Index: src/spectral/ooura_fft8g.c
===================================================================
--- src/spectral/ooura_fft8g.c	(revision 447d6cd942936312198a9ecb6335fa23c8da3a52)
+++ src/spectral/ooura_fft8g.c	(revision 447d6cd942936312198a9ecb6335fa23c8da3a52)
@@ -0,0 +1,1642 @@
+/*
+Fast Fourier/Cosine/Sine Transform
+    dimension   :one
+    data length :power of 2
+    decimation  :frequency
+    radix       :8, 4, 2
+    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 *);
+
+
+-------- 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 bitrv2(int n, int *ip, double *a);
+    void bitrv2conj(int n, int *ip, double *a);
+    void cftfsub(int n, double *a, double *w);
+    void cftbsub(int n, double *a, double *w);
+    
+    if (n > (ip[0] << 2)) {
+        makewt(n >> 2, ip, w);
+    }
+    if (n > 4) {
+        if (isgn >= 0) {
+            bitrv2(n, ip + 2, a);
+            cftfsub(n, a, w);
+        } else {
+            bitrv2conj(n, ip + 2, a);
+            cftbsub(n, a, w);
+        }
+    } else if (n == 4) {
+        cftfsub(n, a, 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 bitrv2(int n, int *ip, double *a);
+    void cftfsub(int n, double *a, double *w);
+    void cftbsub(int n, double *a, 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) {
+            bitrv2(n, ip + 2, a);
+            cftfsub(n, a, w);
+            rftfsub(n, a, nc, w + nw);
+        } else if (n == 4) {
+            cftfsub(n, a, 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);
+            bitrv2(n, ip + 2, a);
+            cftbsub(n, a, w);
+        } else if (n == 4) {
+            cftfsub(n, a, 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 bitrv2(int n, int *ip, double *a);
+    void cftfsub(int n, double *a, double *w);
+    void cftbsub(int n, double *a, 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);
+            bitrv2(n, ip + 2, a);
+            cftbsub(n, a, w);
+        } else if (n == 4) {
+            cftfsub(n, a, w);
+        }
+    }
+    dctsub(n, a, nc, w + nw);
+    if (isgn >= 0) {
+        if (n > 4) {
+            bitrv2(n, ip + 2, a);
+            cftfsub(n, a, w);
+            rftfsub(n, a, nc, w + nw);
+        } else if (n == 4) {
+            cftfsub(n, a, 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 bitrv2(int n, int *ip, double *a);
+    void cftfsub(int n, double *a, double *w);
+    void cftbsub(int n, double *a, 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);
+            bitrv2(n, ip + 2, a);
+            cftbsub(n, a, w);
+        } else if (n == 4) {
+            cftfsub(n, a, w);
+        }
+    }
+    dstsub(n, a, nc, w + nw);
+    if (isgn >= 0) {
+        if (n > 4) {
+            bitrv2(n, ip + 2, a);
+            cftfsub(n, a, w);
+            rftfsub(n, a, nc, w + nw);
+        } else if (n == 4) {
+            cftfsub(n, a, 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 bitrv2(int n, int *ip, double *a);
+    void cftfsub(int n, double *a, 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) {
+            bitrv2(m, ip + 2, a);
+            cftfsub(m, a, w);
+            rftfsub(m, a, nc, w + nw);
+        } else if (m == 4) {
+            cftfsub(m, a, 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) {
+                bitrv2(m, ip + 2, t);
+                cftfsub(m, t, w);
+                rftfsub(m, t, nc, w + nw);
+            } else if (m == 4) {
+                cftfsub(m, t, 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 bitrv2(int n, int *ip, double *a);
+    void cftfsub(int n, double *a, 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) {
+            bitrv2(m, ip + 2, a);
+            cftfsub(m, a, w);
+            rftfsub(m, a, nc, w + nw);
+        } else if (m == 4) {
+            cftfsub(m, a, 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) {
+                bitrv2(m, ip + 2, t);
+                cftfsub(m, t, w);
+                rftfsub(m, t, nc, w + nw);
+            } else if (m == 4) {
+                cftfsub(m, t, 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 bitrv2(int n, int *ip, double *a);
+    int j, nwh;
+    double delta, x, y;
+    
+    ip[0] = nw;
+    ip[1] = 1;
+    if (nw > 2) {
+        nwh = nw >> 1;
+        delta = atan(1.0) / nwh;
+        w[0] = 1;
+        w[1] = 0;
+        w[nwh] = cos(delta * nwh);
+        w[nwh + 1] = w[nwh];
+        if (nwh > 2) {
+            for (j = 2; j < nwh; j += 2) {
+                x = cos(delta * j);
+                y = sin(delta * j);
+                w[j] = x;
+                w[j + 1] = y;
+                w[nw - j] = y;
+                w[nw - j + 1] = x;
+            }
+            for (j = nwh - 2; j >= 2; j -= 2) {
+                x = w[2 * j];
+                y = w[2 * j + 1];
+                w[nwh + j] = x;
+                w[nwh + j + 1] = y;
+            }
+            bitrv2(nw, ip + 2, w);
+        }
+    }
+}
+
+
+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 -------- */
+
+
+void bitrv2(int n, int *ip, double *a)
+{
+    int j, j1, k, k1, l, m, m2;
+    double xr, xi, yr, yi;
+    
+    ip[0] = 0;
+    l = n;
+    m = 1;
+    while ((m << 3) < l) {
+        l >>= 1;
+        for (j = 0; j < m; j++) {
+            ip[m + j] = ip[j] + l;
+        }
+        m <<= 1;
+    }
+    m2 = 2 * m;
+    if ((m << 3) == l) {
+        for (k = 0; k < m; k++) {
+            for (j = 0; j < k; j++) {
+                j1 = 2 * j + ip[k];
+                k1 = 2 * k + ip[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 += m2;
+                k1 += 2 * m2;
+                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 += m2;
+                k1 -= m2;
+                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 += m2;
+                k1 += 2 * m2;
+                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 * k + m2 + ip[k];
+            k1 = j1 + m2;
+            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 = 1; k < m; k++) {
+            for (j = 0; j < k; j++) {
+                j1 = 2 * j + ip[k];
+                k1 = 2 * k + ip[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 += m2;
+                k1 += m2;
+                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, m2;
+    double xr, xi, yr, yi;
+    
+    ip[0] = 0;
+    l = n;
+    m = 1;
+    while ((m << 3) < l) {
+        l >>= 1;
+        for (j = 0; j < m; j++) {
+            ip[m + j] = ip[j] + l;
+        }
+        m <<= 1;
+    }
+    m2 = 2 * m;
+    if ((m << 3) == l) {
+        for (k = 0; k < m; k++) {
+            for (j = 0; j < k; j++) {
+                j1 = 2 * j + ip[k];
+                k1 = 2 * k + ip[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 += m2;
+                k1 += 2 * m2;
+                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 += m2;
+                k1 -= m2;
+                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 += m2;
+                k1 += 2 * m2;
+                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 = 2 * k + ip[k];
+            a[k1 + 1] = -a[k1 + 1];
+            j1 = k1 + m2;
+            k1 = j1 + m2;
+            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 += m2;
+            a[k1 + 1] = -a[k1 + 1];
+        }
+    } else {
+        a[1] = -a[1];
+        a[m2 + 1] = -a[m2 + 1];
+        for (k = 1; k < m; k++) {
+            for (j = 0; j < k; j++) {
+                j1 = 2 * j + ip[k];
+                k1 = 2 * k + ip[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 += m2;
+                k1 += m2;
+                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 = 2 * k + ip[k];
+            a[k1 + 1] = -a[k1 + 1];
+            a[k1 + m2 + 1] = -a[k1 + m2 + 1];
+        }
+    }
+}
+
+
+void cftfsub(int n, double *a, double *w)
+{
+    void cft1st(int n, double *a, double *w);
+    void cftmdl(int n, int l, double *a, double *w);
+    int j, j1, j2, j3, l;
+    double x0r, x0i, x1r, x1i, x2r, x2i, x3r, x3i;
+    
+    l = 2;
+    if (n >= 16) {
+        cft1st(n, a, w);
+        l = 16;
+        while ((l << 3) <= n) {
+            cftmdl(n, l, a, w);
+            l <<= 3;
+        }
+    }
+    if ((l << 1) < n) {
+        for (j = 0; j < l; j += 2) {
+            j1 = j + l;
+            j2 = j1 + l;
+            j3 = j2 + l;
+            x0r = a[j] + a[j1];
+            x0i = a[j + 1] + a[j1 + 1];
+            x1r = a[j] - a[j1];
+            x1i = a[j + 1] - a[j1 + 1];
+            x2r = a[j2] + a[j3];
+            x2i = a[j2 + 1] + a[j3 + 1];
+            x3r = a[j2] - a[j3];
+            x3i = a[j2 + 1] - a[j3 + 1];
+            a[j] = x0r + x2r;
+            a[j + 1] = x0i + x2i;
+            a[j2] = x0r - x2r;
+            a[j2 + 1] = x0i - x2i;
+            a[j1] = x1r - x3i;
+            a[j1 + 1] = x1i + x3r;
+            a[j3] = x1r + x3i;
+            a[j3 + 1] = x1i - x3r;
+        }
+    } else if ((l << 1) == n) {
+        for (j = 0; j < l; j += 2) {
+            j1 = j + l;
+            x0r = a[j] - a[j1];
+            x0i = a[j + 1] - a[j1 + 1];
+            a[j] += a[j1];
+            a[j + 1] += a[j1 + 1];
+            a[j1] = x0r;
+            a[j1 + 1] = x0i;
+        }
+    }
+}
+
+
+void cftbsub(int n, double *a, double *w)
+{
+    void cft1st(int n, double *a, double *w);
+    void cftmdl(int n, int l, double *a, double *w);
+    int j, j1, j2, j3, j4, j5, j6, j7, l;
+    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;
+    
+    l = 2;
+    if (n > 16) {
+        cft1st(n, a, w);
+        l = 16;
+        while ((l << 3) < n) {
+            cftmdl(n, l, a, w);
+            l <<= 3;
+        }
+    }
+    if ((l << 2) < n) {
+        wn4r = w[2];
+        for (j = 0; j < l; j += 2) {
+            j1 = j + l;
+            j2 = j1 + l;
+            j3 = j2 + l;
+            j4 = j3 + l;
+            j5 = j4 + l;
+            j6 = j5 + l;
+            j7 = j6 + l;
+            x0r = a[j] + a[j1];
+            x0i = -a[j + 1] - a[j1 + 1];
+            x1r = a[j] - a[j1];
+            x1i = -a[j + 1] + a[j1 + 1];
+            x2r = a[j2] + a[j3];
+            x2i = a[j2 + 1] + a[j3 + 1];
+            x3r = a[j2] - a[j3];
+            x3i = a[j2 + 1] - a[j3 + 1];
+            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[j4] + a[j5];
+            x0i = a[j4 + 1] + a[j5 + 1];
+            x1r = a[j4] - a[j5];
+            x1i = a[j4 + 1] - a[j5 + 1];
+            x2r = a[j6] + a[j7];
+            x2i = a[j6 + 1] + a[j7 + 1];
+            x3r = a[j6] - a[j7];
+            x3i = a[j6 + 1] - a[j7 + 1];
+            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[j1] = y1r + y5r;
+            a[j1 + 1] = y1i - y5i;
+            a[j5] = y1r - y5r;
+            a[j5 + 1] = y1i + y5i;
+            a[j3] = y3r - y7i;
+            a[j3 + 1] = y3i - y7r;
+            a[j7] = y3r + y7i;
+            a[j7 + 1] = y3i + y7r;
+            a[j] = y0r + y4r;
+            a[j + 1] = y0i - y4i;
+            a[j4] = y0r - y4r;
+            a[j4 + 1] = y0i + y4i;
+            a[j2] = y2r - y6i;
+            a[j2 + 1] = y2i - y6r;
+            a[j6] = y2r + y6i;
+            a[j6 + 1] = y2i + y6r;
+        }
+    } else if ((l << 2) == n) {
+        for (j = 0; j < l; j += 2) {
+            j1 = j + l;
+            j2 = j1 + l;
+            j3 = j2 + l;
+            x0r = a[j] + a[j1];
+            x0i = -a[j + 1] - a[j1 + 1];
+            x1r = a[j] - a[j1];
+            x1i = -a[j + 1] + a[j1 + 1];
+            x2r = a[j2] + a[j3];
+            x2i = a[j2 + 1] + a[j3 + 1];
+            x3r = a[j2] - a[j3];
+            x3i = a[j2 + 1] - a[j3 + 1];
+            a[j] = x0r + x2r;
+            a[j + 1] = x0i - x2i;
+            a[j2] = x0r - x2r;
+            a[j2 + 1] = x0i + x2i;
+            a[j1] = x1r - x3i;
+            a[j1 + 1] = x1i - x3r;
+            a[j3] = x1r + x3i;
+            a[j3 + 1] = x1i + x3r;
+        }
+    } else {
+        for (j = 0; j < l; j += 2) {
+            j1 = j + l;
+            x0r = a[j] - a[j1];
+            x0i = -a[j + 1] + a[j1 + 1];
+            a[j] += a[j1];
+            a[j + 1] = -a[j + 1] - a[j1 + 1];
+            a[j1] = x0r;
+            a[j1 + 1] = x0i;
+        }
+    }
+}
+
+
+void cft1st(int n, double *a, double *w)
+{
+    int j, k1;
+    double wn4r, wtmp, wk1r, wk1i, wk2r, wk2i, wk3r, wk3i, 
+        wk4r, wk4i, wk5r, wk5i, wk6r, wk6i, wk7r, wk7i;
+    double 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[2];
+    x0r = a[0] + a[2];
+    x0i = a[1] + a[3];
+    x1r = a[0] - a[2];
+    x1i = a[1] - a[3];
+    x2r = a[4] + a[6];
+    x2i = a[5] + a[7];
+    x3r = a[4] - a[6];
+    x3i = a[5] - a[7];
+    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[8] + a[10];
+    x0i = a[9] + a[11];
+    x1r = a[8] - a[10];
+    x1i = a[9] - a[11];
+    x2r = a[12] + a[14];
+    x2i = a[13] + a[15];
+    x3r = a[12] - a[14];
+    x3i = a[13] - 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[2] = y1r + y5r;
+    a[3] = y1i + y5i;
+    a[10] = y1r - y5r;
+    a[11] = y1i - y5i;
+    a[6] = y3r - y7i;
+    a[7] = y3i + y7r;
+    a[14] = y3r + y7i;
+    a[15] = y3i - y7r;
+    a[0] = y0r + y4r;
+    a[1] = y0i + y4i;
+    a[8] = y0r - y4r;
+    a[9] = y0i - y4i;
+    a[4] = y2r - y6i;
+    a[5] = y2i + y6r;
+    a[12] = y2r + y6i;
+    a[13] = y2i - y6r;
+    if (n > 16) {
+        wk1r = w[4];
+        wk1i = w[5];
+        x0r = a[16] + a[18];
+        x0i = a[17] + a[19];
+        x1r = a[16] - a[18];
+        x1i = a[17] - a[19];
+        x2r = a[20] + a[22];
+        x2i = a[21] + a[23];
+        x3r = a[20] - a[22];
+        x3i = a[21] - a[23];
+        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[24] + a[26];
+        x0i = a[25] + a[27];
+        x1r = a[24] - a[26];
+        x1i = a[25] - a[27];
+        x2r = a[28] + a[30];
+        x2i = a[29] + a[31];
+        x3r = a[28] - a[30];
+        x3i = a[29] - a[31];
+        y4r = x0r + x2r;
+        y4i = x0i + x2i;
+        y6r = x0r - x2r;
+        y6i = x0i - x2i;
+        x0r = x1r - x3i;
+        x0i = x1i + x3r;
+        x2r = x1r + x3i;
+        x2i = x3r - x1i;
+        y5r = wk1i * x0r - wk1r * x0i;
+        y5i = wk1i * x0i + wk1r * x0r;
+        y7r = wk1r * x2r + wk1i * x2i;
+        y7i = wk1r * x2i - wk1i * x2r;
+        x0r = wk1r * y1r - wk1i * y1i;
+        x0i = wk1r * y1i + wk1i * y1r;
+        a[18] = x0r + y5r;
+        a[19] = x0i + y5i;
+        a[26] = y5i - x0i;
+        a[27] = x0r - y5r;
+        x0r = wk1i * y3r - wk1r * y3i;
+        x0i = wk1i * y3i + wk1r * y3r;
+        a[22] = x0r - y7r;
+        a[23] = x0i + y7i;
+        a[30] = y7i - x0i;
+        a[31] = x0r + y7r;
+        a[16] = y0r + y4r;
+        a[17] = y0i + y4i;
+        a[24] = y4i - y0i;
+        a[25] = y0r - y4r;
+        x0r = y2r - y6i;
+        x0i = y2i + y6r;
+        a[20] = wn4r * (x0r - x0i);
+        a[21] = wn4r * (x0i + x0r);
+        x0r = y6r - y2i;
+        x0i = y2r + y6i;
+        a[28] = wn4r * (x0r - x0i);
+        a[29] = wn4r * (x0i + x0r);
+        k1 = 4;
+        for (j = 32; j < n; j += 16) {
+            k1 += 4;
+            wk1r = w[k1];
+            wk1i = w[k1 + 1];
+            wk2r = w[k1 + 2];
+            wk2i = w[k1 + 3];
+            wtmp = 2 * wk2i;
+            wk3r = wk1r - wtmp * wk1i;
+            wk3i = wtmp * wk1r - wk1i;
+            wk4r = 1 - wtmp * wk2i;
+            wk4i = wtmp * wk2r;
+            wtmp = 2 * wk4i;
+            wk5r = wk3r - wtmp * wk1i;
+            wk5i = wtmp * wk1r - wk3i;
+            wk6r = wk2r - wtmp * wk2i;
+            wk6i = wtmp * wk2r - wk2i;
+            wk7r = wk1r - wtmp * wk3i;
+            wk7i = wtmp * wk3r - wk1i;
+            x0r = a[j] + a[j + 2];
+            x0i = a[j + 1] + a[j + 3];
+            x1r = a[j] - a[j + 2];
+            x1i = a[j + 1] - a[j + 3];
+            x2r = a[j + 4] + a[j + 6];
+            x2i = a[j + 5] + a[j + 7];
+            x3r = a[j + 4] - a[j + 6];
+            x3i = a[j + 5] - a[j + 7];
+            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[j + 8] + a[j + 10];
+            x0i = a[j + 9] + a[j + 11];
+            x1r = a[j + 8] - a[j + 10];
+            x1i = a[j + 9] - a[j + 11];
+            x2r = a[j + 12] + a[j + 14];
+            x2i = a[j + 13] + a[j + 15];
+            x3r = a[j + 12] - a[j + 14];
+            x3i = a[j + 13] - a[j + 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);
+            x0r = y1r + y5r;
+            x0i = y1i + y5i;
+            a[j + 2] = wk1r * x0r - wk1i * x0i;
+            a[j + 3] = wk1r * x0i + wk1i * x0r;
+            x0r = y1r - y5r;
+            x0i = y1i - y5i;
+            a[j + 10] = wk5r * x0r - wk5i * x0i;
+            a[j + 11] = wk5r * x0i + wk5i * x0r;
+            x0r = y3r - y7i;
+            x0i = y3i + y7r;
+            a[j + 6] = wk3r * x0r - wk3i * x0i;
+            a[j + 7] = wk3r * x0i + wk3i * x0r;
+            x0r = y3r + y7i;
+            x0i = y3i - y7r;
+            a[j + 14] = wk7r * x0r - wk7i * x0i;
+            a[j + 15] = wk7r * x0i + wk7i * x0r;
+            a[j] = y0r + y4r;
+            a[j + 1] = y0i + y4i;
+            x0r = y0r - y4r;
+            x0i = y0i - y4i;
+            a[j + 8] = wk4r * x0r - wk4i * x0i;
+            a[j + 9] = wk4r * x0i + wk4i * x0r;
+            x0r = y2r - y6i;
+            x0i = y2i + y6r;
+            a[j + 4] = wk2r * x0r - wk2i * x0i;
+            a[j + 5] = wk2r * x0i + wk2i * x0r;
+            x0r = y2r + y6i;
+            x0i = y2i - y6r;
+            a[j + 12] = wk6r * x0r - wk6i * x0i;
+            a[j + 13] = wk6r * x0i + wk6i * x0r;
+        }
+    }
+}
+
+
+void cftmdl(int n, int l, double *a, double *w)
+{
+    int j, j1, j2, j3, j4, j5, j6, j7, k, k1, m;
+    double wn4r, wtmp, wk1r, wk1i, wk2r, wk2i, wk3r, wk3i, 
+        wk4r, wk4i, wk5r, wk5i, wk6r, wk6i, wk7r, wk7i;
+    double x0r, x0i, x1r, x1i, x2r, x2i, x3r, x3i, 
+        y0r, y0i, y1r, y1i, y2r, y2i, y3r, y3i, 
+        y4r, y4i, y5r, y5i, y6r, y6i, y7r, y7i;
+    
+    m = l << 3;
+    wn4r = w[2];
+    for (j = 0; j < l; j += 2) {
+        j1 = j + l;
+        j2 = j1 + l;
+        j3 = j2 + l;
+        j4 = j3 + l;
+        j5 = j4 + l;
+        j6 = j5 + l;
+        j7 = j6 + l;
+        x0r = a[j] + a[j1];
+        x0i = a[j + 1] + a[j1 + 1];
+        x1r = a[j] - a[j1];
+        x1i = a[j + 1] - a[j1 + 1];
+        x2r = a[j2] + a[j3];
+        x2i = a[j2 + 1] + a[j3 + 1];
+        x3r = a[j2] - a[j3];
+        x3i = a[j2 + 1] - a[j3 + 1];
+        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[j4] + a[j5];
+        x0i = a[j4 + 1] + a[j5 + 1];
+        x1r = a[j4] - a[j5];
+        x1i = a[j4 + 1] - a[j5 + 1];
+        x2r = a[j6] + a[j7];
+        x2i = a[j6 + 1] + a[j7 + 1];
+        x3r = a[j6] - a[j7];
+        x3i = a[j6 + 1] - a[j7 + 1];
+        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[j1] = y1r + y5r;
+        a[j1 + 1] = y1i + y5i;
+        a[j5] = y1r - y5r;
+        a[j5 + 1] = y1i - y5i;
+        a[j3] = y3r - y7i;
+        a[j3 + 1] = y3i + y7r;
+        a[j7] = y3r + y7i;
+        a[j7 + 1] = y3i - y7r;
+        a[j] = y0r + y4r;
+        a[j + 1] = y0i + y4i;
+        a[j4] = y0r - y4r;
+        a[j4 + 1] = y0i - y4i;
+        a[j2] = y2r - y6i;
+        a[j2 + 1] = y2i + y6r;
+        a[j6] = y2r + y6i;
+        a[j6 + 1] = y2i - y6r;
+    }
+    if (m < n) {
+        wk1r = w[4];
+        wk1i = w[5];
+        for (j = m; j < l + m; j += 2) {
+            j1 = j + l;
+            j2 = j1 + l;
+            j3 = j2 + l;
+            j4 = j3 + l;
+            j5 = j4 + l;
+            j6 = j5 + l;
+            j7 = j6 + l;
+            x0r = a[j] + a[j1];
+            x0i = a[j + 1] + a[j1 + 1];
+            x1r = a[j] - a[j1];
+            x1i = a[j + 1] - a[j1 + 1];
+            x2r = a[j2] + a[j3];
+            x2i = a[j2 + 1] + a[j3 + 1];
+            x3r = a[j2] - a[j3];
+            x3i = a[j2 + 1] - a[j3 + 1];
+            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[j4] + a[j5];
+            x0i = a[j4 + 1] + a[j5 + 1];
+            x1r = a[j4] - a[j5];
+            x1i = a[j4 + 1] - a[j5 + 1];
+            x2r = a[j6] + a[j7];
+            x2i = a[j6 + 1] + a[j7 + 1];
+            x3r = a[j6] - a[j7];
+            x3i = a[j6 + 1] - a[j7 + 1];
+            y4r = x0r + x2r;
+            y4i = x0i + x2i;
+            y6r = x0r - x2r;
+            y6i = x0i - x2i;
+            x0r = x1r - x3i;
+            x0i = x1i + x3r;
+            x2r = x1r + x3i;
+            x2i = x3r - x1i;
+            y5r = wk1i * x0r - wk1r * x0i;
+            y5i = wk1i * x0i + wk1r * x0r;
+            y7r = wk1r * x2r + wk1i * x2i;
+            y7i = wk1r * x2i - wk1i * x2r;
+            x0r = wk1r * y1r - wk1i * y1i;
+            x0i = wk1r * y1i + wk1i * y1r;
+            a[j1] = x0r + y5r;
+            a[j1 + 1] = x0i + y5i;
+            a[j5] = y5i - x0i;
+            a[j5 + 1] = x0r - y5r;
+            x0r = wk1i * y3r - wk1r * y3i;
+            x0i = wk1i * y3i + wk1r * y3r;
+            a[j3] = x0r - y7r;
+            a[j3 + 1] = x0i + y7i;
+            a[j7] = y7i - x0i;
+            a[j7 + 1] = x0r + y7r;
+            a[j] = y0r + y4r;
+            a[j + 1] = y0i + y4i;
+            a[j4] = y4i - y0i;
+            a[j4 + 1] = y0r - y4r;
+            x0r = y2r - y6i;
+            x0i = y2i + y6r;
+            a[j2] = wn4r * (x0r - x0i);
+            a[j2 + 1] = wn4r * (x0i + x0r);
+            x0r = y6r - y2i;
+            x0i = y2r + y6i;
+            a[j6] = wn4r * (x0r - x0i);
+            a[j6 + 1] = wn4r * (x0i + x0r);
+        }
+        k1 = 4;
+        for (k = 2 * m; k < n; k += m) {
+            k1 += 4;
+            wk1r = w[k1];
+            wk1i = w[k1 + 1];
+            wk2r = w[k1 + 2];
+            wk2i = w[k1 + 3];
+            wtmp = 2 * wk2i;
+            wk3r = wk1r - wtmp * wk1i;
+            wk3i = wtmp * wk1r - wk1i;
+            wk4r = 1 - wtmp * wk2i;
+            wk4i = wtmp * wk2r;
+            wtmp = 2 * wk4i;
+            wk5r = wk3r - wtmp * wk1i;
+            wk5i = wtmp * wk1r - wk3i;
+            wk6r = wk2r - wtmp * wk2i;
+            wk6i = wtmp * wk2r - wk2i;
+            wk7r = wk1r - wtmp * wk3i;
+            wk7i = wtmp * wk3r - wk1i;
+            for (j = k; j < l + k; j += 2) {
+                j1 = j + l;
+                j2 = j1 + l;
+                j3 = j2 + l;
+                j4 = j3 + l;
+                j5 = j4 + l;
+                j6 = j5 + l;
+                j7 = j6 + l;
+                x0r = a[j] + a[j1];
+                x0i = a[j + 1] + a[j1 + 1];
+                x1r = a[j] - a[j1];
+                x1i = a[j + 1] - a[j1 + 1];
+                x2r = a[j2] + a[j3];
+                x2i = a[j2 + 1] + a[j3 + 1];
+                x3r = a[j2] - a[j3];
+                x3i = a[j2 + 1] - a[j3 + 1];
+                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[j4] + a[j5];
+                x0i = a[j4 + 1] + a[j5 + 1];
+                x1r = a[j4] - a[j5];
+                x1i = a[j4 + 1] - a[j5 + 1];
+                x2r = a[j6] + a[j7];
+                x2i = a[j6 + 1] + a[j7 + 1];
+                x3r = a[j6] - a[j7];
+                x3i = a[j6 + 1] - a[j7 + 1];
+                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);
+                x0r = y1r + y5r;
+                x0i = y1i + y5i;
+                a[j1] = wk1r * x0r - wk1i * x0i;
+                a[j1 + 1] = wk1r * x0i + wk1i * x0r;
+                x0r = y1r - y5r;
+                x0i = y1i - y5i;
+                a[j5] = wk5r * x0r - wk5i * x0i;
+                a[j5 + 1] = wk5r * x0i + wk5i * x0r;
+                x0r = y3r - y7i;
+                x0i = y3i + y7r;
+                a[j3] = wk3r * x0r - wk3i * x0i;
+                a[j3 + 1] = wk3r * x0i + wk3i * x0r;
+                x0r = y3r + y7i;
+                x0i = y3i - y7r;
+                a[j7] = wk7r * x0r - wk7i * x0i;
+                a[j7 + 1] = wk7r * x0i + wk7i * x0r;
+                a[j] = y0r + y4r;
+                a[j + 1] = y0i + y4i;
+                x0r = y0r - y4r;
+                x0i = y0i - y4i;
+                a[j4] = wk4r * x0r - wk4i * x0i;
+                a[j4 + 1] = wk4r * x0i + wk4i * x0r;
+                x0r = y2r - y6i;
+                x0i = y2i + y6r;
+                a[j2] = wk2r * x0r - wk2i * x0i;
+                a[j2 + 1] = wk2r * x0i + wk2i * x0r;
+                x0r = y2r + y6i;
+                x0i = y2i - y6r;
+                a[j6] = wk6r * x0r - wk6i * x0i;
+                a[j6 + 1] = wk6r * x0i + wk6i * x0r;
+            }
+        }
+    }
+}
+
+
+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;
+    
+    a[1] = -a[1];
+    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[j + 1];
+        a[k] += yr;
+        a[k + 1] = yi - a[k + 1];
+    }
+    a[m + 1] = -a[m + 1];
+}
+
+
+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];
+}
+
