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/*
* Copyright (c) 1997-1999 Massachusetts Institute of Technology
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation; either version 2 of the License, or
* (at your option) any later version.
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this program; if not, write to the Free Software
* Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
*
*/
/*
* Compute transforms of prime sizes using Rader's trick: turn them
* into convolutions of size n - 1, which you then perform via a pair
* of FFTs.
*/
#include <stdlib.h>
#include <math.h>
#include <ports/fftw-int.h>
#ifdef FFTW_USING_CILK
#include <ports/cilk.h>
#include <ports/cilk-compat.h>
#endif
#ifdef FFTW_DEBUG
#define WHEN_DEBUG(a) a
#else
#define WHEN_DEBUG(a)
#endif
/* compute n^m mod p, where m >= 0 and p > 0. */
static int power_mod(int n, int m, int p)
{
if (m == 0)
return 1;
else if (m % 2 == 0) {
int x = power_mod(n, m / 2, p);
return ((x * x) % p);
} else
return ((n * power_mod(n, m - 1, p)) % p);
}
/*
* Find the period of n in the multiplicative group mod p (p prime).
* That is, return the smallest m such that n^m == 1 mod p.
*/
static int period(int n, int p)
{
int prod = n, period = 1;
while (prod != 1) {
prod = (prod * n) % p;
++period;
if (prod == 0)
fftw_die("non-prime order in Rader\n");
}
return period;
}
/* find a generator for the multiplicative group mod p, where p is prime */
static int find_generator(int p)
{
int g;
for (g = 1; g < p; ++g)
if (period(g, p) == p - 1)
break;
if (g == p)
fftw_die("couldn't find generator for Rader\n");
return g;
}
/***************************************************************************/
static fftw_rader_data *create_rader_aux(int p, int flags)
{
fftw_complex *omega, *work;
int g, ginv, gpower;
int i;
FFTW_TRIG_REAL twoPiOverN;
fftw_real scale = 1.0 / (p - 1); /* for convolution */
fftw_plan plan;
fftw_rader_data *d;
if (p < 2)
fftw_die("non-prime order in Rader\n");
flags &= ~FFTW_IN_PLACE;
d = (fftw_rader_data *) fftw_malloc(sizeof(fftw_rader_data));
g = find_generator(p);
ginv = power_mod(g, p - 2, p);
omega = (fftw_complex *) fftw_malloc((p - 1) * sizeof(fftw_complex));
plan = fftw_create_plan(p - 1, FFTW_FORWARD, flags);
work = (fftw_complex *) fftw_malloc((p - 1) * sizeof(fftw_complex));
twoPiOverN = FFTW_K2PI / (FFTW_TRIG_REAL) p;
gpower = 1;
for (i = 0; i < p - 1; ++i) {
c_re(work[i]) = scale * FFTW_TRIG_COS(twoPiOverN * gpower);
c_im(work[i]) = FFTW_FORWARD * scale * FFTW_TRIG_SIN(twoPiOverN * gpower);
gpower = (gpower * ginv) % p;
}
/* fft permuted roots of unity */
fftw_executor_simple(p - 1, work, omega, plan->root, 1, 1);
fftw_free(work);
d->plan = plan;
d->omega = omega;
d->g = g;
d->ginv = ginv;
d->p = p;
d->flags = flags;
d->refcount = 1;
d->next = NULL;
d->cdesc = (fftw_codelet_desc *) fftw_malloc(sizeof(fftw_codelet_desc));
d->cdesc->name = NULL;
d->cdesc->codelet = NULL;
d->cdesc->size = p;
d->cdesc->dir = FFTW_FORWARD;
d->cdesc->type = FFTW_RADER;
d->cdesc->signature = g;
d->cdesc->ntwiddle = 0;
d->cdesc->twiddle_order = NULL;
return d;
}
/***************************************************************************/
static fftw_rader_data *fftw_create_rader(int p, int flags)
{
fftw_rader_data *d = fftw_rader_top;
flags &= ~FFTW_IN_PLACE;
while (d && (d->p != p || d->flags != flags))
d = d->next;
if (d) {
d->refcount++;
return d;
}
d = create_rader_aux(p, flags);
d->next = fftw_rader_top;
fftw_rader_top = d;
return d;
}
/***************************************************************************/
/* Compute the prime FFTs, premultiplied by twiddle factors. Below, we
* extensively use the identity that fft(x*)* = ifft(x) in order to
* share data between forward and backward transforms and to obviate
* the necessity of having separate forward and backward plans. */
void fftw_twiddle_rader(fftw_complex *A, const fftw_complex *W,
int m, int r, int stride,
fftw_rader_data * d)
{
fftw_complex *tmp = (fftw_complex *)
fftw_malloc((r - 1) * sizeof(fftw_complex));
int i, k, gpower = 1, g = d->g, ginv = d->ginv;
fftw_real a0r, a0i;
fftw_complex *omega = d->omega;
for (i = 0; i < m; ++i, A += stride, W += r - 1) {
/*
* Here, we fft W[k-1] * A[k*(m*stride)], using Rader.
* (Actually, W is pre-permuted to match the permutation that we
* will do on A.)
*/
/* First, permute the input and multiply by W, storing in tmp: */
/* gpower == g^k mod r in the following loop */
for (k = 0; k < r - 1; ++k, gpower = (gpower * g) % r) {
fftw_real rA, iA, rW, iW;
rW = c_re(W[k]);
iW = c_im(W[k]);
rA = c_re(A[gpower * (m * stride)]);
iA = c_im(A[gpower * (m * stride)]);
c_re(tmp[k]) = rW * rA - iW * iA;
c_im(tmp[k]) = rW * iA + iW * rA;
}
WHEN_DEBUG( {
if (gpower != 1)
fftw_die("incorrect generator in Rader\n");
}
);
/* FFT tmp to A: */
fftw_executor_simple(r - 1, tmp, A + (m * stride),
d->plan->root, 1, m * stride);
/* set output DC component: */
a0r = c_re(A[0]);
a0i = c_im(A[0]);
c_re(A[0]) += c_re(A[(m * stride)]);
c_im(A[0]) += c_im(A[(m * stride)]);
/* now, multiply by omega: */
for (k = 0; k < r - 1; ++k) {
fftw_real rA, iA, rW, iW;
rW = c_re(omega[k]);
iW = c_im(omega[k]);
rA = c_re(A[(k + 1) * (m * stride)]);
iA = c_im(A[(k + 1) * (m * stride)]);
c_re(A[(k + 1) * (m * stride)]) = rW * rA - iW * iA;
c_im(A[(k + 1) * (m * stride)]) = -(rW * iA + iW * rA);
}
/* this will add A[0] to all of the outputs after the ifft */
c_re(A[(m * stride)]) += a0r;
c_im(A[(m * stride)]) -= a0i;
/* inverse FFT: */
fftw_executor_simple(r - 1, A + (m * stride), tmp,
d->plan->root, m * stride, 1);
/* finally, do inverse permutation to unshuffle the output: */
for (k = 0; k < r - 1; ++k, gpower = (gpower * ginv) % r) {
c_re(A[gpower * (m * stride)]) = c_re(tmp[k]);
c_im(A[gpower * (m * stride)]) = -c_im(tmp[k]);
}
WHEN_DEBUG( {
if (gpower != 1)
fftw_die("incorrect generator in Rader\n");
}
);
}
fftw_free(tmp);
}
void fftwi_twiddle_rader(fftw_complex *A, const fftw_complex *W,
int m, int r, int stride,
fftw_rader_data * d)
{
fftw_complex *tmp = (fftw_complex *)
fftw_malloc((r - 1) * sizeof(fftw_complex));
int i, k, gpower = 1, g = d->g, ginv = d->ginv;
fftw_real a0r, a0i;
fftw_complex *omega = d->omega;
for (i = 0; i < m; ++i, A += stride, W += r - 1) {
/*
* Here, we fft W[k-1]* * A[k*(m*stride)], using Rader.
* (Actually, W is pre-permuted to match the permutation that
* we will do on A.)
*/
/* First, permute the input and multiply by W*, storing in tmp: */
/* gpower == g^k mod r in the following loop */
for (k = 0; k < r - 1; ++k, gpower = (gpower * g) % r) {
fftw_real rA, iA, rW, iW;
rW = c_re(W[k]);
iW = c_im(W[k]);
rA = c_re(A[gpower * (m * stride)]);
iA = c_im(A[gpower * (m * stride)]);
c_re(tmp[k]) = rW * rA + iW * iA;
c_im(tmp[k]) = iW * rA - rW * iA;
}
WHEN_DEBUG( {
if (gpower != 1)
fftw_die("incorrect generator in Rader\n");
}
);
/* FFT tmp to A: */
fftw_executor_simple(r - 1, tmp, A + (m * stride),
d->plan->root, 1, m * stride);
/* set output DC component: */
a0r = c_re(A[0]);
a0i = c_im(A[0]);
c_re(A[0]) += c_re(A[(m * stride)]);
c_im(A[0]) -= c_im(A[(m * stride)]);
/* now, multiply by omega: */
for (k = 0; k < r - 1; ++k) {
fftw_real rA, iA, rW, iW;
rW = c_re(omega[k]);
iW = c_im(omega[k]);
rA = c_re(A[(k + 1) * (m * stride)]);
iA = c_im(A[(k + 1) * (m * stride)]);
c_re(A[(k + 1) * (m * stride)]) = rW * rA - iW * iA;
c_im(A[(k + 1) * (m * stride)]) = -(rW * iA + iW * rA);
}
/* this will add A[0] to all of the outputs after the ifft */
c_re(A[(m * stride)]) += a0r;
c_im(A[(m * stride)]) += a0i;
/* inverse FFT: */
fftw_executor_simple(r - 1, A + (m * stride), tmp,
d->plan->root, m * stride, 1);
/* finally, do inverse permutation to unshuffle the output: */
for (k = 0; k < r - 1; ++k, gpower = (gpower * ginv) % r) {
A[gpower * (m * stride)] = tmp[k];
}
WHEN_DEBUG( {
if (gpower != 1)
fftw_die("incorrect generator in Rader\n");
}
);
}
fftw_free(tmp);
}
/***************************************************************************/
/*
* Make an FFTW_RADER plan node. Note that this function must go
* here, rather than in putils.c, because it indirectly calls the
* fftw_planner. If we included it in putils.c, which is also used
* by rfftw, then any program using rfftw would be linked with all
* of the FFTW codelets, even if they were not needed. I wish that the
* darn linkers operated on a function rather than a file granularity.
*/
fftw_plan_node *fftw_make_node_rader(int n, int size, fftw_direction dir,
fftw_plan_node *recurse,
int flags)
{
fftw_plan_node *p = fftw_make_node();
p->type = FFTW_RADER;
p->nodeu.rader.size = size;
p->nodeu.rader.codelet = dir == FFTW_FORWARD ?
fftw_twiddle_rader : fftwi_twiddle_rader;
p->nodeu.rader.rader_data = fftw_create_rader(size, flags);
p->nodeu.rader.recurse = recurse;
fftw_use_node(recurse);
if (flags & FFTW_MEASURE)
p->nodeu.rader.tw =
fftw_create_twiddle(n, p->nodeu.rader.rader_data->cdesc);
else
p->nodeu.rader.tw = 0;
return p;
}