fftsg.c

/*
Fast Fourier/Cosine/Sine Transform
    dimension   :one
    data length :power of 2
    decimation  :frequency
    radix       :split-radix
    data        :inplace
    table       :use
functions
    cdft: Complex Discrete Fourier Transform
    rdft: Real Discrete Fourier Transform
    ddct: Discrete Cosine Transform
    ddst: Discrete Sine Transform
    dfct: Cosine Transform of RDFT (Real Symmetric DFT)
    dfst: Sine Transform of RDFT (Real Anti-symmetric DFT)
function prototypes
    void cdft(int, int, double *, int *, double *);
    void rdft(int, int, double *, int *, double *);
    void ddct(int, int, double *, int *, double *);
    void ddst(int, int, double *, int *, double *);
    void dfct(int, double *, double *, int *, double *);
    void dfst(int, double *, double *, int *, double *);
macro definitions
    USE_CDFT_PTHREADS : default=not defined
        CDFT_THREADS_BEGIN_N  : must be >= 512, default=8192
        CDFT_4THREADS_BEGIN_N : must be >= 512, default=65536
    USE_CDFT_WINTHREADS : default=not defined
        CDFT_THREADS_BEGIN_N  : must be >= 512, default=32768
        CDFT_4THREADS_BEGIN_N : must be >= 512, default=524288


-------- Complex DFT (Discrete Fourier Transform) --------
    [definition]
        <case1>
            X[k] = sum_j=0^n-1 x[j]*exp(2*pi*i*j*k/n), 0<=k<n
        <case2>
            X[k] = sum_j=0^n-1 x[j]*exp(-2*pi*i*j*k/n), 0<=k<n
        (notes: sum_j=0^n-1 is a summation from j=0 to n-1)
    [usage]
        <case1>
            ip[0] = 0; // first time only
            cdft(2*n, 1, a, ip, w);
        <case2>
            ip[0] = 0; // first time only
            cdft(2*n, -1, a, ip, w);
    [parameters]
        2*n            :data length (int)
                        n >= 1, n = power of 2
        a[0...2*n-1]   :input/output data (double *)
                        input data
                            a[2*j] = Re(x[j]), 
                            a[2*j+1] = Im(x[j]), 0<=j<n
                        output data
                            a[2*k] = Re(X[k]), 
                            a[2*k+1] = Im(X[k]), 0<=k<n
        ip[0...*]      :work area for bit reversal (int *)
                        length of ip >= 2+sqrt(n)
                        strictly, 
                        length of ip >= 
                            2+(1<<(int)(log(n+0.5)/log(2))/2).
                        ip[0],ip[1] are pointers of the cos/sin table.
        w[0...n/2-1]   :cos/sin table (double *)
                        w[],ip[] are initialized if ip[0] == 0.
    [remark]
        Inverse of 
            cdft(2*n, -1, a, ip, w);
        is 
            cdft(2*n, 1, a, ip, w);
            for (j = 0; j <= 2 * n - 1; j++) {
                a[j] *= 1.0 / n;
            }
        .


-------- Real DFT / Inverse of Real DFT --------
    [definition]
        <case1> RDFT
            R[k] = sum_j=0^n-1 a[j]*cos(2*pi*j*k/n), 0<=k<=n/2
            I[k] = sum_j=0^n-1 a[j]*sin(2*pi*j*k/n), 0<k<n/2
        <case2> IRDFT (excluding scale)
            a[k] = (R[0] + R[n/2]*cos(pi*k))/2 + 
                   sum_j=1^n/2-1 R[j]*cos(2*pi*j*k/n) + 
                   sum_j=1^n/2-1 I[j]*sin(2*pi*j*k/n), 0<=k<n
    [usage]
        <case1>
            ip[0] = 0; // first time only
            rdft(n, 1, a, ip, w);
        <case2>
            ip[0] = 0; // first time only
            rdft(n, -1, a, ip, w);
    [parameters]
        n              :data length (int)
                        n >= 2, n = power of 2
        a[0...n-1]     :input/output data (double *)
                        <case1>
                            output data
                                a[2*k] = R[k], 0<=k<n/2
                                a[2*k+1] = I[k], 0<k<n/2
                                a[1] = R[n/2]
                        <case2>
                            input data
                                a[2*j] = R[j], 0<=j<n/2
                                a[2*j+1] = I[j], 0<j<n/2
                                a[1] = R[n/2]
        ip[0...*]      :work area for bit reversal (int *)
                        length of ip >= 2+sqrt(n/2)
                        strictly, 
                        length of ip >= 
                            2+(1<<(int)(log(n/2+0.5)/log(2))/2).
                        ip[0],ip[1] are pointers of the cos/sin table.
        w[0...n/2-1]   :cos/sin table (double *)
                        w[],ip[] are initialized if ip[0] == 0.
    [remark]
        Inverse of 
            rdft(n, 1, a, ip, w);
        is 
            rdft(n, -1, a, ip, w);
            for (j = 0; j <= n - 1; j++) {
                a[j] *= 2.0 / n;
            }
        .


-------- DCT (Discrete Cosine Transform) / Inverse of DCT --------
    [definition]
        <case1> IDCT (excluding scale)
            C[k] = sum_j=0^n-1 a[j]*cos(pi*j*(k+1/2)/n), 0<=k<n
        <case2> DCT
            C[k] = sum_j=0^n-1 a[j]*cos(pi*(j+1/2)*k/n), 0<=k<n
    [usage]
        <case1>
            ip[0] = 0; // first time only
            ddct(n, 1, a, ip, w);
        <case2>
            ip[0] = 0; // first time only
            ddct(n, -1, a, ip, w);
    [parameters]
        n              :data length (int)
                        n >= 2, n = power of 2
        a[0...n-1]     :input/output data (double *)
                        output data
                            a[k] = C[k], 0<=k<n
        ip[0...*]      :work area for bit reversal (int *)
                        length of ip >= 2+sqrt(n/2)
                        strictly, 
                        length of ip >= 
                            2+(1<<(int)(log(n/2+0.5)/log(2))/2).
                        ip[0],ip[1] are pointers of the cos/sin table.
        w[0...n*5/4-1] :cos/sin table (double *)
                        w[],ip[] are initialized if ip[0] == 0.
    [remark]
        Inverse of 
            ddct(n, -1, a, ip, w);
        is 
            a[0] *= 0.5;
            ddct(n, 1, a, ip, w);
            for (j = 0; j <= n - 1; j++) {
                a[j] *= 2.0 / n;
            }
        .


-------- DST (Discrete Sine Transform) / Inverse of DST --------
    [definition]
        <case1> IDST (excluding scale)
            S[k] = sum_j=1^n A[j]*sin(pi*j*(k+1/2)/n), 0<=k<n
        <case2> DST
            S[k] = sum_j=0^n-1 a[j]*sin(pi*(j+1/2)*k/n), 0<k<=n
    [usage]
        <case1>
            ip[0] = 0; // first time only
            ddst(n, 1, a, ip, w);
        <case2>
            ip[0] = 0; // first time only
            ddst(n, -1, a, ip, w);
    [parameters]
        n              :data length (int)
                        n >= 2, n = power of 2
        a[0...n-1]     :input/output data (double *)
                        <case1>
                            input data
                                a[j] = A[j], 0<j<n
                                a[0] = A[n]
                            output data
                                a[k] = S[k], 0<=k<n
                        <case2>
                            output data
                                a[k] = S[k], 0<k<n
                                a[0] = S[n]
        ip[0...*]      :work area for bit reversal (int *)
                        length of ip >= 2+sqrt(n/2)
                        strictly, 
                        length of ip >= 
                            2+(1<<(int)(log(n/2+0.5)/log(2))/2).
                        ip[0],ip[1] are pointers of the cos/sin table.
        w[0...n*5/4-1] :cos/sin table (double *)
                        w[],ip[] are initialized if ip[0] == 0.
    [remark]
        Inverse of 
            ddst(n, -1, a, ip, w);
        is 
            a[0] *= 0.5;
            ddst(n, 1, a, ip, w);
            for (j = 0; j <= n - 1; j++) {
                a[j] *= 2.0 / n;
            }
        .


-------- Cosine Transform of RDFT (Real Symmetric DFT) --------
    [definition]
        C[k] = sum_j=0^n a[j]*cos(pi*j*k/n), 0<=k<=n
    [usage]
        ip[0] = 0; // first time only
        dfct(n, a, t, ip, w);
    [parameters]
        n              :data length - 1 (int)
                        n >= 2, n = power of 2
        a[0...n]       :input/output data (double *)
                        output data
                            a[k] = C[k], 0<=k<=n
        t[0...n/2]     :work area (double *)
        ip[0...*]      :work area for bit reversal (int *)
                        length of ip >= 2+sqrt(n/4)
                        strictly, 
                        length of ip >= 
                            2+(1<<(int)(log(n/4+0.5)/log(2))/2).
                        ip[0],ip[1] are pointers of the cos/sin table.
        w[0...n*5/8-1] :cos/sin table (double *)
                        w[],ip[] are initialized if ip[0] == 0.
    [remark]
        Inverse of 
            a[0] *= 0.5;
            a[n] *= 0.5;
            dfct(n, a, t, ip, w);
        is 
            a[0] *= 0.5;
            a[n] *= 0.5;
            dfct(n, a, t, ip, w);
            for (j = 0; j <= n; j++) {
                a[j] *= 2.0 / n;
            }
        .


-------- Sine Transform of RDFT (Real Anti-symmetric DFT) --------
    [definition]
        S[k] = sum_j=1^n-1 a[j]*sin(pi*j*k/n), 0<k<n
    [usage]
        ip[0] = 0; // first time only
        dfst(n, a, t, ip, w);
    [parameters]
        n              :data length + 1 (int)
                        n >= 2, n = power of 2
        a[0...n-1]     :input/output data (double *)
                        output data
                            a[k] = S[k], 0<k<n
                        (a[0] is used for work area)
        t[0...n/2-1]   :work area (double *)
        ip[0...*]      :work area for bit reversal (int *)
                        length of ip >= 2+sqrt(n/4)
                        strictly, 
                        length of ip >= 
                            2+(1<<(int)(log(n/4+0.5)/log(2))/2).
                        ip[0],ip[1] are pointers of the cos/sin table.
        w[0...n*5/8-1] :cos/sin table (double *)
                        w[],ip[] are initialized if ip[0] == 0.
    [remark]
        Inverse of 
            dfst(n, a, t, ip, w);
        is 
            dfst(n, a, t, ip, w);
            for (j = 1; j <= n - 1; j++) {
                a[j] *= 2.0 / n;
            }
        .


Appendix :
    The cos/sin table is recalculated when the larger table required.
    w[] and ip[] are compatible with all routines.
*/


void cdft(int n, int isgn, double *a, int *ip, double *w)
{
	void makewt(int nw, int *ip, double *w);
	void cftfsub(int n, double *a, int *ip, int nw, double *w);
	void cftbsub(int n, double *a, int *ip, int nw, double *w);
	int nw;

	nw = ip[0];
	if (n > (nw << 2)) {
		nw = n >> 2;
		makewt(nw, ip, w);
	}
	if (isgn >= 0) {
		cftfsub(n, a, ip, nw, w);
	} else {
		cftbsub(n, a, ip, nw, w);
	}
}


void rdft(int n, int isgn, double *a, int *ip, double *w)
{
	void makewt(int nw, int *ip, double *w);
	void makect(int nc, int *ip, double *c);
	void cftfsub(int n, double *a, int *ip, int nw, double *w);
	void cftbsub(int n, double *a, int *ip, int nw, double *w);
	void rftfsub(int n, double *a, int nc, double *c);
	void rftbsub(int n, double *a, int nc, double *c);
	int nw, nc;
	double xi;

	nw = ip[0];
	if (n > (nw << 2)) {
		nw = n >> 2;
		makewt(nw, ip, w);
	}
	nc = ip[1];
	if (n > (nc << 2)) {
		nc = n >> 2;
		makect(nc, ip, w + nw);
	}
	if (isgn >= 0) {
		if (n > 4) {
			cftfsub(n, a, ip, nw, w);
			rftfsub(n, a, nc, w + nw);
		} else if (n == 4) {
			cftfsub(n, a, ip, nw, w);
		}
		xi = a[0] - a[1];
		a[0] += a[1];
		a[1] = xi;
	} else {
		a[1] = 0.5 * (a[0] - a[1]);
		a[0] -= a[1];
		if (n > 4) {
			rftbsub(n, a, nc, w + nw);
			cftbsub(n, a, ip, nw, w);
		} else if (n == 4) {
			cftbsub(n, a, ip, nw, w);
		}
	}
}


void ddct(int n, int isgn, double *a, int *ip, double *w)
{
	void makewt(int nw, int *ip, double *w);
	void makect(int nc, int *ip, double *c);
	void cftfsub(int n, double *a, int *ip, int nw, double *w);
	void cftbsub(int n, double *a, int *ip, int nw, double *w);
	void rftfsub(int n, double *a, int nc, double *c);
	void rftbsub(int n, double *a, int nc, double *c);
	void dctsub(int n, double *a, int nc, double *c);
	int j, nw, nc;
	double xr;

	nw = ip[0];
	if (n > (nw << 2)) {
		nw = n >> 2;
		makewt(nw, ip, w);
	}
	nc = ip[1];
	if (n > nc) {
		nc = n;
		makect(nc, ip, w + nw);
	}
	if (isgn < 0) {
		xr = a[n - 1];
		for (j = n - 2; j >= 2; j -= 2) {
			a[j + 1] = a[j] - a[j - 1];
			a[j] += a[j - 1];
		}
		a[1] = a[0] - xr;
		a[0] += xr;
		if (n > 4) {
			rftbsub(n, a, nc, w + nw);
			cftbsub(n, a, ip, nw, w);
		} else if (n == 4) {
			cftbsub(n, a, ip, nw, w);
		}
	}
	dctsub(n, a, nc, w + nw);
	if (isgn >= 0) {
		if (n > 4) {
			cftfsub(n, a, ip, nw, w);
			rftfsub(n, a, nc, w + nw);
		} else if (n == 4) {
			cftfsub(n, a, ip, nw, w);
		}
		xr = a[0] - a[1];
		a[0] += a[1];
		for (j = 2; j < n; j += 2) {
			a[j - 1] = a[j] - a[j + 1];
			a[j] += a[j + 1];
		}
		a[n - 1] = xr;
	}
}


void ddst(int n, int isgn, double *a, int *ip, double *w)
{
	void makewt(int nw, int *ip, double *w);
	void makect(int nc, int *ip, double *c);
	void cftfsub(int n, double *a, int *ip, int nw, double *w);
	void cftbsub(int n, double *a, int *ip, int nw, double *w);
	void rftfsub(int n, double *a, int nc, double *c);
	void rftbsub(int n, double *a, int nc, double *c);
	void dstsub(int n, double *a, int nc, double *c);
	int j, nw, nc;
	double xr;

	nw = ip[0];
	if (n > (nw << 2)) {
		nw = n >> 2;
		makewt(nw, ip, w);
	}
	nc = ip[1];
	if (n > nc) {
		nc = n;
		makect(nc, ip, w + nw);
	}
	if (isgn < 0) {
		xr = a[n - 1];
		for (j = n - 2; j >= 2; j -= 2) {
			a[j + 1] = -a[j] - a[j - 1];
			a[j] -= a[j - 1];
		}
		a[1] = a[0] + xr;
		a[0] -= xr;
		if (n > 4) {
			rftbsub(n, a, nc, w + nw);
			cftbsub(n, a, ip, nw, w);
		} else if (n == 4) {
			cftbsub(n, a, ip, nw, w);
		}
	}
	dstsub(n, a, nc, w + nw);
	if (isgn >= 0) {
		if (n > 4) {
			cftfsub(n, a, ip, nw, w);
			rftfsub(n, a, nc, w + nw);
		} else if (n == 4) {
			cftfsub(n, a, ip, nw, w);
		}
		xr = a[0] - a[1];
		a[0] += a[1];
		for (j = 2; j < n; j += 2) {
			a[j - 1] = -a[j] - a[j + 1];
			a[j] -= a[j + 1];
		}
		a[n - 1] = -xr;
	}
}


void dfct(int n, double *a, double *t, int *ip, double *w)
{
	void makewt(int nw, int *ip, double *w);
	void makect(int nc, int *ip, double *c);
	void cftfsub(int n, double *a, int *ip, int nw, double *w);
	void rftfsub(int n, double *a, int nc, double *c);
	void dctsub(int n, double *a, int nc, double *c);
	int j, k, l, m, mh, nw, nc;
	double xr, xi, yr, yi;

	nw = ip[0];
	if (n > (nw << 3)) {
		nw = n >> 3;
		makewt(nw, ip, w);
	}
	nc = ip[1];
	if (n > (nc << 1)) {
		nc = n >> 1;
		makect(nc, ip, w + nw);
	}
	m = n >> 1;
	yi = a[m];
	xi = a[0] + a[n];
	a[0] -= a[n];
	t[0] = xi - yi;
	t[m] = xi + yi;
	if (n > 2) {
		mh = m >> 1;
		for (j = 1; j < mh; j++) {
			k = m - j;
			xr = a[j] - a[n - j];
			xi = a[j] + a[n - j];
			yr = a[k] - a[n - k];
			yi = a[k] + a[n - k];
			a[j] = xr;
			a[k] = yr;
			t[j] = xi - yi;
			t[k] = xi + yi;
		}
		t[mh] = a[mh] + a[n - mh];
		a[mh] -= a[n - mh];
		dctsub(m, a, nc, w + nw);
		if (m > 4) {
			cftfsub(m, a, ip, nw, w);
			rftfsub(m, a, nc, w + nw);
		} else if (m == 4) {
			cftfsub(m, a, ip, nw, w);
		}
		a[n - 1] = a[0] - a[1];
		a[1] = a[0] + a[1];
		for (j = m - 2; j >= 2; j -= 2) {
			a[2 * j + 1] = a[j] + a[j + 1];
			a[2 * j - 1] = a[j] - a[j + 1];
		}
		l = 2;
		m = mh;
		while (m >= 2) {
			dctsub(m, t, nc, w + nw);
			if (m > 4) {
				cftfsub(m, t, ip, nw, w);
				rftfsub(m, t, nc, w + nw);
			} else if (m == 4) {
				cftfsub(m, t, ip, nw, w);
			}
			a[n - l] = t[0] - t[1];
			a[l] = t[0] + t[1];
			k = 0;
			for (j = 2; j < m; j += 2) {
				k += l << 2;
				a[k - l] = t[j] - t[j + 1];
				a[k + l] = t[j] + t[j + 1];
			}
			l <<= 1;
			mh = m >> 1;
			for (j = 0; j < mh; j++) {
				k = m - j;
				t[j] = t[m + k] - t[m + j];
				t[k] = t[m + k] + t[m + j];
			}
			t[mh] = t[m + mh];
			m = mh;
		}
		a[l] = t[0];
		a[n] = t[2] - t[1];
		a[0] = t[2] + t[1];
	} else {
		a[1] = a[0];
		a[2] = t[0];
		a[0] = t[1];
	}
}


void dfst(int n, double *a, double *t, int *ip, double *w)
{
	void makewt(int nw, int *ip, double *w);
	void makect(int nc, int *ip, double *c);
	void cftfsub(int n, double *a, int *ip, int nw, double *w);
	void rftfsub(int n, double *a, int nc, double *c);
	void dstsub(int n, double *a, int nc, double *c);
	int j, k, l, m, mh, nw, nc;
	double xr, xi, yr, yi;

	nw = ip[0];
	if (n > (nw << 3)) {
		nw = n >> 3;
		makewt(nw, ip, w);
	}
	nc = ip[1];
	if (n > (nc << 1)) {
		nc = n >> 1;
		makect(nc, ip, w + nw);
	}
	if (n > 2) {
		m = n >> 1;
		mh = m >> 1;
		for (j = 1; j < mh; j++) {
			k = m - j;
			xr = a[j] + a[n - j];
			xi = a[j] - a[n - j];
			yr = a[k] + a[n - k];
			yi = a[k] - a[n - k];
			a[j] = xr;
			a[k] = yr;
			t[j] = xi + yi;
			t[k] = xi - yi;
		}
		t[0] = a[mh] - a[n - mh];
		a[mh] += a[n - mh];
		a[0] = a[m];
		dstsub(m, a, nc, w + nw);
		if (m > 4) {
			cftfsub(m, a, ip, nw, w);
			rftfsub(m, a, nc, w + nw);
		} else if (m == 4) {
			cftfsub(m, a, ip, nw, w);
		}
		a[n - 1] = a[1] - a[0];
		a[1] = a[0] + a[1];
		for (j = m - 2; j >= 2; j -= 2) {
			a[2 * j + 1] = a[j] - a[j + 1];
			a[2 * j - 1] = -a[j] - a[j + 1];
		}
		l = 2;
		m = mh;
		while (m >= 2) {
			dstsub(m, t, nc, w + nw);
			if (m > 4) {
				cftfsub(m, t, ip, nw, w);
				rftfsub(m, t, nc, w + nw);
			} else if (m == 4) {
				cftfsub(m, t, ip, nw, w);
			}
			a[n - l] = t[1] - t[0];
			a[l] = t[0] + t[1];
			k = 0;
			for (j = 2; j < m; j += 2) {
				k += l << 2;
				a[k - l] = -t[j] - t[j + 1];
				a[k + l] = t[j] - t[j + 1];
			}
			l <<= 1;
			mh = m >> 1;
			for (j = 1; j < mh; j++) {
				k = m - j;
				t[j] = t[m + k] + t[m + j];
				t[k] = t[m + k] - t[m + j];
			}
			t[0] = t[m + mh];
			m = mh;
		}
		a[l] = t[0];
	}
	a[0] = 0;
}


/* -------- initializing routines -------- */


#include <math.h>

void makewt(int nw, int *ip, double *w)
{
	void makeipt(int nw, int *ip);
	int j, nwh, nw0, nw1;
	double delta, wn4r, wk1r, wk1i, wk3r, wk3i;

	ip[0] = nw;
	ip[1] = 1;
	if (nw > 2) {
		nwh = nw >> 1;
		delta = atan(1.0) / nwh;
		wn4r = cos(delta * nwh);
		w[0] = 1;
		w[1] = wn4r;
		if (nwh == 4) {
			w[2] = cos(delta * 2);
			w[3] = sin(delta * 2);
		} else if (nwh > 4) {
			makeipt(nw, ip);
			w[2] = 0.5 / cos(delta * 2);
			w[3] = 0.5 / cos(delta * 6);
			for (j = 4; j < nwh; j += 4) {
				w[j] = cos(delta * j);
				w[j + 1] = sin(delta * j);
				w[j + 2] = cos(3 * delta * j);
				w[j + 3] = -sin(3 * delta * j);
			}
		}
		nw0 = 0;
		while (nwh > 2) {
			nw1 = nw0 + nwh;
			nwh >>= 1;
			w[nw1] = 1;
			w[nw1 + 1] = wn4r;
			if (nwh == 4) {
				wk1r = w[nw0 + 4];
				wk1i = w[nw0 + 5];
				w[nw1 + 2] = wk1r;
				w[nw1 + 3] = wk1i;
			} else if (nwh > 4) {
				wk1r = w[nw0 + 4];
				wk3r = w[nw0 + 6];
				w[nw1 + 2] = 0.5 / wk1r;
				w[nw1 + 3] = 0.5 / wk3r;
				for (j = 4; j < nwh; j += 4) {
					wk1r = w[nw0 + 2 * j];
					wk1i = w[nw0 + 2 * j + 1];
					wk3r = w[nw0 + 2 * j + 2];
					wk3i = w[nw0 + 2 * j + 3];
					w[nw1 + j] = wk1r;
					w[nw1 + j + 1] = wk1i;
					w[nw1 + j + 2] = wk3r;
					w[nw1 + j + 3] = wk3i;
				}
			}
			nw0 = nw1;
		}
	}
}


void makeipt(int nw, int *ip)
{
	int j, l, m, m2, p, q;

	ip[2] = 0;
	ip[3] = 16;
	m = 2;
	for (l = nw; l > 32; l >>= 2) {
		m2 = m << 1;
		q = m2 << 3;
		for (j = m; j < m2; j++) {
			p = ip[j] << 2;
			ip[m + j] = p;
			ip[m2 + j] = p + q;
		}
		m = m2;
	}
}


void makect(int nc, int *ip, double *c)
{
	int j, nch;
	double delta;

	ip[1] = nc;
	if (nc > 1) {
		nch = nc >> 1;
		delta = atan(1.0) / nch;
		c[0] = cos(delta * nch);
		c[nch] = 0.5 * c[0];
		for (j = 1; j < nch; j++) {
			c[j] = 0.5 * cos(delta * j);
			c[nc - j] = 0.5 * sin(delta * j);
		}
	}
}


/* -------- child routines -------- */


#ifdef USE_CDFT_PTHREADS
#define USE_CDFT_THREADS
#ifndef CDFT_THREADS_BEGIN_N
#define CDFT_THREADS_BEGIN_N 8192
#endif
#ifndef CDFT_4THREADS_BEGIN_N
#define CDFT_4THREADS_BEGIN_N 65536
#endif
#include <pthread.h>
#include <stdio.h>
#include <stdlib.h>
#define cdft_thread_t pthread_t
#define cdft_thread_create(thp,func,argp) { \
    if (pthread_create(thp, NULL, func, (void *) argp) != 0) { \
        fprintf(stderr, "cdft thread error\n"); \
        exit(1); \
    } \
}
#define cdft_thread_wait(th) { \
    if (pthread_join(th, NULL) != 0) { \
        fprintf(stderr, "cdft thread error\n"); \
        exit(1); \
    } \
}
#endif /* USE_CDFT_PTHREADS */


#ifdef USE_CDFT_WINTHREADS
#define USE_CDFT_THREADS
#ifndef CDFT_THREADS_BEGIN_N
#define CDFT_THREADS_BEGIN_N 32768
#endif
#ifndef CDFT_4THREADS_BEGIN_N
#define CDFT_4THREADS_BEGIN_N 524288
#endif
#include <windows.h>
#include <stdio.h>
#include <stdlib.h>
#define cdft_thread_t HANDLE
#define cdft_thread_create(thp,func,argp) { \
    DWORD thid; \
    *(thp) = CreateThread(NULL, 0, (LPTHREAD_START_ROUTINE) func, (LPVOID) argp, 0, &thid); \
    if (*(thp) == 0) { \
        fprintf(stderr, "cdft thread error\n"); \
        exit(1); \
    } \
}
#define cdft_thread_wait(th) { \
    WaitForSingleObject(th, INFINITE); \
    CloseHandle(th); \
}
#endif /* USE_CDFT_WINTHREADS */


void cftfsub(int n, double *a, int *ip, int nw, double *w)
{
	void bitrv2(int n, int *ip, double *a);
	void bitrv216(double *a);
	void bitrv208(double *a);
	void cftf1st(int n, double *a, double *w);
	void cftrec4(int n, double *a, int nw, double *w);
	void cftleaf(int n, int isplt, double *a, int nw, double *w);
	void cftfx41(int n, double *a, int nw, double *w);
	void cftf161(double *a, double *w);
	void cftf081(double *a, double *w);
	void cftf040(double *a);
	void cftx020(double *a);
#ifdef USE_CDFT_THREADS
	void cftrec4_th(int n, double *a, int nw, double *w);
#endif /* USE_CDFT_THREADS */

	if (n > 8) {
		if (n > 32) {
			cftf1st(n, a, &w[nw - (n >> 2)]);
#ifdef USE_CDFT_THREADS
			if (n > CDFT_THREADS_BEGIN_N) {
				cftrec4_th(n, a, nw, w);
			} else
#endif /* USE_CDFT_THREADS */
			if (n > 512) {
				cftrec4(n, a, nw, w);
			} else if (n > 128) {
				cftleaf(n, 1, a, nw, w);
			} else {
				cftfx41(n, a, nw, w);
			}
			bitrv2(n, ip, a);
		} else if (n == 32) {
			cftf161(a, &w[nw - 8]);
			bitrv216(a);
		} else {
			cftf081(a, w);
			bitrv208(a);
		}
	} else if (n == 8) {
		cftf040(a);
	} else if (n == 4) {
		cftx020(a);
	}
}


void cftbsub(int n, double *a, int *ip, int nw, double *w)
{
	void bitrv2conj(int n, int *ip, double *a);
	void bitrv216neg(double *a);
	void bitrv208neg(double *a);
	void cftb1st(int n, double *a, double *w);
	void cftrec4(int n, double *a, int nw, double *w);
	void cftleaf(int n, int isplt, double *a, int nw, double *w);
	void cftfx41(int n, double *a, int nw, double *w);
	void cftf161(double *a, double *w);
	void cftf081(double *a, double *w);
	void cftb040(double *a);
	void cftx020(double *a);
#ifdef USE_CDFT_THREADS
	void cftrec4_th(int n, double *a, int nw, double *w);
#endif /* USE_CDFT_THREADS */

	if (n > 8) {
		if (n > 32) {
			cftb1st(n, a, &w[nw - (n >> 2)]);
#ifdef USE_CDFT_THREADS
			if (n > CDFT_THREADS_BEGIN_N) {
				cftrec4_th(n, a, nw, w);
			} else
#endif /* USE_CDFT_THREADS */
			if (n > 512) {
				cftrec4(n, a, nw, w);
			} else if (n > 128) {
				cftleaf(n, 1, a, nw, w);
			} else {
				cftfx41(n, a, nw, w);
			}
			bitrv2conj(n, ip, a);
		} else if (n == 32) {
			cftf161(a, &w[nw - 8]);
			bitrv216neg(a);
		} else {
			cftf081(a, w);
			bitrv208neg(a);
		}
	} else if (n == 8) {
		cftb040(a);
	} else if (n == 4) {
		cftx020(a);
	}
}


void bitrv2(int n, int *ip, double *a)
{
	int j, j1, k, k1, l, m, nh, nm;
	double xr, xi, yr, yi;

	m = 1;
	for (l = n >> 2; l > 8; l >>= 2) {
		m <<= 1;
	}
	nh = n >> 1;
	nm = 4 * m;
	if (l == 8) {
		for (k = 0; k < m; k++) {
			for (j = 0; j < k; j++) {
				j1 = 4 * j + 2 * ip[m + k];
				k1 = 4 * k + 2 * ip[m + j];
				xr = a[j1];
				xi = a[j1 + 1];
				yr = a[k1];
				yi = a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
				j1 += nm;
				k1 += 2 * nm;
				xr = a[j1];
				xi = a[j1 + 1];
				yr = a[k1];
				yi = a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
				j1 += nm;
				k1 -= nm;
				xr = a[j1];
				xi = a[j1 + 1];
				yr = a[k1];
				yi = a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
				j1 += nm;
				k1 += 2 * nm;
				xr = a[j1];
				xi = a[j1 + 1];
				yr = a[k1];
				yi = a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
				j1 += nh;
				k1 += 2;
				xr = a[j1];
				xi = a[j1 + 1];
				yr = a[k1];
				yi = a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
				j1 -= nm;
				k1 -= 2 * nm;
				xr = a[j1];
				xi = a[j1 + 1];
				yr = a[k1];
				yi = a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
				j1 -= nm;
				k1 += nm;
				xr = a[j1];
				xi = a[j1 + 1];
				yr = a[k1];
				yi = a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
				j1 -= nm;
				k1 -= 2 * nm;
				xr = a[j1];
				xi = a[j1 + 1];
				yr = a[k1];
				yi = a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
				j1 += 2;
				k1 += nh;
				xr = a[j1];
				xi = a[j1 + 1];
				yr = a[k1];
				yi = a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
				j1 += nm;
				k1 += 2 * nm;
				xr = a[j1];
				xi = a[j1 + 1];
				yr = a[k1];
				yi = a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
				j1 += nm;
				k1 -= nm;
				xr = a[j1];
				xi = a[j1 + 1];
				yr = a[k1];
				yi = a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
				j1 += nm;
				k1 += 2 * nm;
				xr = a[j1];
				xi = a[j1 + 1];
				yr = a[k1];
				yi = a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
				j1 -= nh;
				k1 -= 2;
				xr = a[j1];
				xi = a[j1 + 1];
				yr = a[k1];
				yi = a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
				j1 -= nm;
				k1 -= 2 * nm;
				xr = a[j1];
				xi = a[j1 + 1];
				yr = a[k1];
				yi = a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
				j1 -= nm;
				k1 += nm;
				xr = a[j1];
				xi = a[j1 + 1];
				yr = a[k1];
				yi = a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
				j1 -= nm;
				k1 -= 2 * nm;
				xr = a[j1];
				xi = a[j1 + 1];
				yr = a[k1];
				yi = a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
			}
			k1 = 4 * k + 2 * ip[m + k];
			j1 = k1 + 2;
			k1 += nh;
			xr = a[j1];
			xi = a[j1 + 1];
			yr = a[k1];
			yi = a[k1 + 1];
			a[j1] = yr;
			a[j1 + 1] = yi;
			a[k1] = xr;
			a[k1 + 1] = xi;
			j1 += nm;
			k1 += 2 * nm;
			xr = a[j1];
			xi = a[j1 + 1];
			yr = a[k1];
			yi = a[k1 + 1];
			a[j1] = yr;
			a[j1 + 1] = yi;
			a[k1] = xr;
			a[k1 + 1] = xi;
			j1 += nm;
			k1 -= nm;
			xr = a[j1];
			xi = a[j1 + 1];
			yr = a[k1];
			yi = a[k1 + 1];
			a[j1] = yr;
			a[j1 + 1] = yi;
			a[k1] = xr;
			a[k1 + 1] = xi;
			j1 -= 2;
			k1 -= nh;
			xr = a[j1];
			xi = a[j1 + 1];
			yr = a[k1];
			yi = a[k1 + 1];
			a[j1] = yr;
			a[j1 + 1] = yi;
			a[k1] = xr;
			a[k1 + 1] = xi;
			j1 += nh + 2;
			k1 += nh + 2;
			xr = a[j1];
			xi = a[j1 + 1];
			yr = a[k1];
			yi = a[k1 + 1];
			a[j1] = yr;
			a[j1 + 1] = yi;
			a[k1] = xr;
			a[k1 + 1] = xi;
			j1 -= nh - nm;
			k1 += 2 * nm - 2;
			xr = a[j1];
			xi = a[j1 + 1];
			yr = a[k1];
			yi = a[k1 + 1];
			a[j1] = yr;
			a[j1 + 1] = yi;
			a[k1] = xr;
			a[k1 + 1] = xi;
		}
	} else {
		for (k = 0; k < m; k++) {
			for (j = 0; j < k; j++) {
				j1 = 4 * j + ip[m + k];
				k1 = 4 * k + ip[m + j];
				xr = a[j1];
				xi = a[j1 + 1];
				yr = a[k1];
				yi = a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
				j1 += nm;
				k1 += nm;
				xr = a[j1];
				xi = a[j1 + 1];
				yr = a[k1];
				yi = a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
				j1 += nh;
				k1 += 2;
				xr = a[j1];
				xi = a[j1 + 1];
				yr = a[k1];
				yi = a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
				j1 -= nm;
				k1 -= nm;
				xr = a[j1];
				xi = a[j1 + 1];
				yr = a[k1];
				yi = a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
				j1 += 2;
				k1 += nh;
				xr = a[j1];
				xi = a[j1 + 1];
				yr = a[k1];
				yi = a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
				j1 += nm;
				k1 += nm;
				xr = a[j1];
				xi = a[j1 + 1];
				yr = a[k1];
				yi = a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
				j1 -= nh;
				k1 -= 2;
				xr = a[j1];
				xi = a[j1 + 1];
				yr = a[k1];
				yi = a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
				j1 -= nm;
				k1 -= nm;
				xr = a[j1];
				xi = a[j1 + 1];
				yr = a[k1];
				yi = a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
			}
			k1 = 4 * k + ip[m + k];
			j1 = k1 + 2;
			k1 += nh;
			xr = a[j1];
			xi = a[j1 + 1];
			yr = a[k1];
			yi = a[k1 + 1];
			a[j1] = yr;
			a[j1 + 1] = yi;
			a[k1] = xr;
			a[k1 + 1] = xi;
			j1 += nm;
			k1 += nm;
			xr = a[j1];
			xi = a[j1 + 1];
			yr = a[k1];
			yi = a[k1 + 1];
			a[j1] = yr;
			a[j1 + 1] = yi;
			a[k1] = xr;
			a[k1 + 1] = xi;
		}
	}
}


void bitrv2conj(int n, int *ip, double *a)
{
	int j, j1, k, k1, l, m, nh, nm;
	double xr, xi, yr, yi;

	m = 1;
	for (l = n >> 2; l > 8; l >>= 2) {
		m <<= 1;
	}
	nh = n >> 1;
	nm = 4 * m;
	if (l == 8) {
		for (k = 0; k < m; k++) {
			for (j = 0; j < k; j++) {
				j1 = 4 * j + 2 * ip[m + k];
				k1 = 4 * k + 2 * ip[m + j];
				xr = a[j1];
				xi = -a[j1 + 1];
				yr = a[k1];
				yi = -a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
				j1 += nm;
				k1 += 2 * nm;
				xr = a[j1];
				xi = -a[j1 + 1];
				yr = a[k1];
				yi = -a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
				j1 += nm;
				k1 -= nm;
				xr = a[j1];
				xi = -a[j1 + 1];
				yr = a[k1];
				yi = -a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
				j1 += nm;
				k1 += 2 * nm;
				xr = a[j1];
				xi = -a[j1 + 1];
				yr = a[k1];
				yi = -a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
				j1 += nh;
				k1 += 2;
				xr = a[j1];
				xi = -a[j1 + 1];
				yr = a[k1];
				yi = -a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
				j1 -= nm;
				k1 -= 2 * nm;
				xr = a[j1];
				xi = -a[j1 + 1];
				yr = a[k1];
				yi = -a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
				j1 -= nm;
				k1 += nm;
				xr = a[j1];
				xi = -a[j1 + 1];
				yr = a[k1];
				yi = -a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
				j1 -= nm;
				k1 -= 2 * nm;
				xr = a[j1];
				xi = -a[j1 + 1];
				yr = a[k1];
				yi = -a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
				j1 += 2;
				k1 += nh;
				xr = a[j1];
				xi = -a[j1 + 1];
				yr = a[k1];
				yi = -a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
				j1 += nm;
				k1 += 2 * nm;
				xr = a[j1];
				xi = -a[j1 + 1];
				yr = a[k1];
				yi = -a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
				j1 += nm;
				k1 -= nm;
				xr = a[j1];
				xi = -a[j1 + 1];
				yr = a[k1];
				yi = -a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
				j1 += nm;
				k1 += 2 * nm;
				xr = a[j1];
				xi = -a[j1 + 1];
				yr = a[k1];
				yi = -a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
				j1 -= nh;
				k1 -= 2;
				xr = a[j1];
				xi = -a[j1 + 1];
				yr = a[k1];
				yi = -a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
				j1 -= nm;
				k1 -= 2 * nm;
				xr = a[j1];
				xi = -a[j1 + 1];
				yr = a[k1];
				yi = -a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
				j1 -= nm;
				k1 += nm;
				xr = a[j1];
				xi = -a[j1 + 1];
				yr = a[k1];
				yi = -a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
				j1 -= nm;
				k1 -= 2 * nm;
				xr = a[j1];
				xi = -a[j1 + 1];
				yr = a[k1];
				yi = -a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
			}
			k1 = 4 * k + 2 * ip[m + k];
			j1 = k1 + 2;
			k1 += nh;
			a[j1 - 1] = -a[j1 - 1];
			xr = a[j1];
			xi = -a[j1 + 1];
			yr = a[k1];
			yi = -a[k1 + 1];
			a[j1] = yr;
			a[j1 + 1] = yi;
			a[k1] = xr;
			a[k1 + 1] = xi;
			a[k1 + 3] = -a[k1 + 3];
			j1 += nm;
			k1 += 2 * nm;
			xr = a[j1];
			xi = -a[j1 + 1];
			yr = a[k1];
			yi = -a[k1 + 1];
			a[j1] = yr;
			a[j1 + 1] = yi;
			a[k1] = xr;
			a[k1 + 1] = xi;
			j1 += nm;
			k1 -= nm;
			xr = a[j1];
			xi = -a[j1 + 1];
			yr = a[k1];
			yi = -a[k1 + 1];
			a[j1] = yr;
			a[j1 + 1] = yi;
			a[k1] = xr;
			a[k1 + 1] = xi;
			j1 -= 2;
			k1 -= nh;
			xr = a[j1];
			xi = -a[j1 + 1];
			yr = a[k1];
			yi = -a[k1 + 1];
			a[j1] = yr;
			a[j1 + 1] = yi;
			a[k1] = xr;
			a[k1 + 1] = xi;
			j1 += nh + 2;
			k1 += nh + 2;
			xr = a[j1];
			xi = -a[j1 + 1];
			yr = a[k1];
			yi = -a[k1 + 1];
			a[j1] = yr;
			a[j1 + 1] = yi;
			a[k1] = xr;
			a[k1 + 1] = xi;
			j1 -= nh - nm;
			k1 += 2 * nm - 2;
			a[j1 - 1] = -a[j1 - 1];
			xr = a[j1];
			xi = -a[j1 + 1];
			yr = a[k1];
			yi = -a[k1 + 1];
			a[j1] = yr;
			a[j1 + 1] = yi;
			a[k1] = xr;
			a[k1 + 1] = xi;
			a[k1 + 3] = -a[k1 + 3];
		}
	} else {
		for (k = 0; k < m; k++) {
			for (j = 0; j < k; j++) {
				j1 = 4 * j + ip[m + k];
				k1 = 4 * k + ip[m + j];
				xr = a[j1];
				xi = -a[j1 + 1];
				yr = a[k1];
				yi = -a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
				j1 += nm;
				k1 += nm;
				xr = a[j1];
				xi = -a[j1 + 1];
				yr = a[k1];
				yi = -a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
				j1 += nh;
				k1 += 2;
				xr = a[j1];
				xi = -a[j1 + 1];
				yr = a[k1];
				yi = -a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
				j1 -= nm;
				k1 -= nm;
				xr = a[j1];
				xi = -a[j1 + 1];
				yr = a[k1];
				yi = -a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
				j1 += 2;
				k1 += nh;
				xr = a[j1];
				xi = -a[j1 + 1];
				yr = a[k1];
				yi = -a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
				j1 += nm;
				k1 += nm;
				xr = a[j1];
				xi = -a[j1 + 1];
				yr = a[k1];
				yi = -a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
				j1 -= nh;
				k1 -= 2;
				xr = a[j1];
				xi = -a[j1 + 1];
				yr = a[k1];
				yi = -a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
				j1 -= nm;
				k1 -= nm;
				xr = a[j1];
				xi = -a[j1 + 1];
				yr = a[k1];
				yi = -a[k1 + 1];
				a[j1] = yr;
				a[j1 + 1] = yi;
				a[k1] = xr;
				a[k1 + 1] = xi;
			}
			k1 = 4 * k + ip[m + k];
			j1 = k1 + 2;
			k1 += nh;
			a[j1 - 1] = -a[j1 - 1];
			xr = a[j1];
			xi = -a[j1 + 1];
			yr = a[k1];
			yi = -a[k1 + 1];
			a[j1] = yr;
			a[j1 + 1] = yi;
			a[k1] = xr;
			a[k1 + 1] = xi;
			a[k1 + 3] = -a[k1 + 3];
			j1 += nm;
			k1 += nm;
			a[j1 - 1] = -a[j1 - 1];
			xr = a[j1];
			xi = -a[j1 + 1];
			yr = a[k1];
			yi = -a[k1 + 1];
			a[j1] = yr;
			a[j1 + 1] = yi;
			a[k1] = xr;
			a[k1 + 1] = xi;
			a[k1 + 3] = -a[k1 + 3];
		}
	}
}


void bitrv216(double *a)
{
	double x1r, x1i, x2r, x2i, x3r, x3i, x4r, x4i,
		x5r, x5i, x7r, x7i, x8r, x8i, x10r, x10i,
		x11r, x11i, x12r, x12i, x13r, x13i, x14r, x14i;

	x1r = a[2];
	x1i = a[3];
	x2r = a[4];
	x2i = a[5];
	x3r = a[6];
	x3i = a[7];
	x4r = a[8];
	x4i = a[9];
	x5r = a[10];
	x5i = a[11];
	x7r = a[14];
	x7i = a[15];
	x8r = a[16];
	x8i = a[17];
	x10r = a[20];
	x10i = a[21];
	x11r = a[22];
	x11i = a[23];
	x12r = a[24];
	x12i = a[25];
	x13r = a[26];
	x13i = a[27];
	x14r = a[28];
	x14i = a[29];
	a[2] = x8r;
	a[3] = x8i;
	a[4] = x4r;
	a[5] = x4i;
	a[6] = x12r;
	a[7] = x12i;
	a[8] = x2r;
	a[9] = x2i;
	a[10] = x10r;
	a[11] = x10i;
	a[14] = x14r;
	a[15] = x14i;
	a[16] = x1r;
	a[17] = x1i;
	a[20] = x5r;
	a[21] = x5i;
	a[22] = x13r;
	a[23] = x13i;
	a[24] = x3r;
	a[25] = x3i;
	a[26] = x11r;
	a[27] = x11i;
	a[28] = x7r;
	a[29] = x7i;
}


void bitrv216neg(double *a)
{
	double x1r, x1i, x2r, x2i, x3r, x3i, x4r, x4i,
		x5r, x5i, x6r, x6i, x7r, x7i, x8r, x8i,
		x9r, x9i, x10r, x10i, x11r, x11i, x12r, x12i, x13r, x13i, x14r, x14i, x15r, x15i;

	x1r = a[2];
	x1i = a[3];
	x2r = a[4];
	x2i = a[5];
	x3r = a[6];
	x3i = a[7];
	x4r = a[8];
	x4i = a[9];
	x5r = a[10];
	x5i = a[11];
	x6r = a[12];
	x6i = a[13];
	x7r = a[14];
	x7i = a[15];
	x8r = a[16];
	x8i = a[17];
	x9r = a[18];
	x9i = a[19];
	x10r = a[20];
	x10i = a[21];
	x11r = a[22];
	x11i = a[23];
	x12r = a[24];
	x12i = a[25];
	x13r = a[26];
	x13i = a[27];
	x14r = a[28];
	x14i = a[29];
	x15r = a[30];
	x15i = a[31];
	a[2] = x15r;
	a[3] = x15i;
	a[4] = x7r;
	a[5] = x7i;
	a[6] = x11r;
	a[7] = x11i;
	a[8] = x3r;
	a[9] = x3i;
	a[10] = x13r;
	a[11] = x13i;
	a[12] = x5r;
	a[13] = x5i;
	a[14] = x9r;
	a[15] = x9i;
	a[16] = x1r;
	a[17] = x1i;
	a[18] = x14r;
	a[19] = x14i;
	a[20] = x6r;
	a[21] = x6i;
	a[22] = x10r;
	a[23] = x10i;
	a[24] = x2r;
	a[25] = x2i;
	a[26] = x12r;
	a[27] = x12i;
	a[28] = x4r;
	a[29] = x4i;
	a[30] = x8r;
	a[31] = x8i;
}


void bitrv208(double *a)
{
	double x1r, x1i, x3r, x3i, x4r, x4i, x6r, x6i;

	x1r = a[2];
	x1i = a[3];
	x3r = a[6];
	x3i = a[7];
	x4r = a[8];
	x4i = a[9];
	x6r = a[12];
	x6i = a[13];
	a[2] = x4r;
	a[3] = x4i;
	a[6] = x6r;
	a[7] = x6i;
	a[8] = x1r;
	a[9] = x1i;
	a[12] = x3r;
	a[13] = x3i;
}


void bitrv208neg(double *a)
{
	double x1r, x1i, x2r, x2i, x3r, x3i, x4r, x4i, x5r, x5i, x6r, x6i, x7r, x7i;

	x1r = a[2];
	x1i = a[3];
	x2r = a[4];
	x2i = a[5];
	x3r = a[6];
	x3i = a[7];
	x4r = a[8];
	x4i = a[9];
	x5r = a[10];
	x5i = a[11];
	x6r = a[12];
	x6i = a[13];
	x7r = a[14];
	x7i = a[15];
	a[2] = x7r;
	a[3] = x7i;
	a[4] = x3r;
	a[5] = x3i;
	a[6] = x5r;
	a[7] = x5i;
	a[8] = x1r;
	a[9] = x1i;
	a[10] = x6r;
	a[11] = x6i;
	a[12] = x2r;
	a[13] = x2i;
	a[14] = x4r;
	a[15] = x4i;
}


void cftf1st(int n, double *a, double *w)
{
	int j, j0, j1, j2, j3, k, m, mh;
	double wn4r, csc1, csc3, wk1r, wk1i, wk3r, wk3i, wd1r, wd1i, wd3r, wd3i;
	double x0r, x0i, x1r, x1i, x2r, x2i, x3r, x3i, y0r, y0i, y1r, y1i, y2r, y2i, y3r, y3i;

	mh = n >> 3;
	m = 2 * mh;
	j1 = m;
	j2 = j1 + m;
	j3 = j2 + m;
	x0r = a[0] + a[j2];
	x0i = a[1] + a[j2 + 1];
	x1r = a[0] - a[j2];
	x1i = a[1] - a[j2 + 1];
	x2r = a[j1] + a[j3];
	x2i = a[j1 + 1] + a[j3 + 1];
	x3r = a[j1] - a[j3];
	x3i = a[j1 + 1] - a[j3 + 1];
	a[0] = x0r + x2r;
	a[1] = x0i + x2i;
	a[j1] = x0r - x2r;
	a[j1 + 1] = x0i - x2i;
	a[j2] = x1r - x3i;
	a[j2 + 1] = x1i + x3r;
	a[j3] = x1r + x3i;
	a[j3 + 1] = x1i - x3r;
	wn4r = w[1];
	csc1 = w[2];
	csc3 = w[3];
	wd1r = 1;
	wd1i = 0;
	wd3r = 1;
	wd3i = 0;
	k = 0;
	for (j = 2; j < mh - 2; j += 4) {
		k += 4;
		wk1r = csc1 * (wd1r + w[k]);
		wk1i = csc1 * (wd1i + w[k + 1]);
		wk3r = csc3 * (wd3r + w[k + 2]);
		wk3i = csc3 * (wd3i + w[k + 3]);
		wd1r = w[k];
		wd1i = w[k + 1];
		wd3r = w[k + 2];
		wd3i = w[k + 3];
		j1 = j + m;
		j2 = j1 + m;
		j3 = j2 + m;
		x0r = a[j] + a[j2];
		x0i = a[j + 1] + a[j2 + 1];
		x1r = a[j] - a[j2];
		x1i = a[j + 1] - a[j2 + 1];
		y0r = a[j + 2] + a[j2 + 2];
		y0i = a[j + 3] + a[j2 + 3];
		y1r = a[j + 2] - a[j2 + 2];
		y1i = a[j + 3] - a[j2 + 3];
		x2r = a[j1] + a[j3];
		x2i = a[j1 + 1] + a[j3 + 1];
		x3r = a[j1] - a[j3];
		x3i = a[j1 + 1] - a[j3 + 1];
		y2r = a[j1 + 2] + a[j3 + 2];
		y2i = a[j1 + 3] + a[j3 + 3];
		y3r = a[j1 + 2] - a[j3 + 2];
		y3i = a[j1 + 3] - a[j3 + 3];
		a[j] = x0r + x2r;
		a[j + 1] = x0i + x2i;
		a[j + 2] = y0r + y2r;
		a[j + 3] = y0i + y2i;
		a[j1] = x0r - x2r;
		a[j1 + 1] = x0i - x2i;
		a[j1 + 2] = y0r - y2r;
		a[j1 + 3] = y0i - y2i;
		x0r = x1r - x3i;
		x0i = x1i + x3r;
		a[j2] = wk1r * x0r - wk1i * x0i;
		a[j2 + 1] = wk1r * x0i + wk1i * x0r;
		x0r = y1r - y3i;
		x0i = y1i + y3r;
		a[j2 + 2] = wd1r * x0r - wd1i * x0i;
		a[j2 + 3] = wd1r * x0i + wd1i * x0r;
		x0r = x1r + x3i;
		x0i = x1i - x3r;
		a[j3] = wk3r * x0r + wk3i * x0i;
		a[j3 + 1] = wk3r * x0i - wk3i * x0r;
		x0r = y1r + y3i;
		x0i = y1i - y3r;
		a[j3 + 2] = wd3r * x0r + wd3i * x0i;
		a[j3 + 3] = wd3r * x0i - wd3i * x0r;
		j0 = m - j;
		j1 = j0 + m;
		j2 = j1 + m;
		j3 = j2 + m;
		x0r = a[j0] + a[j2];
		x0i = a[j0 + 1] + a[j2 + 1];
		x1r = a[j0] - a[j2];
		x1i = a[j0 + 1] - a[j2 + 1];
		y0r = a[j0 - 2] + a[j2 - 2];
		y0i = a[j0 - 1] + a[j2 - 1];
		y1r = a[j0 - 2] - a[j2 - 2];
		y1i = a[j0 - 1] - a[j2 - 1];
		x2r = a[j1] + a[j3];
		x2i = a[j1 + 1] + a[j3 + 1];
		x3r = a[j1] - a[j3];
		x3i = a[j1 + 1] - a[j3 + 1];
		y2r = a[j1 - 2] + a[j3 - 2];
		y2i = a[j1 - 1] + a[j3 - 1];
		y3r = a[j1 - 2] - a[j3 - 2];
		y3i = a[j1 - 1] - a[j3 - 1];
		a[j0] = x0r + x2r;
		a[j0 + 1] = x0i + x2i;
		a[j0 - 2] = y0r + y2r;
		a[j0 - 1] = y0i + y2i;
		a[j1] = x0r - x2r;
		a[j1 + 1] = x0i - x2i;
		a[j1 - 2] = y0r - y2r;
		a[j1 - 1] = y0i - y2i;
		x0r = x1r - x3i;
		x0i = x1i + x3r;
		a[j2] = wk1i * x0r - wk1r * x0i;
		a[j2 + 1] = wk1i * x0i + wk1r * x0r;
		x0r = y1r - y3i;
		x0i = y1i + y3r;
		a[j2 - 2] = wd1i * x0r - wd1r * x0i;
		a[j2 - 1] = wd1i * x0i + wd1r * x0r;
		x0r = x1r + x3i;
		x0i = x1i - x3r;
		a[j3] = wk3i * x0r + wk3r * x0i;
		a[j3 + 1] = wk3i * x0i - wk3r * x0r;
		x0r = y1r + y3i;
		x0i = y1i - y3r;
		a[j3 - 2] = wd3i * x0r + wd3r * x0i;
		a[j3 - 1] = wd3i * x0i - wd3r * x0r;
	}
	wk1r = csc1 * (wd1r + wn4r);
	wk1i = csc1 * (wd1i + wn4r);
	wk3r = csc3 * (wd3r - wn4r);
	wk3i = csc3 * (wd3i - wn4r);
	j0 = mh;
	j1 = j0 + m;
	j2 = j1 + m;
	j3 = j2 + m;
	x0r = a[j0 - 2] + a[j2 - 2];
	x0i = a[j0 - 1] + a[j2 - 1];
	x1r = a[j0 - 2] - a[j2 - 2];
	x1i = a[j0 - 1] - a[j2 - 1];
	x2r = a[j1 - 2] + a[j3 - 2];
	x2i = a[j1 - 1] + a[j3 - 1];
	x3r = a[j1 - 2] - a[j3 - 2];
	x3i = a[j1 - 1] - a[j3 - 1];
	a[j0 - 2] = x0r + x2r;
	a[j0 - 1] = x0i + x2i;
	a[j1 - 2] = x0r - x2r;
	a[j1 - 1] = x0i - x2i;
	x0r = x1r - x3i;
	x0i = x1i + x3r;
	a[j2 - 2] = wk1r * x0r - wk1i * x0i;
	a[j2 - 1] = wk1r * x0i + wk1i * x0r;
	x0r = x1r + x3i;
	x0i = x1i - x3r;
	a[j3 - 2] = wk3r * x0r + wk3i * x0i;
	a[j3 - 1] = wk3r * x0i - wk3i * x0r;
	x0r = a[j0] + a[j2];
	x0i = a[j0 + 1] + a[j2 + 1];
	x1r = a[j0] - a[j2];
	x1i = a[j0 + 1] - a[j2 + 1];
	x2r = a[j1] + a[j3];
	x2i = a[j1 + 1] + a[j3 + 1];
	x3r = a[j1] - a[j3];
	x3i = a[j1 + 1] - a[j3 + 1];
	a[j0] = x0r + x2r;
	a[j0 + 1] = x0i + x2i;
	a[j1] = x0r - x2r;
	a[j1 + 1] = x0i - x2i;
	x0r = x1r - x3i;
	x0i = x1i + x3r;
	a[j2] = wn4r * (x0r - x0i);
	a[j2 + 1] = wn4r * (x0i + x0r);
	x0r = x1r + x3i;
	x0i = x1i - x3r;
	a[j3] = -wn4r * (x0r + x0i);
	a[j3 + 1] = -wn4r * (x0i - x0r);
	x0r = a[j0 + 2] + a[j2 + 2];
	x0i = a[j0 + 3] + a[j2 + 3];
	x1r = a[j0 + 2] - a[j2 + 2];
	x1i = a[j0 + 3] - a[j2 + 3];
	x2r = a[j1 + 2] + a[j3 + 2];
	x2i = a[j1 + 3] + a[j3 + 3];
	x3r = a[j1 + 2] - a[j3 + 2];
	x3i = a[j1 + 3] - a[j3 + 3];
	a[j0 + 2] = x0r + x2r;
	a[j0 + 3] = x0i + x2i;
	a[j1 + 2] = x0r - x2r;
	a[j1 + 3] = x0i - x2i;
	x0r = x1r - x3i;
	x0i = x1i + x3r;
	a[j2 + 2] = wk1i * x0r - wk1r * x0i;
	a[j2 + 3] = wk1i * x0i + wk1r * x0r;
	x0r = x1r + x3i;
	x0i = x1i - x3r;
	a[j3 + 2] = wk3i * x0r + wk3r * x0i;
	a[j3 + 3] = wk3i * x0i - wk3r * x0r;
}


void cftb1st(int n, double *a, double *w)
{
	int j, j0, j1, j2, j3, k, m, mh;
	double wn4r, csc1, csc3, wk1r, wk1i, wk3r, wk3i, wd1r, wd1i, wd3r, wd3i;
	double x0r, x0i, x1r, x1i, x2r, x2i, x3r, x3i, y0r, y0i, y1r, y1i, y2r, y2i, y3r, y3i;

	mh = n >> 3;
	m = 2 * mh;
	j1 = m;
	j2 = j1 + m;
	j3 = j2 + m;
	x0r = a[0] + a[j2];
	x0i = -a[1] - a[j2 + 1];
	x1r = a[0] - a[j2];
	x1i = -a[1] + a[j2 + 1];
	x2r = a[j1] + a[j3];
	x2i = a[j1 + 1] + a[j3 + 1];
	x3r = a[j1] - a[j3];
	x3i = a[j1 + 1] - a[j3 + 1];
	a[0] = x0r + x2r;
	a[1] = x0i - x2i;
	a[j1] = x0r - x2r;
	a[j1 + 1] = x0i + x2i;
	a[j2] = x1r + x3i;
	a[j2 + 1] = x1i + x3r;
	a[j3] = x1r - x3i;
	a[j3 + 1] = x1i - x3r;
	wn4r = w[1];
	csc1 = w[2];
	csc3 = w[3];
	wd1r = 1;
	wd1i = 0;
	wd3r = 1;
	wd3i = 0;
	k = 0;
	for (j = 2; j < mh - 2; j += 4) {
		k += 4;
		wk1r = csc1 * (wd1r + w[k]);
		wk1i = csc1 * (wd1i + w[k + 1]);
		wk3r = csc3 * (wd3r + w[k + 2]);
		wk3i = csc3 * (wd3i + w[k + 3]);
		wd1r = w[k];
		wd1i = w[k + 1];
		wd3r = w[k + 2];
		wd3i = w[k + 3];
		j1 = j + m;
		j2 = j1 + m;
		j3 = j2 + m;
		x0r = a[j] + a[j2];
		x0i = -a[j + 1] - a[j2 + 1];
		x1r = a[j] - a[j2];
		x1i = -a[j + 1] + a[j2 + 1];
		y0r = a[j + 2] + a[j2 + 2];
		y0i = -a[j + 3] - a[j2 + 3];
		y1r = a[j + 2] - a[j2 + 2];
		y1i = -a[j + 3] + a[j2 + 3];
		x2r = a[j1] + a[j3];
		x2i = a[j1 + 1] + a[j3 + 1];
		x3r = a[j1] - a[j3];
		x3i = a[j1 + 1] - a[j3 + 1];
		y2r = a[j1 + 2] + a[j3 + 2];
		y2i = a[j1 + 3] + a[j3 + 3];
		y3r = a[j1 + 2] - a[j3 + 2];
		y3i = a[j1 + 3] - a[j3 + 3];
		a[j] = x0r + x2r;
		a[j + 1] = x0i - x2i;
		a[j + 2] = y0r + y2r;
		a[j + 3] = y0i - y2i;
		a[j1] = x0r - x2r;
		a[j1 + 1] = x0i + x2i;
		a[j1 + 2] = y0r - y2r;
		a[j1 + 3] = y0i + y2i;
		x0r = x1r + x3i;
		x0i = x1i + x3r;
		a[j2] = wk1r * x0r - wk1i * x0i;
		a[j2 + 1] = wk1r * x0i + wk1i * x0r;
		x0r = y1r + y3i;
		x0i = y1i + y3r;
		a[j2 + 2] = wd1r * x0r - wd1i * x0i;
		a[j2 + 3] = wd1r * x0i + wd1i * x0r;
		x0r = x1r - x3i;
		x0i = x1i - x3r;
		a[j3] = wk3r * x0r + wk3i * x0i;
		a[j3 + 1] = wk3r * x0i - wk3i * x0r;
		x0r = y1r - y3i;
		x0i = y1i - y3r;
		a[j3 + 2] = wd3r * x0r + wd3i * x0i;
		a[j3 + 3] = wd3r * x0i - wd3i * x0r;
		j0 = m - j;
		j1 = j0 + m;
		j2 = j1 + m;
		j3 = j2 + m;
		x0r = a[j0] + a[j2];
		x0i = -a[j0 + 1] - a[j2 + 1];
		x1r = a[j0] - a[j2];
		x1i = -a[j0 + 1] + a[j2 + 1];
		y0r = a[j0 - 2] + a[j2 - 2];
		y0i = -a[j0 - 1] - a[j2 - 1];
		y1r = a[j0 - 2] - a[j2 - 2];
		y1i = -a[j0 - 1] + a[j2 - 1];
		x2r = a[j1] + a[j3];
		x2i = a[j1 + 1] + a[j3 + 1];
		x3r = a[j1] - a[j3];
		x3i = a[j1 + 1] - a[j3 + 1];
		y2r = a[j1 - 2] + a[j3 - 2];
		y2i = a[j1 - 1] + a[j3 - 1];
		y3r = a[j1 - 2] - a[j3 - 2];
		y3i = a[j1 - 1] - a[j3 - 1];
		a[j0] = x0r + x2r;
		a[j0 + 1] = x0i - x2i;
		a[j0 - 2] = y0r + y2r;
		a[j0 - 1] = y0i - y2i;
		a[j1] = x0r - x2r;
		a[j1 + 1] = x0i + x2i;
		a[j1 - 2] = y0r - y2r;
		a[j1 - 1] = y0i + y2i;
		x0r = x1r + x3i;
		x0i = x1i + x3r;
		a[j2] = wk1i * x0r - wk1r * x0i;
		a[j2 + 1] = wk1i * x0i + wk1r * x0r;
		x0r = y1r + y3i;
		x0i = y1i + y3r;
		a[j2 - 2] = wd1i * x0r - wd1r * x0i;
		a[j2 - 1] = wd1i * x0i + wd1r * x0r;
		x0r = x1r - x3i;
		x0i = x1i - x3r;
		a[j3] = wk3i * x0r + wk3r * x0i;
		a[j3 + 1] = wk3i * x0i - wk3r * x0r;
		x0r = y1r - y3i;
		x0i = y1i - y3r;
		a[j3 - 2] = wd3i * x0r + wd3r * x0i;
		a[j3 - 1] = wd3i * x0i - wd3r * x0r;
	}
	wk1r = csc1 * (wd1r + wn4r);
	wk1i = csc1 * (wd1i + wn4r);
	wk3r = csc3 * (wd3r - wn4r);
	wk3i = csc3 * (wd3i - wn4r);
	j0 = mh;
	j1 = j0 + m;
	j2 = j1 + m;
	j3 = j2 + m;
	x0r = a[j0 - 2] + a[j2 - 2];
	x0i = -a[j0 - 1] - a[j2 - 1];
	x1r = a[j0 - 2] - a[j2 - 2];
	x1i = -a[j0 - 1] + a[j2 - 1];
	x2r = a[j1 - 2] + a[j3 - 2];
	x2i = a[j1 - 1] + a[j3 - 1];
	x3r = a[j1 - 2] - a[j3 - 2];
	x3i = a[j1 - 1] - a[j3 - 1];
	a[j0 - 2] = x0r + x2r;
	a[j0 - 1] = x0i - x2i;
	a[j1 - 2] = x0r - x2r;
	a[j1 - 1] = x0i + x2i;
	x0r = x1r + x3i;
	x0i = x1i + x3r;
	a[j2 - 2] = wk1r * x0r - wk1i * x0i;
	a[j2 - 1] = wk1r * x0i + wk1i * x0r;
	x0r = x1r - x3i;
	x0i = x1i - x3r;
	a[j3 - 2] = wk3r * x0r + wk3i * x0i;
	a[j3 - 1] = wk3r * x0i - wk3i * x0r;
	x0r = a[j0] + a[j2];
	x0i = -a[j0 + 1] - a[j2 + 1];
	x1r = a[j0] - a[j2];
	x1i = -a[j0 + 1] + a[j2 + 1];
	x2r = a[j1] + a[j3];
	x2i = a[j1 + 1] + a[j3 + 1];
	x3r = a[j1] - a[j3];
	x3i = a[j1 + 1] - a[j3 + 1];
	a[j0] = x0r + x2r;
	a[j0 + 1] = x0i - x2i;
	a[j1] = x0r - x2r;
	a[j1 + 1] = x0i + x2i;
	x0r = x1r + x3i;
	x0i = x1i + x3r;
	a[j2] = wn4r * (x0r - x0i);
	a[j2 + 1] = wn4r * (x0i + x0r);
	x0r = x1r - x3i;
	x0i = x1i - x3r;
	a[j3] = -wn4r * (x0r + x0i);
	a[j3 + 1] = -wn4r * (x0i - x0r);
	x0r = a[j0 + 2] + a[j2 + 2];
	x0i = -a[j0 + 3] - a[j2 + 3];
	x1r = a[j0 + 2] - a[j2 + 2];
	x1i = -a[j0 + 3] + a[j2 + 3];
	x2r = a[j1 + 2] + a[j3 + 2];
	x2i = a[j1 + 3] + a[j3 + 3];
	x3r = a[j1 + 2] - a[j3 + 2];
	x3i = a[j1 + 3] - a[j3 + 3];
	a[j0 + 2] = x0r + x2r;
	a[j0 + 3] = x0i - x2i;
	a[j1 + 2] = x0r - x2r;
	a[j1 + 3] = x0i + x2i;
	x0r = x1r + x3i;
	x0i = x1i + x3r;
	a[j2 + 2] = wk1i * x0r - wk1r * x0i;
	a[j2 + 3] = wk1i * x0i + wk1r * x0r;
	x0r = x1r - x3i;
	x0i = x1i - x3r;
	a[j3 + 2] = wk3i * x0r + wk3r * x0i;
	a[j3 + 3] = wk3i * x0i - wk3r * x0r;
}


#ifdef USE_CDFT_THREADS
struct cdft_arg_st {
	int n0;
	int n;
	double *a;
	int nw;
	double *w;
};
typedef struct cdft_arg_st cdft_arg_t;


void cftrec4_th(int n, double *a, int nw, double *w)
{
	void *cftrec1_th(void *p);
	void *cftrec2_th(void *p);
	int i, idiv4, m, nthread;
	cdft_thread_t th[4];
	cdft_arg_t ag[4];

	nthread = 2;
	idiv4 = 0;
	m = n >> 1;
	if (n > CDFT_4THREADS_BEGIN_N) {
		nthread = 4;
		idiv4 = 1;
		m >>= 1;
	}
	for (i = 0; i < nthread; i++) {
		ag[i].n0 = n;
		ag[i].n = m;
		ag[i].a = &a[i * m];
		ag[i].nw = nw;
		ag[i].w = w;
		if (i != idiv4) {
			cdft_thread_create(&th[i], cftrec1_th, &ag[i]);
		} else {
			cdft_thread_create(&th[i], cftrec2_th, &ag[i]);
		}
	}
	for (i = 0; i < nthread; i++) {
		cdft_thread_wait(th[i]);
	}
}


void *cftrec1_th(void *p)
{
	int cfttree(int n, int j, int k, double *a, int nw, double *w);
	void cftleaf(int n, int isplt, double *a, int nw, double *w);
	void cftmdl1(int n, double *a, double *w);
	int isplt, j, k, m, n, n0, nw;
	double *a, *w;

	n0 = ((cdft_arg_t *) p)->n0;
	n = ((cdft_arg_t *) p)->n;
	a = ((cdft_arg_t *) p)->a;
	nw = ((cdft_arg_t *) p)->nw;
	w = ((cdft_arg_t *) p)->w;
	m = n0;
	while (m > 512) {
		m >>= 2;
		cftmdl1(m, &a[n - m], &w[nw - (m >> 1)]);
	}
	cftleaf(m, 1, &a[n - m], nw, w);
	k = 0;
	for (j = n - m; j > 0; j -= m) {
		k++;
		isplt = cfttree(m, j, k, a, nw, w);
		cftleaf(m, isplt, &a[j - m], nw, w);
	}
	return (void *) 0;
}


void *cftrec2_th(void *p)
{
	int cfttree(int n, int j, int k, double *a, int nw, double *w);
	void cftleaf(int n, int isplt, double *a, int nw, double *w);
	void cftmdl2(int n, double *a, double *w);
	int isplt, j, k, m, n, n0, nw;
	double *a, *w;

	n0 = ((cdft_arg_t *) p)->n0;
	n = ((cdft_arg_t *) p)->n;
	a = ((cdft_arg_t *) p)->a;
	nw = ((cdft_arg_t *) p)->nw;
	w = ((cdft_arg_t *) p)->w;
	k = 1;
	m = n0;
	while (m > 512) {
		m >>= 2;
		k <<= 2;
		cftmdl2(m, &a[n - m], &w[nw - m]);
	}
	cftleaf(m, 0, &a[n - m], nw, w);
	k >>= 1;
	for (j = n - m; j > 0; j -= m) {
		k++;
		isplt = cfttree(m, j, k, a, nw, w);
		cftleaf(m, isplt, &a[j - m], nw, w);
	}
	return (void *) 0;
}
#endif /* USE_CDFT_THREADS */


void cftrec4(int n, double *a, int nw, double *w)
{
	int cfttree(int n, int j, int k, double *a, int nw, double *w);
	void cftleaf(int n, int isplt, double *a, int nw, double *w);
	void cftmdl1(int n, double *a, double *w);
	int isplt, j, k, m;

	m = n;
	while (m > 512) {
		m >>= 2;
		cftmdl1(m, &a[n - m], &w[nw - (m >> 1)]);
	}
	cftleaf(m, 1, &a[n - m], nw, w);
	k = 0;
	for (j = n - m; j > 0; j -= m) {
		k++;
		isplt = cfttree(m, j, k, a, nw, w);
		cftleaf(m, isplt, &a[j - m], nw, w);
	}
}


int cfttree(int n, int j, int k, double *a, int nw, double *w)
{
	void cftmdl1(int n, double *a, double *w);
	void cftmdl2(int n, double *a, double *w);
	int i, isplt, m;

	if ((k & 3) != 0) {
		isplt = k & 1;
		if (isplt != 0) {
			cftmdl1(n, &a[j - n], &w[nw - (n >> 1)]);
		} else {
			cftmdl2(n, &a[j - n], &w[nw - n]);
		}
	} else {
		m = n;
		for (i = k; (i & 3) == 0; i >>= 2) {
			m <<= 2;
		}
		isplt = i & 1;
		if (isplt != 0) {
			while (m > 128) {
				cftmdl1(m, &a[j - m], &w[nw - (m >> 1)]);
				m >>= 2;
			}
		} else {
			while (m > 128) {
				cftmdl2(m, &a[j - m], &w[nw - m]);
				m >>= 2;
			}
		}
	}
	return isplt;
}


void cftleaf(int n, int isplt, double *a, int nw, double *w)
{
	void cftmdl1(int n, double *a, double *w);
	void cftmdl2(int n, double *a, double *w);
	void cftf161(double *a, double *w);
	void cftf162(double *a, double *w);
	void cftf081(double *a, double *w);
	void cftf082(double *a, double *w);

	if (n == 512) {
		cftmdl1(128, a, &w[nw - 64]);
		cftf161(a, &w[nw - 8]);
		cftf162(&a[32], &w[nw - 32]);
		cftf161(&a[64], &w[nw - 8]);
		cftf161(&a[96], &w[nw - 8]);
		cftmdl2(128, &a[128], &w[nw - 128]);
		cftf161(&a[128], &w[nw - 8]);
		cftf162(&a[160], &w[nw - 32]);
		cftf161(&a[192], &w[nw - 8]);
		cftf162(&a[224], &w[nw - 32]);
		cftmdl1(128, &a[256], &w[nw - 64]);
		cftf161(&a[256], &w[nw - 8]);
		cftf162(&a[288], &w[nw - 32]);
		cftf161(&a[320], &w[nw - 8]);
		cftf161(&a[352], &w[nw - 8]);
		if (isplt != 0) {
			cftmdl1(128, &a[384], &w[nw - 64]);
			cftf161(&a[480], &w[nw - 8]);
		} else {
			cftmdl2(128, &a[384], &w[nw - 128]);
			cftf162(&a[480], &w[nw - 32]);
		}
		cftf161(&a[384], &w[nw - 8]);
		cftf162(&a[416], &w[nw - 32]);
		cftf161(&a[448], &w[nw - 8]);
	} else {
		cftmdl1(64, a, &w[nw - 32]);
		cftf081(a, &w[nw - 8]);
		cftf082(&a[16], &w[nw - 8]);
		cftf081(&a[32], &w[nw - 8]);
		cftf081(&a[48], &w[nw - 8]);
		cftmdl2(64, &a[64], &w[nw - 64]);
		cftf081(&a[64], &w[nw - 8]);
		cftf082(&a[80], &w[nw - 8]);
		cftf081(&a[96], &w[nw - 8]);
		cftf082(&a[112], &w[nw - 8]);
		cftmdl1(64, &a[128], &w[nw - 32]);
		cftf081(&a[128], &w[nw - 8]);
		cftf082(&a[144], &w[nw - 8]);
		cftf081(&a[160], &w[nw - 8]);
		cftf081(&a[176], &w[nw - 8]);
		if (isplt != 0) {
			cftmdl1(64, &a[192], &w[nw - 32]);
			cftf081(&a[240], &w[nw - 8]);
		} else {
			cftmdl2(64, &a[192], &w[nw - 64]);
			cftf082(&a[240], &w[nw - 8]);
		}
		cftf081(&a[192], &w[nw - 8]);
		cftf082(&a[208], &w[nw - 8]);
		cftf081(&a[224], &w[nw - 8]);
	}
}


void cftmdl1(int n, double *a, double *w)
{
	int j, j0, j1, j2, j3, k, m, mh;
	double wn4r, wk1r, wk1i, wk3r, wk3i;
	double x0r, x0i, x1r, x1i, x2r, x2i, x3r, x3i;

	mh = n >> 3;
	m = 2 * mh;
	j1 = m;
	j2 = j1 + m;
	j3 = j2 + m;
	x0r = a[0] + a[j2];
	x0i = a[1] + a[j2 + 1];
	x1r = a[0] - a[j2];
	x1i = a[1] - a[j2 + 1];
	x2r = a[j1] + a[j3];
	x2i = a[j1 + 1] + a[j3 + 1];
	x3r = a[j1] - a[j3];
	x3i = a[j1 + 1] - a[j3 + 1];
	a[0] = x0r + x2r;
	a[1] = x0i + x2i;
	a[j1] = x0r - x2r;
	a[j1 + 1] = x0i - x2i;
	a[j2] = x1r - x3i;
	a[j2 + 1] = x1i + x3r;
	a[j3] = x1r + x3i;
	a[j3 + 1] = x1i - x3r;
	wn4r = w[1];
	k = 0;
	for (j = 2; j < mh; j += 2) {
		k += 4;
		wk1r = w[k];
		wk1i = w[k + 1];
		wk3r = w[k + 2];
		wk3i = w[k + 3];
		j1 = j + m;
		j2 = j1 + m;
		j3 = j2 + m;
		x0r = a[j] + a[j2];
		x0i = a[j + 1] + a[j2 + 1];
		x1r = a[j] - a[j2];
		x1i = a[j + 1] - a[j2 + 1];
		x2r = a[j1] + a[j3];
		x2i = a[j1 + 1] + a[j3 + 1];
		x3r = a[j1] - a[j3];
		x3i = a[j1 + 1] - a[j3 + 1];
		a[j] = x0r + x2r;
		a[j + 1] = x0i + x2i;
		a[j1] = x0r - x2r;
		a[j1 + 1] = x0i - x2i;
		x0r = x1r - x3i;
		x0i = x1i + x3r;
		a[j2] = wk1r * x0r - wk1i * x0i;
		a[j2 + 1] = wk1r * x0i + wk1i * x0r;
		x0r = x1r + x3i;
		x0i = x1i - x3r;
		a[j3] = wk3r * x0r + wk3i * x0i;
		a[j3 + 1] = wk3r * x0i - wk3i * x0r;
		j0 = m - j;
		j1 = j0 + m;
		j2 = j1 + m;
		j3 = j2 + m;
		x0r = a[j0] + a[j2];
		x0i = a[j0 + 1] + a[j2 + 1];
		x1r = a[j0] - a[j2];
		x1i = a[j0 + 1] - a[j2 + 1];
		x2r = a[j1] + a[j3];
		x2i = a[j1 + 1] + a[j3 + 1];
		x3r = a[j1] - a[j3];
		x3i = a[j1 + 1] - a[j3 + 1];
		a[j0] = x0r + x2r;
		a[j0 + 1] = x0i + x2i;
		a[j1] = x0r - x2r;
		a[j1 + 1] = x0i - x2i;
		x0r = x1r - x3i;
		x0i = x1i + x3r;
		a[j2] = wk1i * x0r - wk1r * x0i;
		a[j2 + 1] = wk1i * x0i + wk1r * x0r;
		x0r = x1r + x3i;
		x0i = x1i - x3r;
		a[j3] = wk3i * x0r + wk3r * x0i;
		a[j3 + 1] = wk3i * x0i - wk3r * x0r;
	}
	j0 = mh;
	j1 = j0 + m;
	j2 = j1 + m;
	j3 = j2 + m;
	x0r = a[j0] + a[j2];
	x0i = a[j0 + 1] + a[j2 + 1];
	x1r = a[j0] - a[j2];
	x1i = a[j0 + 1] - a[j2 + 1];
	x2r = a[j1] + a[j3];
	x2i = a[j1 + 1] + a[j3 + 1];
	x3r = a[j1] - a[j3];
	x3i = a[j1 + 1] - a[j3 + 1];
	a[j0] = x0r + x2r;
	a[j0 + 1] = x0i + x2i;
	a[j1] = x0r - x2r;
	a[j1 + 1] = x0i - x2i;
	x0r = x1r - x3i;
	x0i = x1i + x3r;
	a[j2] = wn4r * (x0r - x0i);
	a[j2 + 1] = wn4r * (x0i + x0r);
	x0r = x1r + x3i;
	x0i = x1i - x3r;
	a[j3] = -wn4r * (x0r + x0i);
	a[j3 + 1] = -wn4r * (x0i - x0r);
}


void cftmdl2(int n, double *a, double *w)
{
	int j, j0, j1, j2, j3, k, kr, m, mh;
	double wn4r, wk1r, wk1i, wk3r, wk3i, wd1r, wd1i, wd3r, wd3i;
	double x0r, x0i, x1r, x1i, x2r, x2i, x3r, x3i, y0r, y0i, y2r, y2i;

	mh = n >> 3;
	m = 2 * mh;
	wn4r = w[1];
	j1 = m;
	j2 = j1 + m;
	j3 = j2 + m;
	x0r = a[0] - a[j2 + 1];
	x0i = a[1] + a[j2];
	x1r = a[0] + a[j2 + 1];
	x1i = a[1] - a[j2];
	x2r = a[j1] - a[j3 + 1];
	x2i = a[j1 + 1] + a[j3];
	x3r = a[j1] + a[j3 + 1];
	x3i = a[j1 + 1] - a[j3];
	y0r = wn4r * (x2r - x2i);
	y0i = wn4r * (x2i + x2r);
	a[0] = x0r + y0r;
	a[1] = x0i + y0i;
	a[j1] = x0r - y0r;
	a[j1 + 1] = x0i - y0i;
	y0r = wn4r * (x3r - x3i);
	y0i = wn4r * (x3i + x3r);
	a[j2] = x1r - y0i;
	a[j2 + 1] = x1i + y0r;
	a[j3] = x1r + y0i;
	a[j3 + 1] = x1i - y0r;
	k = 0;
	kr = 2 * m;
	for (j = 2; j < mh; j += 2) {
		k += 4;
		wk1r = w[k];
		wk1i = w[k + 1];
		wk3r = w[k + 2];
		wk3i = w[k + 3];
		kr -= 4;
		wd1i = w[kr];
		wd1r = w[kr + 1];
		wd3i = w[kr + 2];
		wd3r = w[kr + 3];
		j1 = j + m;
		j2 = j1 + m;
		j3 = j2 + m;
		x0r = a[j] - a[j2 + 1];
		x0i = a[j + 1] + a[j2];
		x1r = a[j] + a[j2 + 1];
		x1i = a[j + 1] - a[j2];
		x2r = a[j1] - a[j3 + 1];
		x2i = a[j1 + 1] + a[j3];
		x3r = a[j1] + a[j3 + 1];
		x3i = a[j1 + 1] - a[j3];
		y0r = wk1r * x0r - wk1i * x0i;
		y0i = wk1r * x0i + wk1i * x0r;
		y2r = wd1r * x2r - wd1i * x2i;
		y2i = wd1r * x2i + wd1i * x2r;
		a[j] = y0r + y2r;
		a[j + 1] = y0i + y2i;
		a[j1] = y0r - y2r;
		a[j1 + 1] = y0i - y2i;
		y0r = wk3r * x1r + wk3i * x1i;
		y0i = wk3r * x1i - wk3i * x1r;
		y2r = wd3r * x3r + wd3i * x3i;
		y2i = wd3r * x3i - wd3i * x3r;
		a[j2] = y0r + y2r;
		a[j2 + 1] = y0i + y2i;
		a[j3] = y0r - y2r;
		a[j3 + 1] = y0i - y2i;
		j0 = m - j;
		j1 = j0 + m;
		j2 = j1 + m;
		j3 = j2 + m;
		x0r = a[j0] - a[j2 + 1];
		x0i = a[j0 + 1] + a[j2];
		x1r = a[j0] + a[j2 + 1];
		x1i = a[j0 + 1] - a[j2];
		x2r = a[j1] - a[j3 + 1];
		x2i = a[j1 + 1] + a[j3];
		x3r = a[j1] + a[j3 + 1];
		x3i = a[j1 + 1] - a[j3];
		y0r = wd1i * x0r - wd1r * x0i;
		y0i = wd1i * x0i + wd1r * x0r;
		y2r = wk1i * x2r - wk1r * x2i;
		y2i = wk1i * x2i + wk1r * x2r;
		a[j0] = y0r + y2r;
		a[j0 + 1] = y0i + y2i;
		a[j1] = y0r - y2r;
		a[j1 + 1] = y0i - y2i;
		y0r = wd3i * x1r + wd3r * x1i;
		y0i = wd3i * x1i - wd3r * x1r;
		y2r = wk3i * x3r + wk3r * x3i;
		y2i = wk3i * x3i - wk3r * x3r;
		a[j2] = y0r + y2r;
		a[j2 + 1] = y0i + y2i;
		a[j3] = y0r - y2r;
		a[j3 + 1] = y0i - y2i;
	}
	wk1r = w[m];
	wk1i = w[m + 1];
	j0 = mh;
	j1 = j0 + m;
	j2 = j1 + m;
	j3 = j2 + m;
	x0r = a[j0] - a[j2 + 1];
	x0i = a[j0 + 1] + a[j2];
	x1r = a[j0] + a[j2 + 1];
	x1i = a[j0 + 1] - a[j2];
	x2r = a[j1] - a[j3 + 1];
	x2i = a[j1 + 1] + a[j3];
	x3r = a[j1] + a[j3 + 1];
	x3i = a[j1 + 1] - a[j3];
	y0r = wk1r * x0r - wk1i * x0i;
	y0i = wk1r * x0i + wk1i * x0r;
	y2r = wk1i * x2r - wk1r * x2i;
	y2i = wk1i * x2i + wk1r * x2r;
	a[j0] = y0r + y2r;
	a[j0 + 1] = y0i + y2i;
	a[j1] = y0r - y2r;
	a[j1 + 1] = y0i - y2i;
	y0r = wk1i * x1r - wk1r * x1i;
	y0i = wk1i * x1i + wk1r * x1r;
	y2r = wk1r * x3r - wk1i * x3i;
	y2i = wk1r * x3i + wk1i * x3r;
	a[j2] = y0r - y2r;
	a[j2 + 1] = y0i - y2i;
	a[j3] = y0r + y2r;
	a[j3 + 1] = y0i + y2i;
}


void cftfx41(int n, double *a, int nw, double *w)
{
	void cftf161(double *a, double *w);
	void cftf162(double *a, double *w);
	void cftf081(double *a, double *w);
	void cftf082(double *a, double *w);

	if (n == 128) {
		cftf161(a, &w[nw - 8]);
		cftf162(&a[32], &w[nw - 32]);
		cftf161(&a[64], &w[nw - 8]);
		cftf161(&a[96], &w[nw - 8]);
	} else {
		cftf081(a, &w[nw - 8]);
		cftf082(&a[16], &w[nw - 8]);
		cftf081(&a[32], &w[nw - 8]);
		cftf081(&a[48], &w[nw - 8]);
	}
}


void cftf161(double *a, double *w)
{
	double wn4r, wk1r, wk1i,
		x0r, x0i, x1r, x1i, x2r, x2i, x3r, x3i,
		y0r, y0i, y1r, y1i, y2r, y2i, y3r, y3i,
		y4r, y4i, y5r, y5i, y6r, y6i, y7r, y7i,
		y8r, y8i, y9r, y9i, y10r, y10i, y11r, y11i,
		y12r, y12i, y13r, y13i, y14r, y14i, y15r, y15i;

	wn4r = w[1];
	wk1r = w[2];
	wk1i = w[3];
	x0r = a[0] + a[16];
	x0i = a[1] + a[17];
	x1r = a[0] - a[16];
	x1i = a[1] - a[17];
	x2r = a[8] + a[24];
	x2i = a[9] + a[25];
	x3r = a[8] - a[24];
	x3i = a[9] - a[25];
	y0r = x0r + x2r;
	y0i = x0i + x2i;
	y4r = x0r - x2r;
	y4i = x0i - x2i;
	y8r = x1r - x3i;
	y8i = x1i + x3r;
	y12r = x1r + x3i;
	y12i = x1i - x3r;
	x0r = a[2] + a[18];
	x0i = a[3] + a[19];
	x1r = a[2] - a[18];
	x1i = a[3] - a[19];
	x2r = a[10] + a[26];
	x2i = a[11] + a[27];
	x3r = a[10] - a[26];
	x3i = a[11] - a[27];
	y1r = x0r + x2r;
	y1i = x0i + x2i;
	y5r = x0r - x2r;
	y5i = x0i - x2i;
	x0r = x1r - x3i;
	x0i = x1i + x3r;
	y9r = wk1r * x0r - wk1i * x0i;
	y9i = wk1r * x0i + wk1i * x0r;
	x0r = x1r + x3i;
	x0i = x1i - x3r;
	y13r = wk1i * x0r - wk1r * x0i;
	y13i = wk1i * x0i + wk1r * x0r;
	x0r = a[4] + a[20];
	x0i = a[5] + a[21];
	x1r = a[4] - a[20];
	x1i = a[5] - a[21];
	x2r = a[12] + a[28];
	x2i = a[13] + a[29];
	x3r = a[12] - a[28];
	x3i = a[13] - a[29];
	y2r = x0r + x2r;
	y2i = x0i + x2i;
	y6r = x0r - x2r;
	y6i = x0i - x2i;
	x0r = x1r - x3i;
	x0i = x1i + x3r;
	y10r = wn4r * (x0r - x0i);
	y10i = wn4r * (x0i + x0r);
	x0r = x1r + x3i;
	x0i = x1i - x3r;
	y14r = wn4r * (x0r + x0i);
	y14i = wn4r * (x0i - x0r);
	x0r = a[6] + a[22];
	x0i = a[7] + a[23];
	x1r = a[6] - a[22];
	x1i = a[7] - a[23];
	x2r = a[14] + a[30];
	x2i = a[15] + a[31];
	x3r = a[14] - a[30];
	x3i = a[15] - a[31];
	y3r = x0r + x2r;
	y3i = x0i + x2i;
	y7r = x0r - x2r;
	y7i = x0i - x2i;
	x0r = x1r - x3i;
	x0i = x1i + x3r;
	y11r = wk1i * x0r - wk1r * x0i;
	y11i = wk1i * x0i + wk1r * x0r;
	x0r = x1r + x3i;
	x0i = x1i - x3r;
	y15r = wk1r * x0r - wk1i * x0i;
	y15i = wk1r * x0i + wk1i * x0r;
	x0r = y12r - y14r;
	x0i = y12i - y14i;
	x1r = y12r + y14r;
	x1i = y12i + y14i;
	x2r = y13r - y15r;
	x2i = y13i - y15i;
	x3r = y13r + y15r;
	x3i = y13i + y15i;
	a[24] = x0r + x2r;
	a[25] = x0i + x2i;
	a[26] = x0r - x2r;
	a[27] = x0i - x2i;
	a[28] = x1r - x3i;
	a[29] = x1i + x3r;
	a[30] = x1r + x3i;
	a[31] = x1i - x3r;
	x0r = y8r + y10r;
	x0i = y8i + y10i;
	x1r = y8r - y10r;
	x1i = y8i - y10i;
	x2r = y9r + y11r;
	x2i = y9i + y11i;
	x3r = y9r - y11r;
	x3i = y9i - y11i;
	a[16] = x0r + x2r;
	a[17] = x0i + x2i;
	a[18] = x0r - x2r;
	a[19] = x0i - x2i;
	a[20] = x1r - x3i;
	a[21] = x1i + x3r;
	a[22] = x1r + x3i;
	a[23] = x1i - x3r;
	x0r = y5r - y7i;
	x0i = y5i + y7r;
	x2r = wn4r * (x0r - x0i);
	x2i = wn4r * (x0i + x0r);
	x0r = y5r + y7i;
	x0i = y5i - y7r;
	x3r = wn4r * (x0r - x0i);
	x3i = wn4r * (x0i + x0r);
	x0r = y4r - y6i;
	x0i = y4i + y6r;
	x1r = y4r + y6i;
	x1i = y4i - y6r;
	a[8] = x0r + x2r;
	a[9] = x0i + x2i;
	a[10] = x0r - x2r;
	a[11] = x0i - x2i;
	a[12] = x1r - x3i;
	a[13] = x1i + x3r;
	a[14] = x1r + x3i;
	a[15] = x1i - x3r;
	x0r = y0r + y2r;
	x0i = y0i + y2i;
	x1r = y0r - y2r;
	x1i = y0i - y2i;
	x2r = y1r + y3r;
	x2i = y1i + y3i;
	x3r = y1r - y3r;
	x3i = y1i - y3i;
	a[0] = x0r + x2r;
	a[1] = x0i + x2i;
	a[2] = x0r - x2r;
	a[3] = x0i - x2i;
	a[4] = x1r - x3i;
	a[5] = x1i + x3r;
	a[6] = x1r + x3i;
	a[7] = x1i - x3r;
}


void cftf162(double *a, double *w)
{
	double wn4r, wk1r, wk1i, wk2r, wk2i, wk3r, wk3i,
		x0r, x0i, x1r, x1i, x2r, x2i,
		y0r, y0i, y1r, y1i, y2r, y2i, y3r, y3i,
		y4r, y4i, y5r, y5i, y6r, y6i, y7r, y7i,
		y8r, y8i, y9r, y9i, y10r, y10i, y11r, y11i,
		y12r, y12i, y13r, y13i, y14r, y14i, y15r, y15i;

	wn4r = w[1];
	wk1r = w[4];
	wk1i = w[5];
	wk3r = w[6];
	wk3i = -w[7];
	wk2r = w[8];
	wk2i = w[9];
	x1r = a[0] - a[17];
	x1i = a[1] + a[16];
	x0r = a[8] - a[25];
	x0i = a[9] + a[24];
	x2r = wn4r * (x0r - x0i);
	x2i = wn4r * (x0i + x0r);
	y0r = x1r + x2r;
	y0i = x1i + x2i;
	y4r = x1r - x2r;
	y4i = x1i - x2i;
	x1r = a[0] + a[17];
	x1i = a[1] - a[16];
	x0r = a[8] + a[25];
	x0i = a[9] - a[24];
	x2r = wn4r * (x0r - x0i);
	x2i = wn4r * (x0i + x0r);
	y8r = x1r - x2i;
	y8i = x1i + x2r;
	y12r = x1r + x2i;
	y12i = x1i - x2r;
	x0r = a[2] - a[19];
	x0i = a[3] + a[18];
	x1r = wk1r * x0r - wk1i * x0i;
	x1i = wk1r * x0i + wk1i * x0r;
	x0r = a[10] - a[27];
	x0i = a[11] + a[26];
	x2r = wk3i * x0r - wk3r * x0i;
	x2i = wk3i * x0i + wk3r * x0r;
	y1r = x1r + x2r;
	y1i = x1i + x2i;
	y5r = x1r - x2r;
	y5i = x1i - x2i;
	x0r = a[2] + a[19];
	x0i = a[3] - a[18];
	x1r = wk3r * x0r - wk3i * x0i;
	x1i = wk3r * x0i + wk3i * x0r;
	x0r = a[10] + a[27];
	x0i = a[11] - a[26];
	x2r = wk1r * x0r + wk1i * x0i;
	x2i = wk1r * x0i - wk1i * x0r;
	y9r = x1r - x2r;
	y9i = x1i - x2i;
	y13r = x1r + x2r;
	y13i = x1i + x2i;
	x0r = a[4] - a[21];
	x0i = a[5] + a[20];
	x1r = wk2r * x0r - wk2i * x0i;
	x1i = wk2r * x0i + wk2i * x0r;
	x0r = a[12] - a[29];
	x0i = a[13] + a[28];
	x2r = wk2i * x0r - wk2r * x0i;
	x2i = wk2i * x0i + wk2r * x0r;
	y2r = x1r + x2r;
	y2i = x1i + x2i;
	y6r = x1r - x2r;
	y6i = x1i - x2i;
	x0r = a[4] + a[21];
	x0i = a[5] - a[20];
	x1r = wk2i * x0r - wk2r * x0i;
	x1i = wk2i * x0i + wk2r * x0r;
	x0r = a[12] + a[29];
	x0i = a[13] - a[28];
	x2r = wk2r * x0r - wk2i * x0i;
	x2i = wk2r * x0i + wk2i * x0r;
	y10r = x1r - x2r;
	y10i = x1i - x2i;
	y14r = x1r + x2r;
	y14i = x1i + x2i;
	x0r = a[6] - a[23];
	x0i = a[7] + a[22];
	x1r = wk3r * x0r - wk3i * x0i;
	x1i = wk3r * x0i + wk3i * x0r;
	x0r = a[14] - a[31];
	x0i = a[15] + a[30];
	x2r = wk1i * x0r - wk1r * x0i;
	x2i = wk1i * x0i + wk1r * x0r;
	y3r = x1r + x2r;
	y3i = x1i + x2i;
	y7r = x1r - x2r;
	y7i = x1i - x2i;
	x0r = a[6] + a[23];
	x0i = a[7] - a[22];
	x1r = wk1i * x0r + wk1r * x0i;
	x1i = wk1i * x0i - wk1r * x0r;
	x0r = a[14] + a[31];
	x0i = a[15] - a[30];
	x2r = wk3i * x0r - wk3r * x0i;
	x2i = wk3i * x0i + wk3r * x0r;
	y11r = x1r + x2r;
	y11i = x1i + x2i;
	y15r = x1r - x2r;
	y15i = x1i - x2i;
	x1r = y0r + y2r;
	x1i = y0i + y2i;
	x2r = y1r + y3r;
	x2i = y1i + y3i;
	a[0] = x1r + x2r;
	a[1] = x1i + x2i;
	a[2] = x1r - x2r;
	a[3] = x1i - x2i;
	x1r = y0r - y2r;
	x1i = y0i - y2i;
	x2r = y1r - y3r;
	x2i = y1i - y3i;
	a[4] = x1r - x2i;
	a[5] = x1i + x2r;
	a[6] = x1r + x2i;
	a[7] = x1i - x2r;
	x1r = y4r - y6i;
	x1i = y4i + y6r;
	x0r = y5r - y7i;
	x0i = y5i + y7r;
	x2r = wn4r * (x0r - x0i);
	x2i = wn4r * (x0i + x0r);
	a[8] = x1r + x2r;
	a[9] = x1i + x2i;
	a[10] = x1r - x2r;
	a[11] = x1i - x2i;
	x1r = y4r + y6i;
	x1i = y4i - y6r;
	x0r = y5r + y7i;
	x0i = y5i - y7r;
	x2r = wn4r * (x0r - x0i);
	x2i = wn4r * (x0i + x0r);
	a[12] = x1r - x2i;
	a[13] = x1i + x2r;
	a[14] = x1r + x2i;
	a[15] = x1i - x2r;
	x1r = y8r + y10r;
	x1i = y8i + y10i;
	x2r = y9r - y11r;
	x2i = y9i - y11i;
	a[16] = x1r + x2r;
	a[17] = x1i + x2i;
	a[18] = x1r - x2r;
	a[19] = x1i - x2i;
	x1r = y8r - y10r;
	x1i = y8i - y10i;
	x2r = y9r + y11r;
	x2i = y9i + y11i;
	a[20] = x1r - x2i;
	a[21] = x1i + x2r;
	a[22] = x1r + x2i;
	a[23] = x1i - x2r;
	x1r = y12r - y14i;
	x1i = y12i + y14r;
	x0r = y13r + y15i;
	x0i = y13i - y15r;
	x2r = wn4r * (x0r - x0i);
	x2i = wn4r * (x0i + x0r);
	a[24] = x1r + x2r;
	a[25] = x1i + x2i;
	a[26] = x1r - x2r;
	a[27] = x1i - x2i;
	x1r = y12r + y14i;
	x1i = y12i - y14r;
	x0r = y13r - y15i;
	x0i = y13i + y15r;
	x2r = wn4r * (x0r - x0i);
	x2i = wn4r * (x0i + x0r);
	a[28] = x1r - x2i;
	a[29] = x1i + x2r;
	a[30] = x1r + x2i;
	a[31] = x1i - x2r;
}


void cftf081(double *a, double *w)
{
	double wn4r, x0r, x0i, x1r, x1i, x2r, x2i, x3r, x3i,
		y0r, y0i, y1r, y1i, y2r, y2i, y3r, y3i, y4r, y4i, y5r, y5i, y6r, y6i, y7r, y7i;

	wn4r = w[1];
	x0r = a[0] + a[8];
	x0i = a[1] + a[9];
	x1r = a[0] - a[8];
	x1i = a[1] - a[9];
	x2r = a[4] + a[12];
	x2i = a[5] + a[13];
	x3r = a[4] - a[12];
	x3i = a[5] - a[13];
	y0r = x0r + x2r;
	y0i = x0i + x2i;
	y2r = x0r - x2r;
	y2i = x0i - x2i;
	y1r = x1r - x3i;
	y1i = x1i + x3r;
	y3r = x1r + x3i;
	y3i = x1i - x3r;
	x0r = a[2] + a[10];
	x0i = a[3] + a[11];
	x1r = a[2] - a[10];
	x1i = a[3] - a[11];
	x2r = a[6] + a[14];
	x2i = a[7] + a[15];
	x3r = a[6] - a[14];
	x3i = a[7] - a[15];
	y4r = x0r + x2r;
	y4i = x0i + x2i;
	y6r = x0r - x2r;
	y6i = x0i - x2i;
	x0r = x1r - x3i;
	x0i = x1i + x3r;
	x2r = x1r + x3i;
	x2i = x1i - x3r;
	y5r = wn4r * (x0r - x0i);
	y5i = wn4r * (x0r + x0i);
	y7r = wn4r * (x2r - x2i);
	y7i = wn4r * (x2r + x2i);
	a[8] = y1r + y5r;
	a[9] = y1i + y5i;
	a[10] = y1r - y5r;
	a[11] = y1i - y5i;
	a[12] = y3r - y7i;
	a[13] = y3i + y7r;
	a[14] = y3r + y7i;
	a[15] = y3i - y7r;
	a[0] = y0r + y4r;
	a[1] = y0i + y4i;
	a[2] = y0r - y4r;
	a[3] = y0i - y4i;
	a[4] = y2r - y6i;
	a[5] = y2i + y6r;
	a[6] = y2r + y6i;
	a[7] = y2i - y6r;
}


void cftf082(double *a, double *w)
{
	double wn4r, wk1r, wk1i, x0r, x0i, x1r, x1i,
		y0r, y0i, y1r, y1i, y2r, y2i, y3r, y3i, y4r, y4i, y5r, y5i, y6r, y6i, y7r, y7i;

	wn4r = w[1];
	wk1r = w[2];
	wk1i = w[3];
	y0r = a[0] - a[9];
	y0i = a[1] + a[8];
	y1r = a[0] + a[9];
	y1i = a[1] - a[8];
	x0r = a[4] - a[13];
	x0i = a[5] + a[12];
	y2r = wn4r * (x0r - x0i);
	y2i = wn4r * (x0i + x0r);
	x0r = a[4] + a[13];
	x0i = a[5] - a[12];
	y3r = wn4r * (x0r - x0i);
	y3i = wn4r * (x0i + x0r);
	x0r = a[2] - a[11];
	x0i = a[3] + a[10];
	y4r = wk1r * x0r - wk1i * x0i;
	y4i = wk1r * x0i + wk1i * x0r;
	x0r = a[2] + a[11];
	x0i = a[3] - a[10];
	y5r = wk1i * x0r - wk1r * x0i;
	y5i = wk1i * x0i + wk1r * x0r;
	x0r = a[6] - a[15];
	x0i = a[7] + a[14];
	y6r = wk1i * x0r - wk1r * x0i;
	y6i = wk1i * x0i + wk1r * x0r;
	x0r = a[6] + a[15];
	x0i = a[7] - a[14];
	y7r = wk1r * x0r - wk1i * x0i;
	y7i = wk1r * x0i + wk1i * x0r;
	x0r = y0r + y2r;
	x0i = y0i + y2i;
	x1r = y4r + y6r;
	x1i = y4i + y6i;
	a[0] = x0r + x1r;
	a[1] = x0i + x1i;
	a[2] = x0r - x1r;
	a[3] = x0i - x1i;
	x0r = y0r - y2r;
	x0i = y0i - y2i;
	x1r = y4r - y6r;
	x1i = y4i - y6i;
	a[4] = x0r - x1i;
	a[5] = x0i + x1r;
	a[6] = x0r + x1i;
	a[7] = x0i - x1r;
	x0r = y1r - y3i;
	x0i = y1i + y3r;
	x1r = y5r - y7r;
	x1i = y5i - y7i;
	a[8] = x0r + x1r;
	a[9] = x0i + x1i;
	a[10] = x0r - x1r;
	a[11] = x0i - x1i;
	x0r = y1r + y3i;
	x0i = y1i - y3r;
	x1r = y5r + y7r;
	x1i = y5i + y7i;
	a[12] = x0r - x1i;
	a[13] = x0i + x1r;
	a[14] = x0r + x1i;
	a[15] = x0i - x1r;
}


void cftf040(double *a)
{
	double x0r, x0i, x1r, x1i, x2r, x2i, x3r, x3i;

	x0r = a[0] + a[4];
	x0i = a[1] + a[5];
	x1r = a[0] - a[4];
	x1i = a[1] - a[5];
	x2r = a[2] + a[6];
	x2i = a[3] + a[7];
	x3r = a[2] - a[6];
	x3i = a[3] - a[7];
	a[0] = x0r + x2r;
	a[1] = x0i + x2i;
	a[2] = x1r - x3i;
	a[3] = x1i + x3r;
	a[4] = x0r - x2r;
	a[5] = x0i - x2i;
	a[6] = x1r + x3i;
	a[7] = x1i - x3r;
}


void cftb040(double *a)
{
	double x0r, x0i, x1r, x1i, x2r, x2i, x3r, x3i;

	x0r = a[0] + a[4];
	x0i = a[1] + a[5];
	x1r = a[0] - a[4];
	x1i = a[1] - a[5];
	x2r = a[2] + a[6];
	x2i = a[3] + a[7];
	x3r = a[2] - a[6];
	x3i = a[3] - a[7];
	a[0] = x0r + x2r;
	a[1] = x0i + x2i;
	a[2] = x1r + x3i;
	a[3] = x1i - x3r;
	a[4] = x0r - x2r;
	a[5] = x0i - x2i;
	a[6] = x1r - x3i;
	a[7] = x1i + x3r;
}


void cftx020(double *a)
{
	double x0r, x0i;

	x0r = a[0] - a[2];
	x0i = a[1] - a[3];
	a[0] += a[2];
	a[1] += a[3];
	a[2] = x0r;
	a[3] = x0i;
}


void rftfsub(int n, double *a, int nc, double *c)
{
	int j, k, kk, ks, m;
	double wkr, wki, xr, xi, yr, yi;

	m = n >> 1;
	ks = 2 * nc / m;
	kk = 0;
	for (j = 2; j < m; j += 2) {
		k = n - j;
		kk += ks;
		wkr = 0.5 - c[nc - kk];
		wki = c[kk];
		xr = a[j] - a[k];
		xi = a[j + 1] + a[k + 1];
		yr = wkr * xr - wki * xi;
		yi = wkr * xi + wki * xr;
		a[j] -= yr;
		a[j + 1] -= yi;
		a[k] += yr;
		a[k + 1] -= yi;
	}
}


void rftbsub(int n, double *a, int nc, double *c)
{
	int j, k, kk, ks, m;
	double wkr, wki, xr, xi, yr, yi;

	m = n >> 1;
	ks = 2 * nc / m;
	kk = 0;
	for (j = 2; j < m; j += 2) {
		k = n - j;
		kk += ks;
		wkr = 0.5 - c[nc - kk];
		wki = c[kk];
		xr = a[j] - a[k];
		xi = a[j + 1] + a[k + 1];
		yr = wkr * xr + wki * xi;
		yi = wkr * xi - wki * xr;
		a[j] -= yr;
		a[j + 1] -= yi;
		a[k] += yr;
		a[k + 1] -= yi;
	}
}


void dctsub(int n, double *a, int nc, double *c)
{
	int j, k, kk, ks, m;
	double wkr, wki, xr;

	m = n >> 1;
	ks = nc / n;
	kk = 0;
	for (j = 1; j < m; j++) {
		k = n - j;
		kk += ks;
		wkr = c[kk] - c[nc - kk];
		wki = c[kk] + c[nc - kk];
		xr = wki * a[j] - wkr * a[k];
		a[j] = wkr * a[j] + wki * a[k];
		a[k] = xr;
	}
	a[m] *= c[0];
}


void dstsub(int n, double *a, int nc, double *c)
{
	int j, k, kk, ks, m;
	double wkr, wki, xr;

	m = n >> 1;
	ks = nc / n;
	kk = 0;
	for (j = 1; j < m; j++) {
		k = n - j;
		kk += ks;
		wkr = c[kk] - c[nc - kk];
		wki = c[kk] + c[nc - kk];
		xr = wki * a[k] - wkr * a[j];
		a[k] = wkr * a[k] + wki * a[j];
		a[j] = xr;
	}
	a[m] *= c[0];
}