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+/*
+ Interface for the f2c translation of fftpack as found on http://www.netlib.org/fftpack/
+
+ FFTPACK license:
+
+ http://www.cisl.ucar.edu/css/software/fftpack5/ftpk.html
+
+ Copyright (c) 2004 the University Corporation for Atmospheric
+ Research ("UCAR"). All rights reserved. Developed by NCAR's
+ Computational and Information Systems Laboratory, UCAR,
+ www.cisl.ucar.edu.
+
+ Redistribution and use of the Software in source and binary forms,
+ with or without modification, is permitted provided that the
+ following conditions are met:
+
+ - Neither the names of NCAR's Computational and Information Systems
+ Laboratory, the University Corporation for Atmospheric Research,
+ nor the names of its sponsors or contributors may be used to
+ endorse or promote products derived from this Software without
+ specific prior written permission.
+
+ - Redistributions of source code must retain the above copyright
+ notices, this list of conditions, and the disclaimer below.
+
+ - Redistributions in binary form must reproduce the above copyright
+ notice, this list of conditions, and the disclaimer below in the
+ documentation and/or other materials provided with the
+ distribution.
+
+ THIS SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
+ EXPRESS OR IMPLIED, INCLUDING, BUT NOT LIMITED TO THE WARRANTIES OF
+ MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
+ NONINFRINGEMENT. IN NO EVENT SHALL THE CONTRIBUTORS OR COPYRIGHT
+ HOLDERS BE LIABLE FOR ANY CLAIM, INDIRECT, INCIDENTAL, SPECIAL,
+ EXEMPLARY, OR CONSEQUENTIAL DAMAGES OR OTHER LIABILITY, WHETHER IN AN
+ ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN
+ CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS WITH THE
+ SOFTWARE.
+
+ ChangeLog:
+ 2011/10/02: this is my first release of this file.
+*/
+
+#ifndef FFTPACK_H
+#define FFTPACK_H
+
+#ifdef __cplusplus
+extern "C" {
+#endif
+
+// just define FFTPACK_DOUBLE_PRECISION if you want to build it as a double precision fft
+
+#ifndef FFTPACK_DOUBLE_PRECISION
+ typedef float fftpack_real;
+ typedef int fftpack_int;
+#else
+ typedef double fftpack_real;
+ typedef int fftpack_int;
+#endif
+
+ void cffti(fftpack_int n, fftpack_real *wsave);
+
+ void cfftf(fftpack_int n, fftpack_real *c, fftpack_real *wsave);
+
+ void cfftb(fftpack_int n, fftpack_real *c, fftpack_real *wsave);
+
+ void rffti(fftpack_int n, fftpack_real *wsave);
+ void rfftf(fftpack_int n, fftpack_real *r, fftpack_real *wsave);
+ void rfftb(fftpack_int n, fftpack_real *r, fftpack_real *wsave);
+
+ void cosqi(fftpack_int n, fftpack_real *wsave);
+ void cosqf(fftpack_int n, fftpack_real *x, fftpack_real *wsave);
+ void cosqb(fftpack_int n, fftpack_real *x, fftpack_real *wsave);
+
+ void costi(fftpack_int n, fftpack_real *wsave);
+ void cost(fftpack_int n, fftpack_real *x, fftpack_real *wsave);
+
+ void sinqi(fftpack_int n, fftpack_real *wsave);
+ void sinqb(fftpack_int n, fftpack_real *x, fftpack_real *wsave);
+ void sinqf(fftpack_int n, fftpack_real *x, fftpack_real *wsave);
+
+ void sinti(fftpack_int n, fftpack_real *wsave);
+ void sint(fftpack_int n, fftpack_real *x, fftpack_real *wsave);
+
+#ifdef __cplusplus
+}
+#endif
+
+#endif /* FFTPACK_H */
+
+/*
+
+ FFTPACK
+
+* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
+
+ version 4 april 1985
+
+ a package of fortran subprograms for the fast fourier
+ transform of periodic and other symmetric sequences
+
+ by
+
+ paul n swarztrauber
+
+ national center for atmospheric research boulder,colorado 80307
+
+ which is sponsored by the national science foundation
+
+* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
+
+
+this package consists of programs which perform fast fourier
+transforms for both complex and real periodic sequences and
+certain other symmetric sequences that are listed below.
+
+1. rffti initialize rfftf and rfftb
+2. rfftf forward transform of a real periodic sequence
+3. rfftb backward transform of a real coefficient array
+
+4. ezffti initialize ezfftf and ezfftb
+5. ezfftf a simplified real periodic forward transform
+6. ezfftb a simplified real periodic backward transform
+
+7. sinti initialize sint
+8. sint sine transform of a real odd sequence
+
+9. costi initialize cost
+10. cost cosine transform of a real even sequence
+
+11. sinqi initialize sinqf and sinqb
+12. sinqf forward sine transform with odd wave numbers
+13. sinqb unnormalized inverse of sinqf
+
+14. cosqi initialize cosqf and cosqb
+15. cosqf forward cosine transform with odd wave numbers
+16. cosqb unnormalized inverse of cosqf
+
+17. cffti initialize cfftf and cfftb
+18. cfftf forward transform of a complex periodic sequence
+19. cfftb unnormalized inverse of cfftf
+
+
+******************************************************************
+
+subroutine rffti(n,wsave)
+
+ ****************************************************************
+
+subroutine rffti initializes the array wsave which is used in
+both rfftf and rfftb. the prime factorization of n together with
+a tabulation of the trigonometric functions are computed and
+stored in wsave.
+
+input parameter
+
+n the length of the sequence to be transformed.
+
+output parameter
+
+wsave a work array which must be dimensioned at least 2*n+15.
+ the same work array can be used for both rfftf and rfftb
+ as long as n remains unchanged. different wsave arrays
+ are required for different values of n. the contents of
+ wsave must not be changed between calls of rfftf or rfftb.
+
+******************************************************************
+
+subroutine rfftf(n,r,wsave)
+
+******************************************************************
+
+subroutine rfftf computes the fourier coefficients of a real
+perodic sequence (fourier analysis). the transform is defined
+below at output parameter r.
+
+input parameters
+
+n the length of the array r to be transformed. the method
+ is most efficient when n is a product of small primes.
+ n may change so long as different work arrays are provided
+
+r a real array of length n which contains the sequence
+ to be transformed
+
+wsave a work array which must be dimensioned at least 2*n+15.
+ in the program that calls rfftf. the wsave array must be
+ initialized by calling subroutine rffti(n,wsave) and a
+ different wsave array must be used for each different
+ value of n. this initialization does not have to be
+ repeated so long as n remains unchanged thus subsequent
+ transforms can be obtained faster than the first.
+ the same wsave array can be used by rfftf and rfftb.
+
+
+output parameters
+
+r r(1) = the sum from i=1 to i=n of r(i)
+
+ if n is even set l =n/2 , if n is odd set l = (n+1)/2
+
+ then for k = 2,...,l
+
+ r(2*k-2) = the sum from i = 1 to i = n of
+
+ r(i)*cos((k-1)*(i-1)*2*pi/n)
+
+ r(2*k-1) = the sum from i = 1 to i = n of
+
+ -r(i)*sin((k-1)*(i-1)*2*pi/n)
+
+ if n is even
+
+ r(n) = the sum from i = 1 to i = n of
+
+ (-1)**(i-1)*r(i)
+
+ ***** note
+ this transform is unnormalized since a call of rfftf
+ followed by a call of rfftb will multiply the input
+ sequence by n.
+
+wsave contains results which must not be destroyed between
+ calls of rfftf or rfftb.
+
+
+******************************************************************
+
+subroutine rfftb(n,r,wsave)
+
+******************************************************************
+
+subroutine rfftb computes the real perodic sequence from its
+fourier coefficients (fourier synthesis). the transform is defined
+below at output parameter r.
+
+input parameters
+
+n the length of the array r to be transformed. the method
+ is most efficient when n is a product of small primes.
+ n may change so long as different work arrays are provided
+
+r a real array of length n which contains the sequence
+ to be transformed
+
+wsave a work array which must be dimensioned at least 2*n+15.
+ in the program that calls rfftb. the wsave array must be
+ initialized by calling subroutine rffti(n,wsave) and a
+ different wsave array must be used for each different
+ value of n. this initialization does not have to be
+ repeated so long as n remains unchanged thus subsequent
+ transforms can be obtained faster than the first.
+ the same wsave array can be used by rfftf and rfftb.
+
+
+output parameters
+
+r for n even and for i = 1,...,n
+
+ r(i) = r(1)+(-1)**(i-1)*r(n)
+
+ plus the sum from k=2 to k=n/2 of
+
+ 2.*r(2*k-2)*cos((k-1)*(i-1)*2*pi/n)
+
+ -2.*r(2*k-1)*sin((k-1)*(i-1)*2*pi/n)
+
+ for n odd and for i = 1,...,n
+
+ r(i) = r(1) plus the sum from k=2 to k=(n+1)/2 of
+
+ 2.*r(2*k-2)*cos((k-1)*(i-1)*2*pi/n)
+
+ -2.*r(2*k-1)*sin((k-1)*(i-1)*2*pi/n)
+
+ ***** note
+ this transform is unnormalized since a call of rfftf
+ followed by a call of rfftb will multiply the input
+ sequence by n.
+
+wsave contains results which must not be destroyed between
+ calls of rfftb or rfftf.
+
+******************************************************************
+
+subroutine sinti(n,wsave)
+
+******************************************************************
+
+subroutine sinti initializes the array wsave which is used in
+subroutine sint. the prime factorization of n together with
+a tabulation of the trigonometric functions are computed and
+stored in wsave.
+
+input parameter
+
+n the length of the sequence to be transformed. the method
+ is most efficient when n+1 is a product of small primes.
+
+output parameter
+
+wsave a work array with at least int(2.5*n+15) locations.
+ different wsave arrays are required for different values
+ of n. the contents of wsave must not be changed between
+ calls of sint.
+
+******************************************************************
+
+subroutine sint(n,x,wsave)
+
+******************************************************************
+
+subroutine sint computes the discrete fourier sine transform
+of an odd sequence x(i). the transform is defined below at
+output parameter x.
+
+sint is the unnormalized inverse of itself since a call of sint
+followed by another call of sint will multiply the input sequence
+x by 2*(n+1).
+
+the array wsave which is used by subroutine sint must be
+initialized by calling subroutine sinti(n,wsave).
+
+input parameters
+
+n the length of the sequence to be transformed. the method
+ is most efficient when n+1 is the product of small primes.
+
+x an array which contains the sequence to be transformed
+
+
+wsave a work array with dimension at least int(2.5*n+15)
+ in the program that calls sint. the wsave array must be
+ initialized by calling subroutine sinti(n,wsave) and a
+ different wsave array must be used for each different
+ value of n. this initialization does not have to be
+ repeated so long as n remains unchanged thus subsequent
+ transforms can be obtained faster than the first.
+
+output parameters
+
+x for i=1,...,n
+
+ x(i)= the sum from k=1 to k=n
+
+ 2*x(k)*sin(k*i*pi/(n+1))
+
+ a call of sint followed by another call of
+ sint will multiply the sequence x by 2*(n+1).
+ hence sint is the unnormalized inverse
+ of itself.
+
+wsave contains initialization calculations which must not be
+ destroyed between calls of sint.
+
+******************************************************************
+
+subroutine costi(n,wsave)
+
+******************************************************************
+
+subroutine costi initializes the array wsave which is used in
+subroutine cost. the prime factorization of n together with
+a tabulation of the trigonometric functions are computed and
+stored in wsave.
+
+input parameter
+
+n the length of the sequence to be transformed. the method
+ is most efficient when n-1 is a product of small primes.
+
+output parameter
+
+wsave a work array which must be dimensioned at least 3*n+15.
+ different wsave arrays are required for different values
+ of n. the contents of wsave must not be changed between
+ calls of cost.
+
+******************************************************************
+
+subroutine cost(n,x,wsave)
+
+******************************************************************
+
+subroutine cost computes the discrete fourier cosine transform
+of an even sequence x(i). the transform is defined below at output
+parameter x.
+
+cost is the unnormalized inverse of itself since a call of cost
+followed by another call of cost will multiply the input sequence
+x by 2*(n-1). the transform is defined below at output parameter x
+
+the array wsave which is used by subroutine cost must be
+initialized by calling subroutine costi(n,wsave).
+
+input parameters
+
+n the length of the sequence x. n must be greater than 1.
+ the method is most efficient when n-1 is a product of
+ small primes.
+
+x an array which contains the sequence to be transformed
+
+wsave a work array which must be dimensioned at least 3*n+15
+ in the program that calls cost. the wsave array must be
+ initialized by calling subroutine costi(n,wsave) and a
+ different wsave array must be used for each different
+ value of n. this initialization does not have to be
+ repeated so long as n remains unchanged thus subsequent
+ transforms can be obtained faster than the first.
+
+output parameters
+
+x for i=1,...,n
+
+ x(i) = x(1)+(-1)**(i-1)*x(n)
+
+ + the sum from k=2 to k=n-1
+
+ 2*x(k)*cos((k-1)*(i-1)*pi/(n-1))
+
+ a call of cost followed by another call of
+ cost will multiply the sequence x by 2*(n-1)
+ hence cost is the unnormalized inverse
+ of itself.
+
+wsave contains initialization calculations which must not be
+ destroyed between calls of cost.
+
+******************************************************************
+
+subroutine sinqi(n,wsave)
+
+******************************************************************
+
+subroutine sinqi initializes the array wsave which is used in
+both sinqf and sinqb. the prime factorization of n together with
+a tabulation of the trigonometric functions are computed and
+stored in wsave.
+
+input parameter
+
+n the length of the sequence to be transformed. the method
+ is most efficient when n is a product of small primes.
+
+output parameter
+
+wsave a work array which must be dimensioned at least 3*n+15.
+ the same work array can be used for both sinqf and sinqb
+ as long as n remains unchanged. different wsave arrays
+ are required for different values of n. the contents of
+ wsave must not be changed between calls of sinqf or sinqb.
+
+******************************************************************
+
+subroutine sinqf(n,x,wsave)
+
+******************************************************************
+
+subroutine sinqf computes the fast fourier transform of quarter
+wave data. that is , sinqf computes the coefficients in a sine
+series representation with only odd wave numbers. the transform
+is defined below at output parameter x.
+
+sinqb is the unnormalized inverse of sinqf since a call of sinqf
+followed by a call of sinqb will multiply the input sequence x
+by 4*n.
+
+the array wsave which is used by subroutine sinqf must be
+initialized by calling subroutine sinqi(n,wsave).
+
+
+input parameters
+
+n the length of the array x to be transformed. the method
+ is most efficient when n is a product of small primes.
+
+x an array which contains the sequence to be transformed
+
+wsave a work array which must be dimensioned at least 3*n+15.
+ in the program that calls sinqf. the wsave array must be
+ initialized by calling subroutine sinqi(n,wsave) and a
+ different wsave array must be used for each different
+ value of n. this initialization does not have to be
+ repeated so long as n remains unchanged thus subsequent
+ transforms can be obtained faster than the first.
+
+output parameters
+
+x for i=1,...,n
+
+ x(i) = (-1)**(i-1)*x(n)
+
+ + the sum from k=1 to k=n-1 of
+
+ 2*x(k)*sin((2*i-1)*k*pi/(2*n))
+
+ a call of sinqf followed by a call of
+ sinqb will multiply the sequence x by 4*n.
+ therefore sinqb is the unnormalized inverse
+ of sinqf.
+
+wsave contains initialization calculations which must not
+ be destroyed between calls of sinqf or sinqb.
+
+******************************************************************
+
+subroutine sinqb(n,x,wsave)
+
+******************************************************************
+
+subroutine sinqb computes the fast fourier transform of quarter
+wave data. that is , sinqb computes a sequence from its
+representation in terms of a sine series with odd wave numbers.
+the transform is defined below at output parameter x.
+
+sinqf is the unnormalized inverse of sinqb since a call of sinqb
+followed by a call of sinqf will multiply the input sequence x
+by 4*n.
+
+the array wsave which is used by subroutine sinqb must be
+initialized by calling subroutine sinqi(n,wsave).
+
+
+input parameters
+
+n the length of the array x to be transformed. the method
+ is most efficient when n is a product of small primes.
+
+x an array which contains the sequence to be transformed
+
+wsave a work array which must be dimensioned at least 3*n+15.
+ in the program that calls sinqb. the wsave array must be
+ initialized by calling subroutine sinqi(n,wsave) and a
+ different wsave array must be used for each different
+ value of n. this initialization does not have to be
+ repeated so long as n remains unchanged thus subsequent
+ transforms can be obtained faster than the first.
+
+output parameters
+
+x for i=1,...,n
+
+ x(i)= the sum from k=1 to k=n of
+
+ 4*x(k)*sin((2k-1)*i*pi/(2*n))
+
+ a call of sinqb followed by a call of
+ sinqf will multiply the sequence x by 4*n.
+ therefore sinqf is the unnormalized inverse
+ of sinqb.
+
+wsave contains initialization calculations which must not
+ be destroyed between calls of sinqb or sinqf.
+
+******************************************************************
+
+subroutine cosqi(n,wsave)
+
+******************************************************************
+
+subroutine cosqi initializes the array wsave which is used in
+both cosqf and cosqb. the prime factorization of n together with
+a tabulation of the trigonometric functions are computed and
+stored in wsave.
+
+input parameter
+
+n the length of the array to be transformed. the method
+ is most efficient when n is a product of small primes.
+
+output parameter
+
+wsave a work array which must be dimensioned at least 3*n+15.
+ the same work array can be used for both cosqf and cosqb
+ as long as n remains unchanged. different wsave arrays
+ are required for different values of n. the contents of
+ wsave must not be changed between calls of cosqf or cosqb.
+
+******************************************************************
+
+subroutine cosqf(n,x,wsave)
+
+******************************************************************
+
+subroutine cosqf computes the fast fourier transform of quarter
+wave data. that is , cosqf computes the coefficients in a cosine
+series representation with only odd wave numbers. the transform
+is defined below at output parameter x
+
+cosqf is the unnormalized inverse of cosqb since a call of cosqf
+followed by a call of cosqb will multiply the input sequence x
+by 4*n.
+
+the array wsave which is used by subroutine cosqf must be
+initialized by calling subroutine cosqi(n,wsave).
+
+
+input parameters
+
+n the length of the array x to be transformed. the method
+ is most efficient when n is a product of small primes.
+
+x an array which contains the sequence to be transformed
+
+wsave a work array which must be dimensioned at least 3*n+15
+ in the program that calls cosqf. the wsave array must be
+ initialized by calling subroutine cosqi(n,wsave) and a
+ different wsave array must be used for each different
+ value of n. this initialization does not have to be
+ repeated so long as n remains unchanged thus subsequent
+ transforms can be obtained faster than the first.
+
+output parameters
+
+x for i=1,...,n
+
+ x(i) = x(1) plus the sum from k=2 to k=n of
+
+ 2*x(k)*cos((2*i-1)*(k-1)*pi/(2*n))
+
+ a call of cosqf followed by a call of
+ cosqb will multiply the sequence x by 4*n.
+ therefore cosqb is the unnormalized inverse
+ of cosqf.
+
+wsave contains initialization calculations which must not
+ be destroyed between calls of cosqf or cosqb.
+
+******************************************************************
+
+subroutine cosqb(n,x,wsave)
+
+******************************************************************
+
+subroutine cosqb computes the fast fourier transform of quarter
+wave data. that is , cosqb computes a sequence from its
+representation in terms of a cosine series with odd wave numbers.
+the transform is defined below at output parameter x.
+
+cosqb is the unnormalized inverse of cosqf since a call of cosqb
+followed by a call of cosqf will multiply the input sequence x
+by 4*n.
+
+the array wsave which is used by subroutine cosqb must be
+initialized by calling subroutine cosqi(n,wsave).
+
+
+input parameters
+
+n the length of the array x to be transformed. the method
+ is most efficient when n is a product of small primes.
+
+x an array which contains the sequence to be transformed
+
+wsave a work array that must be dimensioned at least 3*n+15
+ in the program that calls cosqb. the wsave array must be
+ initialized by calling subroutine cosqi(n,wsave) and a
+ different wsave array must be used for each different
+ value of n. this initialization does not have to be
+ repeated so long as n remains unchanged thus subsequent
+ transforms can be obtained faster than the first.
+
+output parameters
+
+x for i=1,...,n
+
+ x(i)= the sum from k=1 to k=n of
+
+ 4*x(k)*cos((2*k-1)*(i-1)*pi/(2*n))
+
+ a call of cosqb followed by a call of
+ cosqf will multiply the sequence x by 4*n.
+ therefore cosqf is the unnormalized inverse
+ of cosqb.
+
+wsave contains initialization calculations which must not
+ be destroyed between calls of cosqb or cosqf.
+
+******************************************************************
+
+subroutine cffti(n,wsave)
+
+******************************************************************
+
+subroutine cffti initializes the array wsave which is used in
+both cfftf and cfftb. the prime factorization of n together with
+a tabulation of the trigonometric functions are computed and
+stored in wsave.
+
+input parameter
+
+n the length of the sequence to be transformed
+
+output parameter
+
+wsave a work array which must be dimensioned at least 4*n+15
+ the same work array can be used for both cfftf and cfftb
+ as long as n remains unchanged. different wsave arrays
+ are required for different values of n. the contents of
+ wsave must not be changed between calls of cfftf or cfftb.
+
+******************************************************************
+
+subroutine cfftf(n,c,wsave)
+
+******************************************************************
+
+subroutine cfftf computes the forward complex discrete fourier
+transform (the fourier analysis). equivalently , cfftf computes
+the fourier coefficients of a complex periodic sequence.
+the transform is defined below at output parameter c.
+
+the transform is not normalized. to obtain a normalized transform
+the output must be divided by n. otherwise a call of cfftf
+followed by a call of cfftb will multiply the sequence by n.
+
+the array wsave which is used by subroutine cfftf must be
+initialized by calling subroutine cffti(n,wsave).
+
+input parameters
+
+
+n the length of the complex sequence c. the method is
+ more efficient when n is the product of small primes. n
+
+c a complex array of length n which contains the sequence
+
+wsave a real work array which must be dimensioned at least 4n+15
+ in the program that calls cfftf. the wsave array must be
+ initialized by calling subroutine cffti(n,wsave) and a
+ different wsave array must be used for each different
+ value of n. this initialization does not have to be
+ repeated so long as n remains unchanged thus subsequent
+ transforms can be obtained faster than the first.
+ the same wsave array can be used by cfftf and cfftb.
+
+output parameters
+
+c for j=1,...,n
+
+ c(j)=the sum from k=1,...,n of
+
+ c(k)*exp(-i*(j-1)*(k-1)*2*pi/n)
+
+ where i=sqrt(-1)
+
+wsave contains initialization calculations which must not be
+ destroyed between calls of subroutine cfftf or cfftb
+
+******************************************************************
+
+subroutine cfftb(n,c,wsave)
+
+******************************************************************
+
+subroutine cfftb computes the backward complex discrete fourier
+transform (the fourier synthesis). equivalently , cfftb computes
+a complex periodic sequence from its fourier coefficients.
+the transform is defined below at output parameter c.
+
+a call of cfftf followed by a call of cfftb will multiply the
+sequence by n.
+
+the array wsave which is used by subroutine cfftb must be
+initialized by calling subroutine cffti(n,wsave).
+
+input parameters
+
+
+n the length of the complex sequence c. the method is
+ more efficient when n is the product of small primes.
+
+c a complex array of length n which contains the sequence
+
+wsave a real work array which must be dimensioned at least 4n+15
+ in the program that calls cfftb. the wsave array must be
+ initialized by calling subroutine cffti(n,wsave) and a
+ different wsave array must be used for each different
+ value of n. this initialization does not have to be
+ repeated so long as n remains unchanged thus subsequent
+ transforms can be obtained faster than the first.
+ the same wsave array can be used by cfftf and cfftb.
+
+output parameters
+
+c for j=1,...,n
+
+ c(j)=the sum from k=1,...,n of
+
+ c(k)*exp(i*(j-1)*(k-1)*2*pi/n)
+
+ where i=sqrt(-1)
+
+wsave contains initialization calculations which must not be
+ destroyed between calls of subroutine cfftf or cfftb
+
+*/
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