can sometimes be useful to
examine them.  For example, the chosen factorization of the FFT length
is given and can be used to provide an estimate of the run-time or
numerical error.

   The wavetable structure is declared in the header file
`gsl_fft_complex.h'.

 - Data Type: gsl_fft_complex_wavetable
     This is a structure that holds the factorization and trigonometric
     lookup tables for the mixed radix fft algorithm.  It has the
     following components:

    `size_t n'
          This is the number of complex data points

    `size_t nf'
          This is the number of factors that the length `n' was
          decomposed into.

    `size_t factor[64]'
          This is the array of factors.  Only the first `nf' elements
          are used.

    `gsl_complex * trig'
          This is a pointer to a preallocated trigonometric lookup
          table of `n' complex elements.

    `gsl_complex * twiddle[64]'
          This is an array of pointers into `trig', giving the twiddle
          factors for each pass.

The mixed radix algorithms require an additional working space to hold
the intermediate steps of the transform.

 - Function: gsl_fft_complex_workspace *
          gsl_fft_complex_workspace_alloc (size_t N)
     This function allocates a workspace for a complex transform of
     length N.

 - Function: void gsl_fft_complex_workspace_free
          (gsl_fft_complex_workspace * WORKSPACE)
     This function frees the memory associated with the workspace
     WORKSPACE. The workspace can be freed if no further FFTs of the
     same length will be needed.

The following functions compute the transform,

 - Function: int gsl_fft_complex_forward (gsl_complex_packed_array
          DATA[], size_t STRIDE, size_t N, const
          gsl_fft_complex_wavetable * WAVETABLE,
          gsl_fft_complex_workspace * WORK)
 - Function: int gsl_fft_complex_transform (gsl_complex_packed_array
          DATA[], size_t STRIDE, size_t N, const
          gsl_fft_complex_wavetable * WAVETABLE,
          gsl_fft_complex_workspace * WORK)
 - Function: int gsl_fft_complex_backward (gsl_complex_packed_array
          DATA[], size_t STRIDE, size_t N, const
          gsl_fft_complex_wavetable * WAVETABLE,
          gsl_fft_complex_workspace * WORK)
 - Function: int gsl_fft_complex_inverse (gsl_complex_packed_array
          DATA[], size_t STRIDE, size_t N, const
          gsl_fft_complex_wavetable * WAVETABLE,
          gsl_fft_complex_workspace * WORK)
     These functions compute forward, backward and inverse FFTs of
     length N with stride STRIDE, on the packed complex array DATA,
     using a mixed radix decimation-in-frequency algorithm.  There is
     no restriction on the length N.  Efficient modules are provided
     for subtransforms of length 2, 3, 4, 5, 6 and 7.  Any remaining
     factors are computed with a slow, O(n^2), general-n module. The
     caller must supply a WAVETABLE containing the trigonometric lookup
     tables and a workspace WORK.

     The functions return a value of `0' if no errors were detected. The
     following `gsl_errno' conditions are defined for these functions:

    `GSL_EDOM'
          The length of the data N is not a positive integer (i.e. N is
          zero).

    `GSL_EINVAL'
          The length of the data N and the length used to compute the
          given WAVETABLE do not match.

   Here is an example program which computes the FFT of a short pulse
in a sample of length 630 (=2*3*3*5*7) using the mixed-radix algorithm.

     #include <stdio.h>
     #include <math.h>
     #include <gsl/gsl_errno.h>
     #include <gsl/gsl_fft_complex.h>
     
     #define REAL(z,i) ((z)[2*(i)])
     #define IMAG(z,i) ((z)[2*(i)+1])
     
     int
     main (void)
     {
       int i;
       const int n = 630;
       double data[2*n];
     
       gsl_fft_complex_wavetable * wavetable;
       gsl_fft_complex_workspace * workspace;
     
       for (i = 0; i < n; i++)
         {
           REAL(data,i) = 0.0;
           IMAG(data,i) = 0.0;
         }
     
       data[0].real = 1.0;
     
       for (i = 1; i <= 10; i++)
         {
           REAL(data,i) = REAL(data,n-i) = 1.0;
         }
     
       for (i = 0; i < n; i++)
         {
           printf ("%d: %e %e\n", i, REAL(data,i),
                                     IMAG(data,i));
         }
       printf ("\n");
     
       wavetable = gsl_fft_complex_wavetable_alloc (n);
       workspace = gsl_fft_complex_workspace_alloc (n);
     
       for (i = 0; i < wavetable->nf; i++)
         {
            printf("# factor %d: %d\n", i,
                   wavetable->factor[i]);
         }
     
       gsl_fft_complex_forward (data, 1, n,
                                wavetable, workspace);
     
       for (i = 0; i < n; i++)
         {
           printf ("%d: %e %e\n", i, REAL(data,i),
                                     IMAG(data,i));
         }
     
       gsl_fft_complex_wavetable_free (wavetable);
       gsl_fft_complex_workspace_free (workspace);
       return 0;
     }

Note that we have assumed that the program is using the default `gsl'
error handler (which calls `abort' for any errors).  If you are not
using a safe error handler you would need to check the return status of
all the `gsl' routines.


File: gsl-ref.info,  Node: Overview of real data FFTs,  Next: Radix-2 FFT routines for real data,  Prev: Mixed-radix FFT routines for complex data,  Up: Fast Fourier Transforms

Overview of real data FFTs
==========================

   The functions for real data are similar to those for complex data.
However, there is an important difference between forward and inverse
transforms.  The fourier transform of a real sequence is not real.  It
is a complex sequence with a special symmetry:

     z_k = z_{N-k}^*

A sequence with this symmetry is called "conjugate-complex" or
"half-complex".  This different structure requires different storage
layouts for the forward transform (from real to half-complex) and
inverse transform (from half-complex back to real).  As a consequence
the routines are divided into two sets: functions in `gsl_fft_real'
which operate on real sequences and functions in `gsl_fft_halfcomplex'
which operate on half-complex sequences.

   Functions in `gsl_fft_real' compute the frequency coefficients of a
real sequence.  The half-complex coefficients c of a real sequence x
are given by fourier analysis,

     c_k = \sum_{j=0}^{N-1} x_k \exp(-2 \pi i j k /N)

Functions in `gsl_fft_halfcomplex' compute inverse or backwards
transforms.  They reconstruct real sequences by fourier synthesis from
their half-complex frequency coefficients, c,

     x_j = {1 \over N} \sum_{k=0}^{N-1} c_k \exp(2 \pi i j k /N)

The symmetry of the half-complex sequence implies that only half of the
complex numbers in the output need to be stored.  The remaining half can
be reconstructed using the half-complex symmetry condition. (This works
for all lengths, even and odd.  When the length is even the middle
value, where k=N/2, is also real).  Thus only N real numbers are
required to store the half-complex sequence, and the transform of a real
sequence can be stored in the same size array as the original data.

   The precise storage arrangements depend on the algorithm, and are
different for radix-2 and mixed-radix routines.  The radix-2 function
operates in-place, which constrain the locations where each element can
be stored.  The restriction forces real and imaginary parts to be stored
far apart.  The mixed-radix algorithm does not have this restriction,
and it stores the real and imaginary parts of a given term in
neighboring locations.  This is desirable for better locality of memory
accesses.


File: gsl-ref.info,  Node: Radix-2 FFT routines for real data,  Next: Mixed-radix FFT routines for real data,  Prev: Overview of real data FFTs,  Up: Fast Fourier Transforms

Radix-2 FFT routines for real data
==================================

   This section describes radix-2 FFT algorithms for real data.  They
use the Cooley-Tukey algorithm to compute in-place FFTs for lengths
which are a power of 2.

   The radix-2 FFT functions for real data are declared in the header
files `gsl_fft_real.h'

 - Function: int gsl_fft_real_radix2_transform (double DATA[], size_t
          STRIDE, size_t N)
     This function computes an in-place radix-2 FFT of length N and
     stride STRIDE on the real array DATA.  The output is a
     half-complex sequence, which is stored in-place.  The arrangement
     of the half-complex terms uses the following scheme: for k < N/2
     the real part of the k-th term is stored in location k, and the
     corresponding imaginary part is stored in location N-k.  Terms
     with k > N/2 can be reconstructed using the symmetry z_k =
     z^*_{N-k}.  The terms for k=0 and k=N/2 are both purely real, and
     count as a special case.  Their real parts are stored in locations
     0 and N/2 respectively, while their imaginary parts which are zero
     are not stored.

     The following table shows the correspondence between the output
     DATA and the equivalent results obtained by considering the input
     data as a complex sequence with zero imaginary part,

          complex[0].real    =    data[0]
          complex[0].imag    =    0
          complex[1].real    =    data[1]
          complex[1].imag    =    data[N-1]
          ...............         ................
          complex[k].real    =    data[k]
          complex[k].imag    =    data[N-k]
          ...............         ................
          complex[N/2].real  =    data[N/2]
          complex[N/2].real  =    0
          ...............         ................
          complex[k'].real   =    data[k]        k' = N - k
          complex[k'].imag   =   -data[N-k]
          ...............         ................
          complex[N-1].real  =    data[1]
          complex[N-1].imag  =   -data[N-1]

   The radix-2 FFT functions for halfcomplex data are declared in the
header file `gsl_fft_halfcomplex.h'.

 - Function: int gsl_fft_halfcomplex_radix2_inverse (double DATA[],
          size_t STRIDE, size_t N)
 - Function: int gsl_fft_halfcomplex_radix2_backward (double DATA[],
          size_t STRIDE, size_t N)
     These functions compute the inverse or backwards in-place radix-2
     FFT of length N and stride STRIDE on the half-complex sequence
     DATA stored according the output scheme used by
     `gsl_fft_real_radix2'.  The result is a real array stored in
     natural order.



File: gsl-ref.info,  Node: Mixed-radix FFT routines for real data,  Next: FFT References and Further Reading,  Prev: Radix-2 FFT routines for real data,  Up: Fast Fourier Transforms

Mixed-radix FFT routines for real data
======================================

   This section describes mixed-radix FFT algorithms for real data.  The
mixed-radix functions work for FFTs of any length.  They are a
reimplementation of the real-FFT routines in the Fortran FFTPACK library
by Paul Swarztrauber.  The theory behind the algorithm is explained in
the article `Fast Mixed-Radix Real Fourier Transforms' by Clive
Temperton.  The routines here use the same indexing scheme and basic
algorithms as FFTPACK.

   The functions use the FFTPACK storage convention for half-complex
sequences.  In this convention the half-complex transform of a real
sequence is stored with frequencies in increasing order, starting at
zero, with the real and imaginary parts of each frequency in neighboring
locations.  When a value is known to be real the imaginary part is not
stored.  The imaginary part of the zero-frequency component is never
stored.  It is known to be zero (since the zero frequency component is
simply the sum of the input data (all real)).  For a sequence of even
length the imaginary part of the frequency n/2 is not stored either,
since the symmetry z_k = z_{N-k}^* implies that this is purely real too.

   The storage scheme is best shown by some examples.  The table below
shows the output for an odd-length sequence, n=5.  The two columns give
the correspondence between the 5 values in the half-complex sequence
returned by `gsl_fft_real_transform', HALFCOMPLEX[] and the values
COMPLEX[] that would be returned if the same real input sequence were
passed to `gsl_fft_complex_backward' as a complex sequence (with
imaginary parts set to `0'),

     complex[0].real  =  halfcomplex[0]
     complex[0].imag  =  0
     complex[1].real  =  halfcomplex[1]
     complex[1].imag  =  halfcomplex[2]
     complex[2].real  =  halfcomplex[3]
     complex[2].imag  =  halfcomplex[4]
     complex[3].real  =  halfcomplex[3]
     complex[3].imag  = -halfcomplex[4]
     complex[4].real  =  halfcomplex[1]
     complex[4].imag  = -halfcomplex[2]

The upper elements of the COMPLEX array, `complex[3]' and `complex[4]'
are filled in using the symmetry condition.  The imaginary part of the
zero-frequency term `complex[0].imag' is known to be zero by the
symmetry.

   The next table shows the output for an even-length sequence, n=5 In
the even case both the there are two values which are purely real,

     complex[0].real  =  halfcomplex[0]
     complex[0].imag  =  0
     complex[1].real  =  halfcomplex[1]
     complex[1].imag  =  halfcomplex[2]
     complex[2].real  =  halfcomplex[3]
     complex[2].imag  =  halfcomplex[4]
     complex[3].real  =  halfcomplex[5]
     complex[3].imag  =  0
     complex[4].real  =  halfcomplex[3]
     complex[4].imag  = -halfcomplex[4]
     complex[5].real  =  halfcomplex[1]
     complex[5].imag  = -halfcomplex[2]

The upper elements of the COMPLEX array, `complex[4]' and `complex[5]'
are filled in using the symmetry condition.  Both `complex[0].imag' and
`complex[3].imag' are known to be zero.

   All these functions are declared in the header files
`gsl_fft_real.h' and `gsl_fft_halfcomplex.h'.

 - Function: gsl_fft_real_wavetable * gsl_fft_real_wavetable_alloc
          (size_t N)
 - Function: gsl_fft_halfcomplex_wavetable *
gsl_fft_halfcomplex_wavetable_alloc (size_t N)
     These functions prepare trigonometric lookup tables for an FFT of
     size n real elements.  The functions return a pointer to the newly
     allocated struct if no errors were detected, and a null pointer in
     the case of error.  The length N is factorized into a product of
     subtransforms, and the factors and their trigonometric
     coefficients are stored in the wavetable. The trigonometric
     coefficients are computed using direct calls to `sin' and `cos',
     for accuracy.  Recursion relations could be used to compute the
     lookup table faster, but if an application performs many FFTs of
     the same length then computing the wavetable is a one-off overhead
     which does not affect the final throughput.

     The wavetable structure can be used repeatedly for any transform
     of the same length.  The table is not modified by calls to any of
     the other FFT functions.  The appropriate type of wavetable must
     be used for forward real or inverse half-complex transforms.

 - Function: void gsl_fft_real_wavetable_free (gsl_fft_real_wavetable *
          WAVETABLE)
 - Function: void gsl_fft_halfcomplex_wavetable_free
          (gsl_fft_halfcomplex_wavetable * WAVETABLE)
     These functions free the memory associated with the wavetable
     WAVETABLE. The wavetable can be freed if no further FFTs of the
     same length will be needed.

The mixed radix algorithms require an additional working space to hold
the intermediate steps of the transform,

 - Function: gsl_fft_real_workspace * gsl_fft_real_workspace_alloc
          (size_t N)
     This function allocates a workspace for a real transform of length
     N.  The same workspace is used for both forward real and inverse
     halfcomplex transforms.

 - Function: void gsl_fft_real_workspace_free (gsl_fft_real_workspace *
          WORKSPACE)
     This function frees the memory associated with the workspace
     WORKSPACE. The workspace can be freed if no further FFTs of the
     same length will be needed.

The following functions compute the transforms of real and half-complex
data,

 - Function: int gsl_fft_real_transform (double DATA[], size_t STRIDE,
          size_t N, const gsl_fft_real_wavetable * WAVETABLE,
          gsl_fft_real_workspace * WORK)
 - Function: int gsl_fft_halfcomplex_transform (double DATA[], size_t
          STRIDE, size_t N, const gsl_fft_halfcomplex_wavetable *
          WAVETABLE, gsl_fft_real_workspace * WORK)
     These functions compute the FFT of DATA, a real or half-complex
     array of length N, using a mixed radix decimation-in-frequency
     algorithm.  For `gsl_fft_real_transform' DATA is an array of
     time-ordered real data.  For `gsl_fft_halfcomplex_transform' DATA
     contains fourier coefficients in the half-complex ordering
     described above.  There is no restriction on the length N.
     Efficient modules are provided for subtransforms of length 2, 3, 4
     and 5.  Any remaining factors are computed with a slow, O(n^2),
     general-n module.  The caller must supply a WAVETABLE containing
     trigonometric lookup tables and a workspace WORK.

 - Function: int gsl_fft_real_unpack (const double REAL_COEFFICIENT[],
          gsl_complex_packed_array COMPLEX_COEFFICIENT[], size_t
          STRIDE, size_t N)
     This function converts a single real array, REAL_COEFFICIENT into
     an equivalent complex array, COMPLEX_COEFFICIENT, (with imaginary
     part set to zero), suitable for `gsl_fft_complex' routines.  The
     algorithm for the conversion is simply,

          for (i = 0; i < n; i++)
            {
              complex_coefficient[i].real
                = real_coefficient[i];
              complex_coefficient[i].imag
                = 0.0;
            }

 - Function: int gsl_fft_halfcomplex_unpack (const double
          HALFCOMPLEX_COEFFICIENT[], gsl_complex_packed_array
          COMPLEX_COEFFICIENT[], size_t STRIDE, size_t N)
     This function converts HALFCOMPLEX_COEFFICIENT, an array of
     half-complex coefficients as returned by `gsl_fft_real_transform',
     into an ordinary complex array, COMPLEX_COEFFICIENT.  It fills in
     the complex array using the symmetry z_k = z_{N-k}^* to
     reconstruct the redundant elements.  The algorithm for the
     conversion is,

          complex_coefficient[0].real
            = halfcomplex_coefficient[0];
          complex_coefficient[0].imag
            = 0.0;
          
          for (i = 1; i < n - i; i++)
            {
              double hc_real
                = halfcomplex_coefficient[2 * i - 1];
              double hc_imag
                = halfcomplex_coefficient[2 * i];
              complex_coefficient[i].real = hc_real;
              complex_coefficient[i].imag = hc_imag;
              complex_coefficient[n - i].real = hc_real;
              complex_coefficient[n - i].imag = -hc_imag;
            }
          
          if (i == n - i)
            {
              complex_coefficient[i].real
                = halfcomplex_coefficient[n - 1];
              complex_coefficient[i].imag
                = 0.0;
            }

   Here is an example program using `gsl_fft_real_transform' and
`gsl_fft_halfcomplex_inverse'.  It generates a real signal in the shape
of a square pulse.  The pulse is fourier transformed to frequency
space, and all but the lowest ten frequency components are removed from
the array of fourier coefficients returned by `gsl_fft_real_transform'.

   The remaining fourier coefficients are transformed back to the
time-domain, to give a filtered version of the square pulse.  Since
fourier coefficients are stored using the half-complex symmetry both
positive and negative frequencies are removed and the final filtered
signal is also real.

     #include <stdio.h>
     #include <math.h>
     #include <gsl/gsl_errno.h>
     #include <gsl/gsl_fft_real.h>
     #include <gsl/gsl_fft_halfcomplex.h>
     
     int
     main (void)
     {
       int i, n = 100;
       double data[n];
     
       gsl_fft_real_wavetable * real;
       gsl_fft_halfcomplex_wavetable * hc;
       gsl_fft_real_workspace * work;
     
       for (i = 0; i < n; i++)
         {
           data[i] = 0.0;
         }
     
       for (i = n / 3; i < 2 * n / 3; i++)
         {
           data[i] = 1.0;
         }
     
       for (i = 0; i < n; i++)
         {
           printf ("%d: %e\n", i, data[i]);
         }
       printf ("\n");
     
       work = gsl_fft_real_workspace_alloc (n);
       real = gsl_fft_real_wavetable_alloc (n);
     
       gsl_fft_real_transform (data, 1, n,
                               real, work);
     
       gsl_fft_real_wavetable_free (real);
     
       for (i = 11; i < n; i++)
         {
           data[i] = 0;
         }
     
       hc = gsl_fft_halfcomplex_wavetable_alloc (n);
     
       gsl_fft_halfcomplex_inverse (data, 1, n,
                                    hc, work);
       gsl_fft_halfcomplex_wavetable_free (hc);
     
       for (i = 0; i < n; i++)
         {
           printf ("%d: %e\n", i, data[i]);
         }
     
       gsl_fft_real_workspace_free (work);
       return 0;
     }


File: gsl-ref.info,  Node: FFT References and Further Reading,  Prev: Mixed-radix FFT routines for real data,  Up: Fast Fourier Transforms

References and Further Reading
==============================

A good starting point for learning more about the FFT is the review
article `Fast Fourier Transforms: A Tutorial Review and A State of the
Art' by Duhamel and Vetterli,

     P. Duhamel and M. Vetterli.  Fast fourier transforms: A tutorial
     review and a state of the art.  `Signal Processing', 19:259-299,
     1990.

To find out about the algorithms used in the GSL routines you may want
to consult the latex document `GSL FFT Algorithms' (it is included in
GSL, as `doc/fftalgorithms.tex').  This has general information on FFTs
and explicit derivations of the implementation for each routine.  There
are also references to the relevant literature.  For convenience some
of the more important references are reproduced below.

There are several introductory books on the FFT with example programs,
such as `The Fast Fourier Transform' by Brigham and `DFT/FFT and
Convolution Algorithms' by Burrus and Parks,

     E. Oran Brigham.  `The Fast Fourier Transform'.  Prentice Hall,
     1974.

     C. S. Burrus and T. W. Parks.  `DFT/FFT and Convolution
     Algorithms'.  Wiley, 1984.

Both these introductory books cover the radix-2 FFT in some detail.
The mixed-radix algorithm at the heart of the FFTPACK routines is
reviewed in Clive Temperton's paper,

     Clive Temperton.  Self-sorting mixed-radix fast fourier transforms.
     `Journal of Computational Physics', 52(1):1-23, 1983.

The derivation of FFTs for real-valued data is explained in the
following two articles,

     Henrik V. Sorenson, Douglas L. Jones, Michael T. Heideman, and C.
     Sidney Burrus.  Real-valued fast fourier transform algorithms.
     `IEEE Transactions on Acoustics, Speech, and Signal Processing',
     ASSP-35(6):849-863, 1987.

     Clive Temperton.  Fast mixed-radix real fourier transforms.
     `Journal of Computational Physics', 52:340-350, 1983.

In 1979 the IEEE published a compendium of carefully-reviewed Fortran
FFT programs in `Programs for Digital Signal Processing'.  It is a
useful reference for implementations of many different FFT algorithms,

     Digital Signal Processing Committee and IEEE Acoustics, Speech,
     and Signal Processing Committee, editors.  `Programs for Digital
     Signal Processing'.  IEEE Press, 1979.

For serious FFT work we recommend the use of the dedicated FFTW library
by Frigo and Johnson.  The FFTW library is self-optimizing -- it
automatically tunes itself for each hardware platform in order to
achieve maximum performance.  It is available under the GNU GPL.

     FFTW Website, <http://www.fftw.org/>


File: gsl-ref.info,  Node: Numerical Integration,  Next: Random Number Generation,  Prev: Fast Fourier Transforms,  Up: Top

Numerical Integration
*********************

   This chapter describes routines for performing numerical integration
(quadrature) of a function in one dimension.  There are routines for
adaptive and non-adaptive integration of general functions, with
specialised routines for specific cases.  These include integration over
infinite and semi-infinite ranges, singular integrals, including
logarithmic singularities, computation of Cauchy principal values and
oscillatory integrals.  The library reimplements the algorithms used in
QUADPACK, a numerical integration package written by Piessens,
Doncker-Kapenga, Uberhuber and Kahaner.  Fortran code for QUADPACK is
available on Netlib.

   The functions described in this chapter are declared in the header
file `gsl_integration.h'.

* Menu:

* Numerical Integration Introduction::
* QNG non-adaptive Gauss-Kronrod integration::
* QAG adaptive integration::
* QAGS adaptive integration with singularities::
* QAGP adaptive integration with known singular points::
* QAGI adaptive integration on infinite intervals::
* QAWC adaptive integration for Cauchy principal values::
* QAWS adaptive integration for singular functions::
* QAWO adaptive integration for oscillatory functions::
* QAWF adaptive integration for Fourier integrals::
* Numerical integration error codes::
* Numerical integration examples::
* Numerical integration References and Further Reading::

