From e76b5fa6896c09257181675bbf4cf47789d32927 Mon Sep 17 00:00:00 2001
From: Travis Oliphant <oliphant@enthought.com>
Date: Sat, 15 Dec 2007 18:54:52 +0000
Subject: Create a branch for io work in NumPy

---
 numpy/fft/fftpack.c | 1501 ---------------------------------------------------
 1 file changed, 1501 deletions(-)
 delete mode 100644 numpy/fft/fftpack.c

(limited to 'numpy/fft/fftpack.c')

diff --git a/numpy/fft/fftpack.c b/numpy/fft/fftpack.c
deleted file mode 100644
index 9c8fd118a..000000000
--- a/numpy/fft/fftpack.c
+++ /dev/null
@@ -1,1501 +0,0 @@
-/*
-fftpack.c : A set of FFT routines in C.
-Algorithmically based on Fortran-77 FFTPACK by Paul N. Swarztrauber (Version 4, 1985).
-
-*/
-
-/* isign is +1 for backward and -1 for forward transforms */
-
-#include <math.h>
-#include <stdio.h>
-#define DOUBLE
-
-#ifdef DOUBLE
-#define Treal double
-#else
-#define Treal float
-#endif
-
-
-#define ref(u,a) u[a]
-
-#define MAXFAC 13    /* maximum number of factors in factorization of n */
-#define NSPECIAL 4   /* number of factors for which we have special-case routines */
-
-#ifdef __cplusplus
-extern "C" {
-#endif
-
-
-/* ----------------------------------------------------------------------
-   passf2, passf3, passf4, passf5, passf. Complex FFT passes fwd and bwd.
----------------------------------------------------------------------- */
-
-static void passf2(int ido, int l1, const Treal cc[], Treal ch[], const Treal wa1[], int isign)
-  /* isign==+1 for backward transform */
-  {
-    int i, k, ah, ac;
-    Treal ti2, tr2;
-    if (ido <= 2) {
-      for (k=0; k<l1; k++) {
-        ah = k*ido;
-        ac = 2*k*ido;
-        ch[ah]              = ref(cc,ac) + ref(cc,ac + ido);
-        ch[ah + ido*l1]     = ref(cc,ac) - ref(cc,ac + ido);
-        ch[ah+1]            = ref(cc,ac+1) + ref(cc,ac + ido + 1);
-        ch[ah + ido*l1 + 1] = ref(cc,ac+1) - ref(cc,ac + ido + 1);
-      }
-    } else {
-      for (k=0; k<l1; k++) {
-        for (i=0; i<ido-1; i+=2) {
-          ah = i + k*ido;
-          ac = i + 2*k*ido;
-          ch[ah]   = ref(cc,ac) + ref(cc,ac + ido);
-          tr2      = ref(cc,ac) - ref(cc,ac + ido);
-          ch[ah+1] = ref(cc,ac+1) + ref(cc,ac + 1 + ido);
-          ti2      = ref(cc,ac+1) - ref(cc,ac + 1 + ido);
-          ch[ah+l1*ido+1] = wa1[i]*ti2 + isign*wa1[i+1]*tr2;
-          ch[ah+l1*ido]   = wa1[i]*tr2 - isign*wa1[i+1]*ti2;
-        }
-      }
-    }
-  } /* passf2 */
-
-
-static void passf3(int ido, int l1, const Treal cc[], Treal ch[],
-      const Treal wa1[], const Treal wa2[], int isign)
-  /* isign==+1 for backward transform */
-  {
-    static const Treal taur = -0.5;
-    static const Treal taui = 0.866025403784439;
-    int i, k, ac, ah;
-    Treal ci2, ci3, di2, di3, cr2, cr3, dr2, dr3, ti2, tr2;
-    if (ido == 2) {
-      for (k=1; k<=l1; k++) {
-        ac = (3*k - 2)*ido;
-        tr2 = ref(cc,ac) + ref(cc,ac + ido);
-        cr2 = ref(cc,ac - ido) + taur*tr2;
-        ah = (k - 1)*ido;
-        ch[ah] = ref(cc,ac - ido) + tr2;
-
-        ti2 = ref(cc,ac + 1) + ref(cc,ac + ido + 1);
-        ci2 = ref(cc,ac - ido + 1) + taur*ti2;
-        ch[ah + 1] = ref(cc,ac - ido + 1) + ti2;
-
-        cr3 = isign*taui*(ref(cc,ac) - ref(cc,ac + ido));
-        ci3 = isign*taui*(ref(cc,ac + 1) - ref(cc,ac + ido + 1));
-        ch[ah + l1*ido] = cr2 - ci3;
-        ch[ah + 2*l1*ido] = cr2 + ci3;
-        ch[ah + l1*ido + 1] = ci2 + cr3;
-        ch[ah + 2*l1*ido + 1] = ci2 - cr3;
-      }
-    } else {
-      for (k=1; k<=l1; k++) {
-        for (i=0; i<ido-1; i+=2) {
-          ac = i + (3*k - 2)*ido;
-          tr2 = ref(cc,ac) + ref(cc,ac + ido);
-          cr2 = ref(cc,ac - ido) + taur*tr2;
-          ah = i + (k-1)*ido;
-          ch[ah] = ref(cc,ac - ido) + tr2;
-          ti2 = ref(cc,ac + 1) + ref(cc,ac + ido + 1);
-          ci2 = ref(cc,ac - ido + 1) + taur*ti2;
-          ch[ah + 1] = ref(cc,ac - ido + 1) + ti2;
-          cr3 = isign*taui*(ref(cc,ac) - ref(cc,ac + ido));
-          ci3 = isign*taui*(ref(cc,ac + 1) - ref(cc,ac + ido + 1));
-          dr2 = cr2 - ci3;
-          dr3 = cr2 + ci3;
-          di2 = ci2 + cr3;
-          di3 = ci2 - cr3;
-          ch[ah + l1*ido + 1] = wa1[i]*di2 + isign*wa1[i+1]*dr2;
-          ch[ah + l1*ido] = wa1[i]*dr2 - isign*wa1[i+1]*di2;
-          ch[ah + 2*l1*ido + 1] = wa2[i]*di3 + isign*wa2[i+1]*dr3;
-          ch[ah + 2*l1*ido] = wa2[i]*dr3 - isign*wa2[i+1]*di3;
-        }
-      }
-    }
-  } /* passf3 */
-
-
-static void passf4(int ido, int l1, const Treal cc[], Treal ch[],
-      const Treal wa1[], const Treal wa2[], const Treal wa3[], int isign)
-  /* isign == -1 for forward transform and +1 for backward transform */
-  {
-    int i, k, ac, ah;
-    Treal ci2, ci3, ci4, cr2, cr3, cr4, ti1, ti2, ti3, ti4, tr1, tr2, tr3, tr4;
-    if (ido == 2) {
-      for (k=0; k<l1; k++) {
-        ac = 4*k*ido + 1;
-        ti1 = ref(cc,ac) - ref(cc,ac + 2*ido);
-        ti2 = ref(cc,ac) + ref(cc,ac + 2*ido);
-        tr4 = ref(cc,ac + 3*ido) - ref(cc,ac + ido);
-        ti3 = ref(cc,ac + ido) + ref(cc,ac + 3*ido);
-        tr1 = ref(cc,ac - 1) - ref(cc,ac + 2*ido - 1);
-        tr2 = ref(cc,ac - 1) + ref(cc,ac + 2*ido - 1);
-        ti4 = ref(cc,ac + ido - 1) - ref(cc,ac + 3*ido - 1);
-        tr3 = ref(cc,ac + ido - 1) + ref(cc,ac + 3*ido - 1);
-        ah = k*ido;
-        ch[ah] = tr2 + tr3;
-        ch[ah + 2*l1*ido] = tr2 - tr3;
-        ch[ah + 1] = ti2 + ti3;
-        ch[ah + 2*l1*ido + 1] = ti2 - ti3;
-        ch[ah + l1*ido] = tr1 + isign*tr4;
-        ch[ah + 3*l1*ido] = tr1 - isign*tr4;
-        ch[ah + l1*ido + 1] = ti1 + isign*ti4;
-        ch[ah + 3*l1*ido + 1] = ti1 - isign*ti4;
-      }
-    } else {
-      for (k=0; k<l1; k++) {
-        for (i=0; i<ido-1; i+=2) {
-          ac = i + 1 + 4*k*ido;
-          ti1 = ref(cc,ac) - ref(cc,ac + 2*ido);
-          ti2 = ref(cc,ac) + ref(cc,ac + 2*ido);
-          ti3 = ref(cc,ac + ido) + ref(cc,ac + 3*ido);
-          tr4 = ref(cc,ac + 3*ido) - ref(cc,ac + ido);
-          tr1 = ref(cc,ac - 1) - ref(cc,ac + 2*ido - 1);
-          tr2 = ref(cc,ac - 1) + ref(cc,ac + 2*ido - 1);
-          ti4 = ref(cc,ac + ido - 1) - ref(cc,ac + 3*ido - 1);
-          tr3 = ref(cc,ac + ido - 1) + ref(cc,ac + 3*ido - 1);
-          ah = i + k*ido;
-          ch[ah] = tr2 + tr3;
-          cr3 = tr2 - tr3;
-          ch[ah + 1] = ti2 + ti3;
-          ci3 = ti2 - ti3;
-          cr2 = tr1 + isign*tr4;
-          cr4 = tr1 - isign*tr4;
-          ci2 = ti1 + isign*ti4;
-          ci4 = ti1 - isign*ti4;
-          ch[ah + l1*ido] = wa1[i]*cr2 - isign*wa1[i + 1]*ci2;
-          ch[ah + l1*ido + 1] = wa1[i]*ci2 + isign*wa1[i + 1]*cr2;
-          ch[ah + 2*l1*ido] = wa2[i]*cr3 - isign*wa2[i + 1]*ci3;
-          ch[ah + 2*l1*ido + 1] = wa2[i]*ci3 + isign*wa2[i + 1]*cr3;
-          ch[ah + 3*l1*ido] = wa3[i]*cr4 -isign*wa3[i + 1]*ci4;
-          ch[ah + 3*l1*ido + 1] = wa3[i]*ci4 + isign*wa3[i + 1]*cr4;
-        }
-      }
-    }
-  } /* passf4 */
-
-
-static void passf5(int ido, int l1, const Treal cc[], Treal ch[],
-      const Treal wa1[], const Treal wa2[], const Treal wa3[], const Treal wa4[], int isign)
-  /* isign == -1 for forward transform and +1 for backward transform */
-  {
-    static const Treal tr11 = 0.309016994374947;
-    static const Treal ti11 = 0.951056516295154;
-    static const Treal tr12 = -0.809016994374947;
-    static const Treal ti12 = 0.587785252292473;
-    int i, k, ac, ah;
-    Treal ci2, ci3, ci4, ci5, di3, di4, di5, di2, cr2, cr3, cr5, cr4, ti2, ti3,
-        ti4, ti5, dr3, dr4, dr5, dr2, tr2, tr3, tr4, tr5;
-    if (ido == 2) {
-      for (k = 1; k <= l1; ++k) {
-        ac = (5*k - 4)*ido + 1;
-        ti5 = ref(cc,ac) - ref(cc,ac + 3*ido);
-        ti2 = ref(cc,ac) + ref(cc,ac + 3*ido);
-        ti4 = ref(cc,ac + ido) - ref(cc,ac + 2*ido);
-        ti3 = ref(cc,ac + ido) + ref(cc,ac + 2*ido);
-        tr5 = ref(cc,ac - 1) - ref(cc,ac + 3*ido - 1);
-        tr2 = ref(cc,ac - 1) + ref(cc,ac + 3*ido - 1);
-        tr4 = ref(cc,ac + ido - 1) - ref(cc,ac + 2*ido - 1);
-        tr3 = ref(cc,ac + ido - 1) + ref(cc,ac + 2*ido - 1);
-        ah = (k - 1)*ido;
-        ch[ah] = ref(cc,ac - ido - 1) + tr2 + tr3;
-        ch[ah + 1] = ref(cc,ac - ido) + ti2 + ti3;
-        cr2 = ref(cc,ac - ido - 1) + tr11*tr2 + tr12*tr3;
-        ci2 = ref(cc,ac - ido) + tr11*ti2 + tr12*ti3;
-        cr3 = ref(cc,ac - ido - 1) + tr12*tr2 + tr11*tr3;
-        ci3 = ref(cc,ac - ido) + tr12*ti2 + tr11*ti3;
-        cr5 = isign*(ti11*tr5 + ti12*tr4);
-        ci5 = isign*(ti11*ti5 + ti12*ti4);
-        cr4 = isign*(ti12*tr5 - ti11*tr4);
-        ci4 = isign*(ti12*ti5 - ti11*ti4);
-        ch[ah + l1*ido] = cr2 - ci5;
-        ch[ah + 4*l1*ido] = cr2 + ci5;
-        ch[ah + l1*ido + 1] = ci2 + cr5;
-        ch[ah + 2*l1*ido + 1] = ci3 + cr4;
-        ch[ah + 2*l1*ido] = cr3 - ci4;
-        ch[ah + 3*l1*ido] = cr3 + ci4;
-        ch[ah + 3*l1*ido + 1] = ci3 - cr4;
-        ch[ah + 4*l1*ido + 1] = ci2 - cr5;
-      }
-    } else {
-      for (k=1; k<=l1; k++) {
-        for (i=0; i<ido-1; i+=2) {
-          ac = i + 1 + (k*5 - 4)*ido;
-          ti5 = ref(cc,ac) - ref(cc,ac + 3*ido);
-          ti2 = ref(cc,ac) + ref(cc,ac + 3*ido);
-          ti4 = ref(cc,ac + ido) - ref(cc,ac + 2*ido);
-          ti3 = ref(cc,ac + ido) + ref(cc,ac + 2*ido);
-          tr5 = ref(cc,ac - 1) - ref(cc,ac + 3*ido - 1);
-          tr2 = ref(cc,ac - 1) + ref(cc,ac + 3*ido - 1);
-          tr4 = ref(cc,ac + ido - 1) - ref(cc,ac + 2*ido - 1);
-          tr3 = ref(cc,ac + ido - 1) + ref(cc,ac + 2*ido - 1);
-          ah = i + (k - 1)*ido;
-          ch[ah] = ref(cc,ac - ido - 1) + tr2 + tr3;
-          ch[ah + 1] = ref(cc,ac - ido) + ti2 + ti3;
-          cr2 = ref(cc,ac - ido - 1) + tr11*tr2 + tr12*tr3;
-
-          ci2 = ref(cc,ac - ido) + tr11*ti2 + tr12*ti3;
-          cr3 = ref(cc,ac - ido - 1) + tr12*tr2 + tr11*tr3;
-
-          ci3 = ref(cc,ac - ido) + tr12*ti2 + tr11*ti3;
-          cr5 = isign*(ti11*tr5 + ti12*tr4);
-          ci5 = isign*(ti11*ti5 + ti12*ti4);
-          cr4 = isign*(ti12*tr5 - ti11*tr4);
-          ci4 = isign*(ti12*ti5 - ti11*ti4);
-          dr3 = cr3 - ci4;
-          dr4 = cr3 + ci4;
-          di3 = ci3 + cr4;
-          di4 = ci3 - cr4;
-          dr5 = cr2 + ci5;
-          dr2 = cr2 - ci5;
-          di5 = ci2 - cr5;
-          di2 = ci2 + cr5;
-          ch[ah + l1*ido] = wa1[i]*dr2 - isign*wa1[i+1]*di2;
-          ch[ah + l1*ido + 1] = wa1[i]*di2 + isign*wa1[i+1]*dr2;
-          ch[ah + 2*l1*ido] = wa2[i]*dr3 - isign*wa2[i+1]*di3;
-          ch[ah + 2*l1*ido + 1] = wa2[i]*di3 + isign*wa2[i+1]*dr3;
-          ch[ah + 3*l1*ido] = wa3[i]*dr4 - isign*wa3[i+1]*di4;
-          ch[ah + 3*l1*ido + 1] = wa3[i]*di4 + isign*wa3[i+1]*dr4;
-          ch[ah + 4*l1*ido] = wa4[i]*dr5 - isign*wa4[i+1]*di5;
-          ch[ah + 4*l1*ido + 1] = wa4[i]*di5 + isign*wa4[i+1]*dr5;
-        }
-      }
-    }
-  } /* passf5 */
-
-
-static void passf(int *nac, int ido, int ip, int l1, int idl1,
-      Treal cc[], Treal ch[],
-      const Treal wa[], int isign)
-  /* isign is -1 for forward transform and +1 for backward transform */
-  {
-    int idij, idlj, idot, ipph, i, j, k, l, jc, lc, ik, idj, idl, inc,idp;
-    Treal wai, war;
-
-    idot = ido / 2;
-    /* nt = ip*idl1;*/
-    ipph = (ip + 1) / 2;
-    idp = ip*ido;
-    if (ido >= l1) {
-      for (j=1; j<ipph; j++) {
-        jc = ip - j;
-        for (k=0; k<l1; k++) {
-          for (i=0; i<ido; i++) {
-            ch[i + (k + j*l1)*ido] =
-                ref(cc,i + (j + k*ip)*ido) + ref(cc,i + (jc + k*ip)*ido);
-            ch[i + (k + jc*l1)*ido] =
-                ref(cc,i + (j + k*ip)*ido) - ref(cc,i + (jc + k*ip)*ido);
-          }
-        }
-      }
-      for (k=0; k<l1; k++)
-        for (i=0; i<ido; i++)
-          ch[i + k*ido] = ref(cc,i + k*ip*ido);
-    } else {
-      for (j=1; j<ipph; j++) {
-        jc = ip - j;
-        for (i=0; i<ido; i++) {
-          for (k=0; k<l1; k++) {
-            ch[i + (k + j*l1)*ido] = ref(cc,i + (j + k*ip)*ido) + ref(cc,i + (jc + k*
-                ip)*ido);
-            ch[i + (k + jc*l1)*ido] = ref(cc,i + (j + k*ip)*ido) - ref(cc,i + (jc + k*
-                ip)*ido);
-          }
-        }
-      }
-      for (i=0; i<ido; i++)
-        for (k=0; k<l1; k++)
-          ch[i + k*ido] = ref(cc,i + k*ip*ido);
-    }
-
-    idl = 2 - ido;
-    inc = 0;
-    for (l=1; l<ipph; l++) {
-      lc = ip - l;
-      idl += ido;
-      for (ik=0; ik<idl1; ik++) {
-        cc[ik + l*idl1] = ch[ik] + wa[idl - 2]*ch[ik + idl1];
-        cc[ik + lc*idl1] = isign*wa[idl-1]*ch[ik + (ip-1)*idl1];
-      }
-      idlj = idl;
-      inc += ido;
-      for (j=2; j<ipph; j++) {
-        jc = ip - j;
-        idlj += inc;
-        if (idlj > idp) idlj -= idp;
-        war = wa[idlj - 2];
-        wai = wa[idlj-1];
-        for (ik=0; ik<idl1; ik++) {
-          cc[ik + l*idl1] += war*ch[ik + j*idl1];
-          cc[ik + lc*idl1] += isign*wai*ch[ik + jc*idl1];
-        }
-      }
-    }
-    for (j=1; j<ipph; j++)
-      for (ik=0; ik<idl1; ik++)
-        ch[ik] += ch[ik + j*idl1];
-    for (j=1; j<ipph; j++) {
-      jc = ip - j;
-      for (ik=1; ik<idl1; ik+=2) {
-        ch[ik - 1 + j*idl1] = cc[ik - 1 + j*idl1] - cc[ik + jc*idl1];
-        ch[ik - 1 + jc*idl1] = cc[ik - 1 + j*idl1] + cc[ik + jc*idl1];
-        ch[ik + j*idl1] = cc[ik + j*idl1] + cc[ik - 1 + jc*idl1];
-        ch[ik + jc*idl1] = cc[ik + j*idl1] - cc[ik - 1 + jc*idl1];
-      }
-    }
-    *nac = 1;
-    if (ido == 2) return;
-    *nac = 0;
-    for (ik=0; ik<idl1; ik++)
-      cc[ik] = ch[ik];
-    for (j=1; j<ip; j++) {
-      for (k=0; k<l1; k++) {
-        cc[(k + j*l1)*ido + 0] = ch[(k + j*l1)*ido + 0];
-        cc[(k + j*l1)*ido + 1] = ch[(k + j*l1)*ido + 1];
-      }
-    }
-    if (idot <= l1) {
-      idij = 0;
-      for (j=1; j<ip; j++) {
-        idij += 2;
-        for (i=3; i<ido; i+=2) {
-          idij += 2;
-          for (k=0; k<l1; k++) {
-            cc[i - 1 + (k + j*l1)*ido] =
-                wa[idij - 2]*ch[i - 1 + (k + j*l1)*ido] -
-                isign*wa[idij-1]*ch[i + (k + j*l1)*ido];
-            cc[i + (k + j*l1)*ido] =
-                wa[idij - 2]*ch[i + (k + j*l1)*ido] +
-                isign*wa[idij-1]*ch[i - 1 + (k + j*l1)*ido];
-          }
-        }
-      }
-    } else {
-      idj = 2 - ido;
-      for (j=1; j<ip; j++) {
-        idj += ido;
-        for (k = 0; k < l1; k++) {
-          idij = idj;
-          for (i=3; i<ido; i+=2) {
-            idij += 2;
-            cc[i - 1 + (k + j*l1)*ido] =
-                wa[idij - 2]*ch[i - 1 + (k + j*l1)*ido] -
-                isign*wa[idij-1]*ch[i + (k + j*l1)*ido];
-            cc[i + (k + j*l1)*ido] =
-                wa[idij - 2]*ch[i + (k + j*l1)*ido] +
-                isign*wa[idij-1]*ch[i - 1 + (k + j*l1)*ido];
-          }
-        }
-      }
-    }
-  } /* passf */
-
-
-  /* ----------------------------------------------------------------------
-radf2,radb2, radf3,radb3, radf4,radb4, radf5,radb5, radfg,radbg.
-Treal FFT passes fwd and bwd.
----------------------------------------------------------------------- */
-
-static void radf2(int ido, int l1, const Treal cc[], Treal ch[], const Treal wa1[])
-  {
-    int i, k, ic;
-    Treal ti2, tr2;
-    for (k=0; k<l1; k++) {
-      ch[2*k*ido] =
-          ref(cc,k*ido) + ref(cc,(k + l1)*ido);
-      ch[(2*k+1)*ido + ido-1] =
-          ref(cc,k*ido) - ref(cc,(k + l1)*ido);
-    }
-    if (ido < 2) return;
-    if (ido != 2) {
-      for (k=0; k<l1; k++) {
-        for (i=2; i<ido; i+=2) {
-          ic = ido - i;
-          tr2 = wa1[i - 2]*ref(cc, i-1 + (k + l1)*ido) + wa1[i - 1]*ref(cc, i + (k + l1)*ido);
-          ti2 = wa1[i - 2]*ref(cc, i + (k + l1)*ido) - wa1[i - 1]*ref(cc, i-1 + (k + l1)*ido);
-          ch[i + 2*k*ido] = ref(cc,i + k*ido) + ti2;
-          ch[ic + (2*k+1)*ido] = ti2 - ref(cc,i + k*ido);
-          ch[i - 1 + 2*k*ido] = ref(cc,i - 1 + k*ido) + tr2;
-          ch[ic - 1 + (2*k+1)*ido] = ref(cc,i - 1 + k*ido) - tr2;
-        }
-      }
-      if (ido % 2 == 1) return;
-    }
-    for (k=0; k<l1; k++) {
-      ch[(2*k+1)*ido] = -ref(cc,ido-1 + (k + l1)*ido);
-      ch[ido-1 + 2*k*ido] = ref(cc,ido-1 + k*ido);
-    }
-  } /* radf2 */
-
-
-static void radb2(int ido, int l1, const Treal cc[], Treal ch[], const Treal wa1[])
-  {
-    int i, k, ic;
-    Treal ti2, tr2;
-    for (k=0; k<l1; k++) {
-      ch[k*ido] =
-          ref(cc,2*k*ido) + ref(cc,ido-1 + (2*k+1)*ido);
-      ch[(k + l1)*ido] =
-          ref(cc,2*k*ido) - ref(cc,ido-1 + (2*k+1)*ido);
-    }
-    if (ido < 2) return;
-    if (ido != 2) {
-      for (k = 0; k < l1; ++k) {
-        for (i = 2; i < ido; i += 2) {
-          ic = ido - i;
-          ch[i-1 + k*ido] =
-              ref(cc,i-1 + 2*k*ido) + ref(cc,ic-1 + (2*k+1)*ido);
-          tr2 = ref(cc,i-1 + 2*k*ido) - ref(cc,ic-1 + (2*k+1)*ido);
-          ch[i + k*ido] =
-              ref(cc,i + 2*k*ido) - ref(cc,ic + (2*k+1)*ido);
-          ti2 = ref(cc,i + (2*k)*ido) + ref(cc,ic + (2*k+1)*ido);
-          ch[i-1 + (k + l1)*ido] =
-              wa1[i - 2]*tr2 - wa1[i - 1]*ti2;
-          ch[i + (k + l1)*ido] =
-              wa1[i - 2]*ti2 + wa1[i - 1]*tr2;
-        }
-      }
-      if (ido % 2 == 1) return;
-    }
-    for (k = 0; k < l1; k++) {
-      ch[ido-1 + k*ido] = 2*ref(cc,ido-1 + 2*k*ido);
-      ch[ido-1 + (k + l1)*ido] = -2*ref(cc,(2*k+1)*ido);
-    }
-  } /* radb2 */
-
-
-static void radf3(int ido, int l1, const Treal cc[], Treal ch[],
-      const Treal wa1[], const Treal wa2[])
-  {
-    static const Treal taur = -0.5;
-    static const Treal taui = 0.866025403784439;
-    int i, k, ic;
-    Treal ci2, di2, di3, cr2, dr2, dr3, ti2, ti3, tr2, tr3;
-    for (k=0; k<l1; k++) {
-      cr2 = ref(cc,(k + l1)*ido) + ref(cc,(k + 2*l1)*ido);
-      ch[3*k*ido] = ref(cc,k*ido) + cr2;
-      ch[(3*k+2)*ido] = taui*(ref(cc,(k + l1*2)*ido) - ref(cc,(k + l1)*ido));
-      ch[ido-1 + (3*k + 1)*ido] = ref(cc,k*ido) + taur*cr2;
-    }
-    if (ido == 1) return;
-    for (k=0; k<l1; k++) {
-      for (i=2; i<ido; i+=2) {
-        ic = ido - i;
-        dr2 = wa1[i - 2]*ref(cc,i - 1 + (k + l1)*ido) +
-            wa1[i - 1]*ref(cc,i + (k + l1)*ido);
-        di2 = wa1[i - 2]*ref(cc,i + (k + l1)*ido) - wa1[i - 1]*ref(cc,i - 1 + (k + l1)*ido);
-        dr3 = wa2[i - 2]*ref(cc,i - 1 + (k + l1*2)*ido) + wa2[i - 1]*ref(cc,i + (k + l1*2)*ido);
-        di3 = wa2[i - 2]*ref(cc,i + (k + l1*2)*ido) - wa2[i - 1]*ref(cc,i - 1 + (k + l1*2)*ido);
-        cr2 = dr2 + dr3;
-        ci2 = di2 + di3;
-        ch[i - 1 + 3*k*ido] = ref(cc,i - 1 + k*ido) + cr2;
-        ch[i + 3*k*ido] = ref(cc,i + k*ido) + ci2;
-        tr2 = ref(cc,i - 1 + k*ido) + taur*cr2;
-        ti2 = ref(cc,i + k*ido) + taur*ci2;
-        tr3 = taui*(di2 - di3);
-        ti3 = taui*(dr3 - dr2);
-        ch[i - 1 + (3*k + 2)*ido] = tr2 + tr3;
-        ch[ic - 1 + (3*k + 1)*ido] = tr2 - tr3;
-        ch[i + (3*k + 2)*ido] = ti2 + ti3;
-        ch[ic + (3*k + 1)*ido] = ti3 - ti2;
-      }
-    }
-  } /* radf3 */
-
-
-static void radb3(int ido, int l1, const Treal cc[], Treal ch[],
-      const Treal wa1[], const Treal wa2[])
-  {
-    static const Treal taur = -0.5;
-    static const Treal taui = 0.866025403784439;
-    int i, k, ic;
-    Treal ci2, ci3, di2, di3, cr2, cr3, dr2, dr3, ti2, tr2;
-    for (k=0; k<l1; k++) {
-      tr2 = 2*ref(cc,ido-1 + (3*k + 1)*ido);
-      cr2 = ref(cc,3*k*ido) + taur*tr2;
-      ch[k*ido] = ref(cc,3*k*ido) + tr2;
-      ci3 = 2*taui*ref(cc,(3*k + 2)*ido);
-      ch[(k + l1)*ido] = cr2 - ci3;
-      ch[(k + 2*l1)*ido] = cr2 + ci3;
-    }
-    if (ido == 1) return;
-    for (k=0; k<l1; k++) {
-      for (i=2; i<ido; i+=2) {
-        ic = ido - i;
-        tr2 = ref(cc,i - 1 + (3*k + 2)*ido) + ref(cc,ic - 1 + (3*k + 1)*ido);
-        cr2 = ref(cc,i - 1 + 3*k*ido) + taur*tr2;
-        ch[i - 1 + k*ido] = ref(cc,i - 1 + 3*k*ido) + tr2;
-        ti2 = ref(cc,i + (3*k + 2)*ido) - ref(cc,ic + (3*k + 1)*ido);
-        ci2 = ref(cc,i + 3*k*ido) + taur*ti2;
-        ch[i + k*ido] = ref(cc,i + 3*k*ido) + ti2;
-        cr3 = taui*(ref(cc,i - 1 + (3*k + 2)*ido) - ref(cc,ic - 1 + (3*k + 1)*ido));
-        ci3 = taui*(ref(cc,i + (3*k + 2)*ido) + ref(cc,ic + (3*k + 1)*ido));
-        dr2 = cr2 - ci3;
-        dr3 = cr2 + ci3;
-        di2 = ci2 + cr3;
-        di3 = ci2 - cr3;
-        ch[i - 1 + (k + l1)*ido] = wa1[i - 2]*dr2 - wa1[i - 1]*di2;
-        ch[i + (k + l1)*ido] = wa1[i - 2]*di2 + wa1[i - 1]*dr2;
-        ch[i - 1 + (k + 2*l1)*ido] = wa2[i - 2]*dr3 - wa2[i - 1]*di3;
-        ch[i + (k + 2*l1)*ido] = wa2[i - 2]*di3 + wa2[i - 1]*dr3;
-      }
-    }
-  } /* radb3 */
-
-
-static void radf4(int ido, int l1, const Treal cc[], Treal ch[],
-      const Treal wa1[], const Treal wa2[], const Treal wa3[])
-  {
-    static const Treal hsqt2 = 0.7071067811865475;
-    int i, k, ic;
-    Treal ci2, ci3, ci4, cr2, cr3, cr4, ti1, ti2, ti3, ti4, tr1, tr2, tr3, tr4;
-    for (k=0; k<l1; k++) {
-      tr1 = ref(cc,(k + l1)*ido) + ref(cc,(k + 3*l1)*ido);
-      tr2 = ref(cc,k*ido) + ref(cc,(k + 2*l1)*ido);
-      ch[4*k*ido] = tr1 + tr2;
-      ch[ido-1 + (4*k + 3)*ido] = tr2 - tr1;
-      ch[ido-1 + (4*k + 1)*ido] = ref(cc,k*ido) - ref(cc,(k + 2*l1)*ido);
-      ch[(4*k + 2)*ido] = ref(cc,(k + 3*l1)*ido) - ref(cc,(k + l1)*ido);
-    }
-    if (ido < 2) return;
-    if (ido != 2) {
-      for (k=0; k<l1; k++) {
-        for (i=2; i<ido; i += 2) {
-          ic = ido - i;
-          cr2 = wa1[i - 2]*ref(cc,i - 1 + (k + l1)*ido) + wa1[i - 1]*ref(cc,i + (k + l1)*ido);
-          ci2 = wa1[i - 2]*ref(cc,i + (k + l1)*ido) - wa1[i - 1]*ref(cc,i - 1 + (k + l1)*ido);
-          cr3 = wa2[i - 2]*ref(cc,i - 1 + (k + 2*l1)*ido) + wa2[i - 1]*ref(cc,i + (k + 2*l1)*
-              ido);
-          ci3 = wa2[i - 2]*ref(cc,i + (k + 2*l1)*ido) - wa2[i - 1]*ref(cc,i - 1 + (k + 2*l1)*
-              ido);
-          cr4 = wa3[i - 2]*ref(cc,i - 1 + (k + 3*l1)*ido) + wa3[i - 1]*ref(cc,i + (k + 3*l1)*
-              ido);
-          ci4 = wa3[i - 2]*ref(cc,i + (k + 3*l1)*ido) - wa3[i - 1]*ref(cc,i - 1 + (k + 3*l1)*
-              ido);
-          tr1 = cr2 + cr4;
-          tr4 = cr4 - cr2;
-          ti1 = ci2 + ci4;
-          ti4 = ci2 - ci4;
-          ti2 = ref(cc,i + k*ido) + ci3;
-          ti3 = ref(cc,i + k*ido) - ci3;
-          tr2 = ref(cc,i - 1 + k*ido) + cr3;
-          tr3 = ref(cc,i - 1 + k*ido) - cr3;
-          ch[i - 1 + 4*k*ido] = tr1 + tr2;
-          ch[ic - 1 + (4*k + 3)*ido] = tr2 - tr1;
-          ch[i + 4*k*ido] = ti1 + ti2;
-          ch[ic + (4*k + 3)*ido] = ti1 - ti2;
-          ch[i - 1 + (4*k + 2)*ido] = ti4 + tr3;
-          ch[ic - 1 + (4*k + 1)*ido] = tr3 - ti4;
-          ch[i + (4*k + 2)*ido] = tr4 + ti3;
-          ch[ic + (4*k + 1)*ido] = tr4 - ti3;
-        }
-      }
-      if (ido % 2 == 1) return;
-    }
-    for (k=0; k<l1; k++) {
-      ti1 = -hsqt2*(ref(cc,ido-1 + (k + l1)*ido) + ref(cc,ido-1 + (k + 3*l1)*ido));
-      tr1 = hsqt2*(ref(cc,ido-1 + (k + l1)*ido) - ref(cc,ido-1 + (k + 3*l1)*ido));
-      ch[ido-1 + 4*k*ido] = tr1 + ref(cc,ido-1 + k*ido);
-      ch[ido-1 + (4*k + 2)*ido] = ref(cc,ido-1 + k*ido) - tr1;
-      ch[(4*k + 1)*ido] = ti1 - ref(cc,ido-1 + (k + 2*l1)*ido);
-      ch[(4*k + 3)*ido] = ti1 + ref(cc,ido-1 + (k + 2*l1)*ido);
-    }
-  } /* radf4 */
-
-
-static void radb4(int ido, int l1, const Treal cc[], Treal ch[],
-      const Treal wa1[], const Treal wa2[], const Treal wa3[])
-  {
-    static const Treal sqrt2 = 1.414213562373095;
-    int i, k, ic;
-    Treal ci2, ci3, ci4, cr2, cr3, cr4, ti1, ti2, ti3, ti4, tr1, tr2, tr3, tr4;
-    for (k = 0; k < l1; k++) {
-      tr1 = ref(cc,4*k*ido) - ref(cc,ido-1 + (4*k + 3)*ido);
-      tr2 = ref(cc,4*k*ido) + ref(cc,ido-1 + (4*k + 3)*ido);
-      tr3 = ref(cc,ido-1 + (4*k + 1)*ido) + ref(cc,ido-1 + (4*k + 1)*ido);
-      tr4 = ref(cc,(4*k + 2)*ido) + ref(cc,(4*k + 2)*ido);
-      ch[k*ido] = tr2 + tr3;
-      ch[(k + l1)*ido] = tr1 - tr4;
-      ch[(k + 2*l1)*ido] = tr2 - tr3;
-      ch[(k + 3*l1)*ido] = tr1 + tr4;
-    }
-    if (ido < 2) return;
-    if (ido != 2) {
-      for (k = 0; k < l1; ++k) {
-        for (i = 2; i < ido; i += 2) {
-          ic = ido - i;
-          ti1 = ref(cc,i + 4*k*ido) + ref(cc,ic + (4*k + 3)*ido);
-          ti2 = ref(cc,i + 4*k*ido) - ref(cc,ic + (4*k + 3)*ido);
-          ti3 = ref(cc,i + (4*k + 2)*ido) - ref(cc,ic + (4*k + 1)*ido);
-          tr4 = ref(cc,i + (4*k + 2)*ido) + ref(cc,ic + (4*k + 1)*ido);
-          tr1 = ref(cc,i - 1 + 4*k*ido) - ref(cc,ic - 1 + (4*k + 3)*ido);
-          tr2 = ref(cc,i - 1 + 4*k*ido) + ref(cc,ic - 1 + (4*k + 3)*ido);
-          ti4 = ref(cc,i - 1 + (4*k + 2)*ido) - ref(cc,ic - 1 + (4*k + 1)*ido);
-          tr3 = ref(cc,i - 1 + (4*k + 2)*ido) + ref(cc,ic - 1 + (4*k + 1)*ido);
-          ch[i - 1 + k*ido] = tr2 + tr3;
-          cr3 = tr2 - tr3;
-          ch[i + k*ido] = ti2 + ti3;
-          ci3 = ti2 - ti3;
-          cr2 = tr1 - tr4;
-          cr4 = tr1 + tr4;
-          ci2 = ti1 + ti4;
-          ci4 = ti1 - ti4;
-          ch[i - 1 + (k + l1)*ido] = wa1[i - 2]*cr2 - wa1[i - 1]*ci2;
-          ch[i + (k + l1)*ido] = wa1[i - 2]*ci2 + wa1[i - 1]*cr2;
-          ch[i - 1 + (k + 2*l1)*ido] = wa2[i - 2]*cr3 - wa2[i - 1]*ci3;
-          ch[i + (k + 2*l1)*ido] = wa2[i - 2]*ci3 + wa2[i - 1]*cr3;
-          ch[i - 1 + (k + 3*l1)*ido] = wa3[i - 2]*cr4 - wa3[i - 1]*ci4;
-          ch[i + (k + 3*l1)*ido] = wa3[i - 2]*ci4 + wa3[i - 1]*cr4;
-        }
-      }
-      if (ido % 2 == 1) return;
-    }
-    for (k = 0; k < l1; k++) {
-      ti1 = ref(cc,(4*k + 1)*ido) + ref(cc,(4*k + 3)*ido);
-      ti2 = ref(cc,(4*k + 3)*ido) - ref(cc,(4*k + 1)*ido);
-      tr1 = ref(cc,ido-1 + 4*k*ido) - ref(cc,ido-1 + (4*k + 2)*ido);
-      tr2 = ref(cc,ido-1 + 4*k*ido) + ref(cc,ido-1 + (4*k + 2)*ido);
-      ch[ido-1 + k*ido] = tr2 + tr2;
-      ch[ido-1 + (k + l1)*ido] = sqrt2*(tr1 - ti1);
-      ch[ido-1 + (k + 2*l1)*ido] = ti2 + ti2;
-      ch[ido-1 + (k + 3*l1)*ido] = -sqrt2*(tr1 + ti1);
-    }
-  } /* radb4 */
-
-
-static void radf5(int ido, int l1, const Treal cc[], Treal ch[],
-      const Treal wa1[], const Treal wa2[], const Treal wa3[], const Treal wa4[])
-  {
-    static const Treal tr11 = 0.309016994374947;
-    static const Treal ti11 = 0.951056516295154;
-    static const Treal tr12 = -0.809016994374947;
-    static const Treal ti12 = 0.587785252292473;
-    int i, k, ic;
-    Treal ci2, di2, ci4, ci5, di3, di4, di5, ci3, cr2, cr3, dr2, dr3, dr4, dr5,
-        cr5, cr4, ti2, ti3, ti5, ti4, tr2, tr3, tr4, tr5;
-    for (k = 0; k < l1; k++) {
-      cr2 = ref(cc,(k + 4*l1)*ido) + ref(cc,(k + l1)*ido);
-      ci5 = ref(cc,(k + 4*l1)*ido) - ref(cc,(k + l1)*ido);
-      cr3 = ref(cc,(k + 3*l1)*ido) + ref(cc,(k + 2*l1)*ido);
-      ci4 = ref(cc,(k + 3*l1)*ido) - ref(cc,(k + 2*l1)*ido);
-      ch[5*k*ido] = ref(cc,k*ido) + cr2 + cr3;
-      ch[ido-1 + (5*k + 1)*ido] = ref(cc,k*ido) + tr11*cr2 + tr12*cr3;
-      ch[(5*k + 2)*ido] = ti11*ci5 + ti12*ci4;
-      ch[ido-1 + (5*k + 3)*ido] = ref(cc,k*ido) + tr12*cr2 + tr11*cr3;
-      ch[(5*k + 4)*ido] = ti12*ci5 - ti11*ci4;
-    }
-    if (ido == 1) return;
-    for (k = 0; k < l1; ++k) {
-      for (i = 2; i < ido; i += 2) {
-        ic = ido - i;
-        dr2 = wa1[i - 2]*ref(cc,i - 1 + (k + l1)*ido) + wa1[i - 1]*ref(cc,i + (k + l1)*ido);
-        di2 = wa1[i - 2]*ref(cc,i + (k + l1)*ido) - wa1[i - 1]*ref(cc,i - 1 + (k + l1)*ido);
-        dr3 = wa2[i - 2]*ref(cc,i - 1 + (k + 2*l1)*ido) + wa2[i - 1]*ref(cc,i + (k + 2*l1)*ido);
-        di3 = wa2[i - 2]*ref(cc,i + (k + 2*l1)*ido) - wa2[i - 1]*ref(cc,i - 1 + (k + 2*l1)*ido);
-        dr4 = wa3[i - 2]*ref(cc,i - 1 + (k + 3*l1)*ido) + wa3[i - 1]*ref(cc,i + (k + 3*l1)*ido);
-        di4 = wa3[i - 2]*ref(cc,i + (k + 3*l1)*ido) - wa3[i - 1]*ref(cc,i - 1 + (k + 3*l1)*ido);
-        dr5 = wa4[i - 2]*ref(cc,i - 1 + (k + 4*l1)*ido) + wa4[i - 1]*ref(cc,i + (k + 4*l1)*ido);
-        di5 = wa4[i - 2]*ref(cc,i + (k + 4*l1)*ido) - wa4[i - 1]*ref(cc,i - 1 + (k + 4*l1)*ido);
-        cr2 = dr2 + dr5;
-        ci5 = dr5 - dr2;
-        cr5 = di2 - di5;
-        ci2 = di2 + di5;
-        cr3 = dr3 + dr4;
-        ci4 = dr4 - dr3;
-        cr4 = di3 - di4;
-        ci3 = di3 + di4;
-        ch[i - 1 + 5*k*ido] = ref(cc,i - 1 + k*ido) + cr2 + cr3;
-        ch[i + 5*k*ido] = ref(cc,i + k*ido) + ci2 + ci3;
-        tr2 = ref(cc,i - 1 + k*ido) + tr11*cr2 + tr12*cr3;
-        ti2 = ref(cc,i + k*ido) + tr11*ci2 + tr12*ci3;
-        tr3 = ref(cc,i - 1 + k*ido) + tr12*cr2 + tr11*cr3;
-        ti3 = ref(cc,i + k*ido) + tr12*ci2 + tr11*ci3;
-        tr5 = ti11*cr5 + ti12*cr4;
-        ti5 = ti11*ci5 + ti12*ci4;
-        tr4 = ti12*cr5 - ti11*cr4;
-        ti4 = ti12*ci5 - ti11*ci4;
-        ch[i - 1 + (5*k + 2)*ido] = tr2 + tr5;
-        ch[ic - 1 + (5*k + 1)*ido] = tr2 - tr5;
-        ch[i + (5*k + 2)*ido] = ti2 + ti5;
-        ch[ic + (5*k + 1)*ido] = ti5 - ti2;
-        ch[i - 1 + (5*k + 4)*ido] = tr3 + tr4;
-        ch[ic - 1 + (5*k + 3)*ido] = tr3 - tr4;
-        ch[i + (5*k + 4)*ido] = ti3 + ti4;
-        ch[ic + (5*k + 3)*ido] = ti4 - ti3;
-      }
-    }
-  } /* radf5 */
-
-
-static void radb5(int ido, int l1, const Treal cc[], Treal ch[],
-      const Treal wa1[], const Treal wa2[], const Treal wa3[], const Treal wa4[])
-  {
-    static const Treal tr11 = 0.309016994374947;
-    static const Treal ti11 = 0.951056516295154;
-    static const Treal tr12 = -0.809016994374947;
-    static const Treal ti12 = 0.587785252292473;
-    int i, k, ic;
-    Treal ci2, ci3, ci4, ci5, di3, di4, di5, di2, cr2, cr3, cr5, cr4, ti2, ti3,
-        ti4, ti5, dr3, dr4, dr5, dr2, tr2, tr3, tr4, tr5;
-    for (k = 0; k < l1; k++) {
-      ti5 = 2*ref(cc,(5*k + 2)*ido);
-      ti4 = 2*ref(cc,(5*k + 4)*ido);
-      tr2 = 2*ref(cc,ido-1 + (5*k + 1)*ido);
-      tr3 = 2*ref(cc,ido-1 + (5*k + 3)*ido);
-      ch[k*ido] = ref(cc,5*k*ido) + tr2 + tr3;
-      cr2 = ref(cc,5*k*ido) + tr11*tr2 + tr12*tr3;
-      cr3 = ref(cc,5*k*ido) + tr12*tr2 + tr11*tr3;
-      ci5 = ti11*ti5 + ti12*ti4;
-      ci4 = ti12*ti5 - ti11*ti4;
-      ch[(k + l1)*ido] = cr2 - ci5;
-      ch[(k + 2*l1)*ido] = cr3 - ci4;
-      ch[(k + 3*l1)*ido] = cr3 + ci4;
-      ch[(k + 4*l1)*ido] = cr2 + ci5;
-    }
-    if (ido == 1) return;
-    for (k = 0; k < l1; ++k) {
-      for (i = 2; i < ido; i += 2) {
-        ic = ido - i;
-        ti5 = ref(cc,i + (5*k + 2)*ido) + ref(cc,ic + (5*k + 1)*ido);
-        ti2 = ref(cc,i + (5*k + 2)*ido) - ref(cc,ic + (5*k + 1)*ido);
-        ti4 = ref(cc,i + (5*k + 4)*ido) + ref(cc,ic + (5*k + 3)*ido);
-        ti3 = ref(cc,i + (5*k + 4)*ido) - ref(cc,ic + (5*k + 3)*ido);
-        tr5 = ref(cc,i - 1 + (5*k + 2)*ido) - ref(cc,ic - 1 + (5*k + 1)*ido);
-        tr2 = ref(cc,i - 1 + (5*k + 2)*ido) + ref(cc,ic - 1 + (5*k + 1)*ido);
-        tr4 = ref(cc,i - 1 + (5*k + 4)*ido) - ref(cc,ic - 1 + (5*k + 3)*ido);
-        tr3 = ref(cc,i - 1 + (5*k + 4)*ido) + ref(cc,ic - 1 + (5*k + 3)*ido);
-        ch[i - 1 + k*ido] = ref(cc,i - 1 + 5*k*ido) + tr2 + tr3;
-        ch[i + k*ido] = ref(cc,i + 5*k*ido) + ti2 + ti3;
-        cr2 = ref(cc,i - 1 + 5*k*ido) + tr11*tr2 + tr12*tr3;
-
-        ci2 = ref(cc,i + 5*k*ido) + tr11*ti2 + tr12*ti3;
-        cr3 = ref(cc,i - 1 + 5*k*ido) + tr12*tr2 + tr11*tr3;
-
-        ci3 = ref(cc,i + 5*k*ido) + tr12*ti2 + tr11*ti3;
-        cr5 = ti11*tr5 + ti12*tr4;
-        ci5 = ti11*ti5 + ti12*ti4;
-        cr4 = ti12*tr5 - ti11*tr4;
-        ci4 = ti12*ti5 - ti11*ti4;
-        dr3 = cr3 - ci4;
-        dr4 = cr3 + ci4;
-        di3 = ci3 + cr4;
-        di4 = ci3 - cr4;
-        dr5 = cr2 + ci5;
-        dr2 = cr2 - ci5;
-        di5 = ci2 - cr5;
-        di2 = ci2 + cr5;
-        ch[i - 1 + (k + l1)*ido] = wa1[i - 2]*dr2 - wa1[i - 1]*di2;
-        ch[i + (k + l1)*ido] = wa1[i - 2]*di2 + wa1[i - 1]*dr2;
-        ch[i - 1 + (k + 2*l1)*ido] = wa2[i - 2]*dr3 - wa2[i - 1]*di3;
-        ch[i + (k + 2*l1)*ido] = wa2[i - 2]*di3 + wa2[i - 1]*dr3;
-        ch[i - 1 + (k + 3*l1)*ido] = wa3[i - 2]*dr4 - wa3[i - 1]*di4;
-        ch[i + (k + 3*l1)*ido] = wa3[i - 2]*di4 + wa3[i - 1]*dr4;
-        ch[i - 1 + (k + 4*l1)*ido] = wa4[i - 2]*dr5 - wa4[i - 1]*di5;
-        ch[i + (k + 4*l1)*ido] = wa4[i - 2]*di5 + wa4[i - 1]*dr5;
-      }
-    }
-  } /* radb5 */
-
-
-static void radfg(int ido, int ip, int l1, int idl1,
-      Treal cc[], Treal ch[], const Treal wa[])
-  {
-    static const Treal twopi = 6.28318530717959;
-    int idij, ipph, i, j, k, l, j2, ic, jc, lc, ik, is, nbd;
-    Treal dc2, ai1, ai2, ar1, ar2, ds2, dcp, arg, dsp, ar1h, ar2h;
-    arg = twopi / ip;
-    dcp = cos(arg);
-    dsp = sin(arg);
-    ipph = (ip + 1) / 2;
-    nbd = (ido - 1) / 2;
-    if (ido != 1) {
-      for (ik=0; ik<idl1; ik++) ch[ik] = cc[ik];
-      for (j=1; j<ip; j++)
-        for (k=0; k<l1; k++)
-          ch[(k + j*l1)*ido] = cc[(k + j*l1)*ido];
-      if (nbd <= l1) {
-        is = -ido;
-        for (j=1; j<ip; j++) {
-          is += ido;
-          idij = is-1;
-          for (i=2; i<ido; i+=2) {
-            idij += 2;
-            for (k=0; k<l1; k++) {
-              ch[i - 1 + (k + j*l1)*ido] =
-                  wa[idij - 1]*cc[i - 1 + (k + j*l1)*ido] + wa[idij]*cc[i + (k + j*l1)*ido];
-              ch[i + (k + j*l1)*ido] =
-                  wa[idij - 1]*cc[i + (k + j*l1)*ido] - wa[idij]*cc[i - 1 + (k + j*l1)*ido];
-            }
-          }
-        }
-      } else {
-        is = -ido;
-        for (j=1; j<ip; j++) {
-          is += ido;
-          for (k=0; k<l1; k++) {
-            idij = is-1;
-            for (i=2; i<ido; i+=2) {
-              idij += 2;
-              ch[i - 1 + (k + j*l1)*ido] =
-                  wa[idij - 1]*cc[i - 1 + (k + j*l1)*ido] + wa[idij]*cc[i + (k + j*l1)*ido];
-              ch[i + (k + j*l1)*ido] =
-                  wa[idij - 1]*cc[i + (k + j*l1)*ido] - wa[idij]*cc[i - 1 + (k + j*l1)*ido];
-            }
-          }
-        }
-      }
-      if (nbd >= l1) {
-        for (j=1; j<ipph; j++) {
-          jc = ip - j;
-          for (k=0; k<l1; k++) {
-            for (i=2; i<ido; i+=2) {
-              cc[i - 1 + (k + j*l1)*ido] = ch[i - 1 + (k + j*l1)*ido] + ch[i - 1 + (k + jc*l1)*ido];
-              cc[i - 1 + (k + jc*l1)*ido] = ch[i + (k + j*l1)*ido] - ch[i + (k + jc*l1)*ido];
-              cc[i + (k + j*l1)*ido] = ch[i + (k + j*l1)*ido] + ch[i + (k + jc*l1)*ido];
-              cc[i + (k + jc*l1)*ido] = ch[i - 1 + (k + jc*l1)*ido] - ch[i - 1 + (k + j*l1)*ido];
-            }
-          }
-        }
-      } else {
-        for (j=1; j<ipph; j++) {
-          jc = ip - j;
-          for (i=2; i<ido; i+=2) {
-            for (k=0; k<l1; k++) {
-              cc[i - 1 + (k + j*l1)*ido] =
-                  ch[i - 1 + (k + j*l1)*ido] + ch[i - 1 + (k + jc*l1)*ido];
-              cc[i - 1 + (k + jc*l1)*ido] = ch[i + (k + j*l1)*ido] - ch[i + (k + jc*l1)*ido];
-              cc[i + (k + j*l1)*ido] = ch[i + (k + j*l1)*ido] + ch[i + (k + jc*l1)*ido];
-              cc[i + (k + jc*l1)*ido] = ch[i - 1 + (k + jc*l1)*ido] - ch[i - 1 + (k + j*l1)*ido];
-            }
-          }
-        }
-      }
-    } else {  /* now ido == 1 */
-      for (ik=0; ik<idl1; ik++) cc[ik] = ch[ik];
-    }
-    for (j=1; j<ipph; j++) {
-      jc = ip - j;
-      for (k=0; k<l1; k++) {
-        cc[(k + j*l1)*ido] = ch[(k + j*l1)*ido] + ch[(k + jc*l1)*ido];
-        cc[(k + jc*l1)*ido] = ch[(k + jc*l1)*ido] - ch[(k + j*l1)*ido];
-      }
-    }
-
-    ar1 = 1;
-    ai1 = 0;
-    for (l=1; l<ipph; l++) {
-      lc = ip - l;
-      ar1h = dcp*ar1 - dsp*ai1;
-      ai1 = dcp*ai1 + dsp*ar1;
-      ar1 = ar1h;
-      for (ik=0; ik<idl1; ik++) {
-        ch[ik + l*idl1] = cc[ik] + ar1*cc[ik + idl1];
-        ch[ik + lc*idl1] = ai1*cc[ik + (ip-1)*idl1];
-      }
-      dc2 = ar1;
-      ds2 = ai1;
-      ar2 = ar1;
-      ai2 = ai1;
-      for (j=2; j<ipph; j++) {
-        jc = ip - j;
-        ar2h = dc2*ar2 - ds2*ai2;
-        ai2 = dc2*ai2 + ds2*ar2;
-        ar2 = ar2h;
-        for (ik=0; ik<idl1; ik++) {
-          ch[ik + l*idl1] += ar2*cc[ik + j*idl1];
-          ch[ik + lc*idl1] += ai2*cc[ik + jc*idl1];
-        }
-      }
-    }
-    for (j=1; j<ipph; j++)
-      for (ik=0; ik<idl1; ik++)
-        ch[ik] += cc[ik + j*idl1];
-
-    if (ido >= l1) {
-      for (k=0; k<l1; k++) {
-        for (i=0; i<ido; i++) {
-          ref(cc,i + k*ip*ido) = ch[i + k*ido];
-        }
-      }
-    } else {
-      for (i=0; i<ido; i++) {
-        for (k=0; k<l1; k++) {
-          ref(cc,i + k*ip*ido) = ch[i + k*ido];
-        }
-      }
-    }
-    for (j=1; j<ipph; j++) {
-      jc = ip - j;
-      j2 = 2*j;
-      for (k=0; k<l1; k++) {
-        ref(cc,ido-1 + (j2 - 1 + k*ip)*ido) =
-            ch[(k + j*l1)*ido];
-        ref(cc,(j2 + k*ip)*ido) =
-            ch[(k + jc*l1)*ido];
-      }
-    }
-    if (ido == 1) return;
-    if (nbd >= l1) {
-      for (j=1; j<ipph; j++) {
-        jc = ip - j;
-        j2 = 2*j;
-        for (k=0; k<l1; k++) {
-          for (i=2; i<ido; i+=2) {
-            ic = ido - i;
-            ref(cc,i - 1 + (j2 + k*ip)*ido) = ch[i - 1 + (k + j*l1)*ido] + ch[i - 1 + (k + jc*l1)*ido];
-            ref(cc,ic - 1 + (j2 - 1 + k*ip)*ido) = ch[i - 1 + (k + j*l1)*ido] - ch[i - 1 + (k + jc*l1)*ido];
-            ref(cc,i + (j2 + k*ip)*ido) = ch[i + (k + j*l1)*ido] + ch[i + (k + jc*l1)*ido];
-            ref(cc,ic + (j2 - 1 + k*ip)*ido) = ch[i + (k + jc*l1)*ido] - ch[i + (k + j*l1)*ido];
-          }
-        }
-      }
-    } else {
-      for (j=1; j<ipph; j++) {
-        jc = ip - j;
-        j2 = 2*j;
-        for (i=2; i<ido; i+=2) {
-          ic = ido - i;
-          for (k=0; k<l1; k++) {
-            ref(cc,i - 1 + (j2 + k*ip)*ido) = ch[i - 1 + (k + j*l1)*ido] + ch[i - 1 + (k + jc*l1)*ido];
-            ref(cc,ic - 1 + (j2 - 1 + k*ip)*ido) = ch[i - 1 + (k + j*l1)*ido] - ch[i - 1 + (k + jc*l1)*ido];
-            ref(cc,i + (j2 + k*ip)*ido) = ch[i + (k + j*l1)*ido] + ch[i + (k + jc*l1)*ido];
-            ref(cc,ic + (j2 - 1 + k*ip)*ido) = ch[i + (k + jc*l1)*ido] - ch[i + (k + j*l1)*ido];
-          }
-        }
-      }
-    }
-  } /* radfg */
-
-
-static void radbg(int ido, int ip, int l1, int idl1,
-      Treal cc[], Treal ch[], const Treal wa[])
-  {
-    static const Treal twopi = 6.28318530717959;
-    int idij, ipph, i, j, k, l, j2, ic, jc, lc, ik, is;
-    Treal dc2, ai1, ai2, ar1, ar2, ds2;
-    int nbd;
-    Treal dcp, arg, dsp, ar1h, ar2h;
-    arg = twopi / ip;
-    dcp = cos(arg);
-    dsp = sin(arg);
-    nbd = (ido - 1) / 2;
-    ipph = (ip + 1) / 2;
-    if (ido >= l1) {
-      for (k=0; k<l1; k++) {
-        for (i=0; i<ido; i++) {
-          ch[i + k*ido] = ref(cc,i + k*ip*ido);
-        }
-      }
-    } else {
-      for (i=0; i<ido; i++) {
-        for (k=0; k<l1; k++) {
-          ch[i + k*ido] = ref(cc,i + k*ip*ido);
-        }
-      }
-    }
-    for (j=1; j<ipph; j++) {
-      jc = ip - j;
-      j2 = 2*j;
-      for (k=0; k<l1; k++) {
-        ch[(k + j*l1)*ido] = ref(cc,ido-1 + (j2 - 1 + k*ip)*ido) + ref(cc,ido-1 + (j2 - 1 + k*ip)*
-            ido);
-        ch[(k + jc*l1)*ido] = ref(cc,(j2 + k*ip)*ido) + ref(cc,(j2 + k*ip)*ido);
-      }
-    }
-
-    if (ido != 1) {
-      if (nbd >= l1) {
-        for (j=1; j<ipph; j++) {
-          jc = ip - j;
-          for (k=0; k<l1; k++) {
-            for (i=2; i<ido; i+=2) {
-              ic = ido - i;
-              ch[i - 1 + (k + j*l1)*ido] = ref(cc,i - 1 + (2*j + k*ip)*ido) + ref(cc,
-                  ic - 1 + (2*j - 1 + k*ip)*ido);
-              ch[i - 1 + (k + jc*l1)*ido] = ref(cc,i - 1 + (2*j + k*ip)*ido) -
-                  ref(cc,ic - 1 + (2*j - 1 + k*ip)*ido);
-              ch[i + (k + j*l1)*ido] = ref(cc,i + (2*j + k*ip)*ido) - ref(cc,ic
-                  + (2*j - 1 + k*ip)*ido);
-              ch[i + (k + jc*l1)*ido] = ref(cc,i + (2*j + k*ip)*ido) + ref(cc,ic
-                  + (2*j - 1 + k*ip)*ido);
-            }
-          }
-        }
-      } else {
-        for (j=1; j<ipph; j++) {
-          jc = ip - j;
-          for (i=2; i<ido; i+=2) {
-            ic = ido - i;
-            for (k=0; k<l1; k++) {
-              ch[i - 1 + (k + j*l1)*ido] = ref(cc,i - 1 + (2*j + k*ip)*ido) + ref(cc,
-                  ic - 1 + (2*j - 1 + k*ip)*ido);
-              ch[i - 1 + (k + jc*l1)*ido] = ref(cc,i - 1 + (2*j + k*ip)*ido) -
-                  ref(cc,ic - 1 + (2*j - 1 + k*ip)*ido);
-              ch[i + (k + j*l1)*ido] = ref(cc,i + (2*j + k*ip)*ido) - ref(cc,ic
-                  + (2*j - 1 + k*ip)*ido);
-              ch[i + (k + jc*l1)*ido] = ref(cc,i + (2*j + k*ip)*ido) + ref(cc,ic
-                  + (2*j - 1 + k*ip)*ido);
-            }
-          }
-        }
-      }
-    }
-
-    ar1 = 1;
-    ai1 = 0;
-    for (l=1; l<ipph; l++) {
-      lc = ip - l;
-      ar1h = dcp*ar1 - dsp*ai1;
-      ai1 = dcp*ai1 + dsp*ar1;
-      ar1 = ar1h;
-      for (ik=0; ik<idl1; ik++) {
-        cc[ik + l*idl1] = ch[ik] + ar1*ch[ik + idl1];
-        cc[ik + lc*idl1] = ai1*ch[ik + (ip-1)*idl1];
-      }
-      dc2 = ar1;
-      ds2 = ai1;
-      ar2 = ar1;
-      ai2 = ai1;
-      for (j=2; j<ipph; j++) {
-        jc = ip - j;
-        ar2h = dc2*ar2 - ds2*ai2;
-        ai2 = dc2*ai2 + ds2*ar2;
-        ar2 = ar2h;
-        for (ik=0; ik<idl1; ik++) {
-          cc[ik + l*idl1] += ar2*ch[ik + j*idl1];
-          cc[ik + lc*idl1] += ai2*ch[ik + jc*idl1];
-        }
-      }
-    }
-    for (j=1; j<ipph; j++) {
-      for (ik=0; ik<idl1; ik++) {
-        ch[ik] += ch[ik + j*idl1];
-      }
-    }
-    for (j=1; j<ipph; j++) {
-      jc = ip - j;
-      for (k=0; k<l1; k++) {
-        ch[(k + j*l1)*ido] = cc[(k + j*l1)*ido] - cc[(k + jc*l1)*ido];
-        ch[(k + jc*l1)*ido] = cc[(k + j*l1)*ido] + cc[(k + jc*l1)*ido];
-      }
-    }
-
-    if (ido == 1) return;
-    if (nbd >= l1) {
-      for (j=1; j<ipph; j++) {
-        jc = ip - j;
-        for (k=0; k<l1; k++) {
-          for (i=2; i<ido; i+=2) {
-            ch[i - 1 + (k + j*l1)*ido] = cc[i - 1 + (k + j*l1)*ido] - cc[i + (k + jc*l1)*ido];
-            ch[i - 1 + (k + jc*l1)*ido] = cc[i - 1 + (k + j*l1)*ido] + cc[i + (k + jc*l1)*ido];
-            ch[i + (k + j*l1)*ido] = cc[i + (k + j*l1)*ido] + cc[i - 1 + (k + jc*l1)*ido];
-            ch[i + (k + jc*l1)*ido] = cc[i + (k + j*l1)*ido] - cc[i - 1 + (k + jc*l1)*ido];
-          }
-        }
-      }
-    } else {
-      for (j=1; j<ipph; j++) {
-        jc = ip - j;
-        for (i=2; i<ido; i+=2) {
-          for (k=0; k<l1; k++) {
-            ch[i - 1 + (k + j*l1)*ido] = cc[i - 1 + (k + j*l1)*ido] - cc[i + (k + jc*l1)*ido];
-            ch[i - 1 + (k + jc*l1)*ido] = cc[i - 1 + (k + j *l1)*ido] + cc[i + (k + jc*l1)*ido];
-            ch[i + (k + j*l1)*ido] = cc[i + (k + j*l1)*ido] + cc[i - 1 + (k + jc*l1)*ido];
-            ch[i + (k + jc*l1)*ido] = cc[i + (k + j*l1)*ido] - cc[i - 1 + (k + jc*l1)*ido];
-          }
-        }
-      }
-    }
-    for (ik=0; ik<idl1; ik++) cc[ik] = ch[ik];
-    for (j=1; j<ip; j++)
-      for (k=0; k<l1; k++)
-        cc[(k + j*l1)*ido] = ch[(k + j*l1)*ido];
-    if (nbd <= l1) {
-      is = -ido;
-      for (j=1; j<ip; j++) {
-        is += ido;
-        idij = is-1;
-        for (i=2; i<ido; i+=2) {
-          idij += 2;
-          for (k=0; k<l1; k++) {
-            cc[i - 1 + (k + j*l1)*ido] = wa[idij - 1]*ch[i - 1 + (k + j*l1)*ido] - wa[idij]*
-                ch[i + (k + j*l1)*ido];
-            cc[i + (k + j*l1)*ido] = wa[idij - 1]*ch[i + (k + j*l1)*ido] + wa[idij]*ch[i - 1 + (k + j*l1)*ido];
-          }
-        }
-      }
-    } else {
-      is = -ido;
-      for (j=1; j<ip; j++) {
-        is += ido;
-        for (k=0; k<l1; k++) {
-          idij = is - 1;
-          for (i=2; i<ido; i+=2) {
-            idij += 2;
-            cc[i - 1 + (k + j*l1)*ido] = wa[idij-1]*ch[i - 1 + (k + j*l1)*ido] - wa[idij]*
-                ch[i + (k + j*l1)*ido];
-            cc[i + (k + j*l1)*ido] = wa[idij-1]*ch[i + (k + j*l1)*ido] + wa[idij]*ch[i - 1 + (k + j*l1)*ido];
-          }
-        }
-      }
-    }
-  } /* radbg */
-
-  /* ----------------------------------------------------------------------
-cfftf1, cfftf, cfftb, cffti1, cffti. Complex FFTs.
----------------------------------------------------------------------- */
-
-static void cfftf1(int n, Treal c[], Treal ch[], const Treal wa[], const int ifac[MAXFAC+2], int isign)
-  {
-    int idot, i;
-    int k1, l1, l2;
-    int na, nf, ip, iw, ix2, ix3, ix4, nac, ido, idl1;
-    Treal *cinput, *coutput;
-    nf = ifac[1];
-    na = 0;
-    l1 = 1;
-    iw = 0;
-    for (k1=2; k1<=nf+1; k1++) {
-      ip = ifac[k1];
-      l2 = ip*l1;
-      ido = n / l2;
-      idot = ido + ido;
-      idl1 = idot*l1;
-      if (na) {
-        cinput = ch;
-        coutput = c;
-      } else {
-        cinput = c;
-        coutput = ch;
-      }
-      switch (ip) {
-      case 4:
-        ix2 = iw + idot;
-        ix3 = ix2 + idot;
-        passf4(idot, l1, cinput, coutput, &wa[iw], &wa[ix2], &wa[ix3], isign);
-        na = !na;
-        break;
-      case 2:
-        passf2(idot, l1, cinput, coutput, &wa[iw], isign);
-        na = !na;
-        break;
-      case 3:
-        ix2 = iw + idot;
-        passf3(idot, l1, cinput, coutput, &wa[iw], &wa[ix2], isign);
-        na = !na;
-        break;
-      case 5:
-        ix2 = iw + idot;
-        ix3 = ix2 + idot;
-        ix4 = ix3 + idot;
-        passf5(idot, l1, cinput, coutput, &wa[iw], &wa[ix2], &wa[ix3], &wa[ix4], isign);
-        na = !na;
-        break;
-      default:
-        passf(&nac, idot, ip, l1, idl1, cinput, coutput, &wa[iw], isign);
-        if (nac != 0) na = !na;
-      }
-      l1 = l2;
-      iw += (ip - 1)*idot;
-    }
-    if (na == 0) return;
-    for (i=0; i<2*n; i++) c[i] = ch[i];
-  } /* cfftf1 */
-
-
-void cfftf(int n, Treal c[], Treal wsave[])
-  {
-    int iw1, iw2;
-    if (n == 1) return;
-    iw1 = 2*n;
-    iw2 = iw1 + 2*n;
-    cfftf1(n, c, wsave, wsave+iw1, (int*)(wsave+iw2), -1);
-  } /* cfftf */
-
-
-void cfftb(int n, Treal c[], Treal wsave[])
-  {
-    int iw1, iw2;
-    if (n == 1) return;
-    iw1 = 2*n;
-    iw2 = iw1 + 2*n;
-    cfftf1(n, c, wsave, wsave+iw1, (int*)(wsave+iw2), +1);
-  } /* cfftb */
-
-
-static void factorize(int n, int ifac[MAXFAC+2], const int ntryh[NSPECIAL])
-  /* Factorize n in factors in ntryh and rest. On exit,
-ifac[0] contains n and ifac[1] contains number of factors,
-the factors start from ifac[2]. */
-  {
-    int ntry=3, i, j=0, ib, nf=0, nl=n, nq, nr;
-startloop:
-    if (j < NSPECIAL)
-      ntry = ntryh[j];
-    else
-      ntry+= 2;
-    j++;
-    do {
-      nq = nl / ntry;
-      nr = nl - ntry*nq;
-      if (nr != 0) goto startloop;
-      nf++;
-      ifac[nf + 1] = ntry;
-      nl = nq;
-      if (ntry == 2 && nf != 1) {
-        for (i=2; i<=nf; i++) {
-          ib = nf - i + 2;
-          ifac[ib + 1] = ifac[ib];
-        }
-        ifac[2] = 2;
-      }
-    } while (nl != 1);
-    ifac[0] = n;
-    ifac[1] = nf;
-  }
-
-
-static void cffti1(int n, Treal wa[], int ifac[MAXFAC+2])
-  {
-    static const Treal twopi = 6.28318530717959;
-    Treal arg, argh, argld, fi;
-    int idot, i, j;
-    int i1, k1, l1, l2;
-    int ld, ii, nf, ip;
-    int ido, ipm;
-
-    static const int ntryh[NSPECIAL] = {
-      3,4,2,5    }; /* Do not change the order of these. */
-
-    factorize(n,ifac,ntryh);
-    nf = ifac[1];
-    argh = twopi/(Treal)n;
-    i = 1;
-    l1 = 1;
-    for (k1=1; k1<=nf; k1++) {
-      ip = ifac[k1+1];
-      ld = 0;
-      l2 = l1*ip;
-      ido = n / l2;
-      idot = ido + ido + 2;
-      ipm = ip - 1;
-      for (j=1; j<=ipm; j++) {
-        i1 = i;
-        wa[i-1] = 1;
-        wa[i] = 0;
-        ld += l1;
-        fi = 0;
-        argld = ld*argh;
-        for (ii=4; ii<=idot; ii+=2) {
-          i+= 2;
-          fi+= 1;
-          arg = fi*argld;
-          wa[i-1] = cos(arg);
-          wa[i] = sin(arg);
-        }
-        if (ip > 5) {
-          wa[i1-1] = wa[i-1];
-          wa[i1] = wa[i];
-        }
-      }
-      l1 = l2;
-    }
-  } /* cffti1 */
-
-
-void cffti(int n, Treal wsave[])
- {
-    int iw1, iw2;
-    if (n == 1) return;
-    iw1 = 2*n;
-    iw2 = iw1 + 2*n;
-    cffti1(n, wsave+iw1, (int*)(wsave+iw2));
-  } /* cffti */
-
-  /* ----------------------------------------------------------------------
-rfftf1, rfftb1, rfftf, rfftb, rffti1, rffti. Treal FFTs.
----------------------------------------------------------------------- */
-
-static void rfftf1(int n, Treal c[], Treal ch[], const Treal wa[], const int ifac[MAXFAC+2])
-  {
-    int i;
-    int k1, l1, l2, na, kh, nf, ip, iw, ix2, ix3, ix4, ido, idl1;
-    Treal *cinput, *coutput;
-    nf = ifac[1];
-    na = 1;
-    l2 = n;
-    iw = n-1;
-    for (k1 = 1; k1 <= nf; ++k1) {
-      kh = nf - k1;
-      ip = ifac[kh + 2];
-      l1 = l2 / ip;
-      ido = n / l2;
-      idl1 = ido*l1;
-      iw -= (ip - 1)*ido;
-      na = !na;
-      if (na) {
-        cinput = ch;
-        coutput = c;
-      } else {
-        cinput = c;
-        coutput = ch;
-      }
-      switch (ip) {
-      case 4:
-        ix2 = iw + ido;
-        ix3 = ix2 + ido;
-        radf4(ido, l1, cinput, coutput, &wa[iw], &wa[ix2], &wa[ix3]);
-        break;
-      case 2:
-        radf2(ido, l1, cinput, coutput, &wa[iw]);
-        break;
-      case 3:
-        ix2 = iw + ido;
-        radf3(ido, l1, cinput, coutput, &wa[iw], &wa[ix2]);
-        break;
-      case 5:
-        ix2 = iw + ido;
-        ix3 = ix2 + ido;
-        ix4 = ix3 + ido;
-        radf5(ido, l1, cinput, coutput, &wa[iw], &wa[ix2], &wa[ix3], &wa[ix4]);
-        break;
-      default:
-        if (ido == 1)
-          na = !na;
-        if (na == 0) {
-          radfg(ido, ip, l1, idl1, c, ch, &wa[iw]);
-          na = 1;
-        } else {
-          radfg(ido, ip, l1, idl1, ch, c, &wa[iw]);
-          na = 0;
-        }
-      }
-      l2 = l1;
-    }
-    if (na == 1) return;
-    for (i = 0; i < n; i++) c[i] = ch[i];
-  } /* rfftf1 */
-
-
-void rfftb1(int n, Treal c[], Treal ch[], const Treal wa[], const int ifac[MAXFAC+2])
-  {
-    int i;
-    int k1, l1, l2, na, nf, ip, iw, ix2, ix3, ix4, ido, idl1;
-    Treal *cinput, *coutput;
-    nf = ifac[1];
-    na = 0;
-    l1 = 1;
-    iw = 0;
-    for (k1=1; k1<=nf; k1++) {
-      ip = ifac[k1 + 1];
-      l2 = ip*l1;
-      ido = n / l2;
-      idl1 = ido*l1;
-      if (na) {
-        cinput = ch;
-        coutput = c;
-      } else {
-        cinput = c;
-        coutput = ch;
-      }
-      switch (ip) {
-      case 4:
-        ix2 = iw + ido;
-        ix3 = ix2 + ido;
-        radb4(ido, l1, cinput, coutput, &wa[iw], &wa[ix2], &wa[ix3]);
-        na = !na;
-        break;
-      case 2:
-        radb2(ido, l1, cinput, coutput, &wa[iw]);
-        na = !na;
-        break;
-      case 3:
-        ix2 = iw + ido;
-        radb3(ido, l1, cinput, coutput, &wa[iw], &wa[ix2]);
-        na = !na;
-        break;
-      case 5:
-        ix2 = iw + ido;
-        ix3 = ix2 + ido;
-        ix4 = ix3 + ido;
-        radb5(ido, l1, cinput, coutput, &wa[iw], &wa[ix2], &wa[ix3], &wa[ix4]);
-        na = !na;
-        break;
-      default:
-        radbg(ido, ip, l1, idl1, cinput, coutput, &wa[iw]);
-        if (ido == 1) na = !na;
-      }
-      l1 = l2;
-      iw += (ip - 1)*ido;
-    }
-    if (na == 0) return;
-    for (i=0; i<n; i++) c[i] = ch[i];
-  } /* rfftb1 */
-
-
-void rfftf(int n, Treal r[], Treal wsave[])
-  {
-    if (n == 1) return;
-    rfftf1(n, r, wsave, wsave+n, (int*)(wsave+2*n));
-  } /* rfftf */
-
-
-void rfftb(int n, Treal r[], Treal wsave[])
-  {
-    if (n == 1) return;
-    rfftb1(n, r, wsave, wsave+n, (int*)(wsave+2*n));
-  } /* rfftb */
-
-
-static void rffti1(int n, Treal wa[], int ifac[MAXFAC+2])
-  {
-    static const Treal twopi = 6.28318530717959;
-    Treal arg, argh, argld, fi;
-    int i, j;
-    int k1, l1, l2;
-    int ld, ii, nf, ip, is;
-    int ido, ipm, nfm1;
-    static const int ntryh[NSPECIAL] = {
-      4,2,3,5    }; /* Do not change the order of these. */
-    factorize(n,ifac,ntryh);
-    nf = ifac[1];
-    argh = twopi / n;
-    is = 0;
-    nfm1 = nf - 1;
-    l1 = 1;
-    if (nfm1 == 0) return;
-    for (k1 = 1; k1 <= nfm1; k1++) {
-      ip = ifac[k1 + 1];
-      ld = 0;
-      l2 = l1*ip;
-      ido = n / l2;
-      ipm = ip - 1;
-      for (j = 1; j <= ipm; ++j) {
-        ld += l1;
-        i = is;
-        argld = (Treal) ld*argh;
-        fi = 0;
-        for (ii = 3; ii <= ido; ii += 2) {
-          i += 2;
-          fi += 1;
-          arg = fi*argld;
-          wa[i - 2] = cos(arg);
-          wa[i - 1] = sin(arg);
-        }
-        is += ido;
-      }
-      l1 = l2;
-    }
-  } /* rffti1 */
-
-
-void rffti(int n, Treal wsave[])
-  {
-    if (n == 1) return;
-    rffti1(n, wsave+n, (int*)(wsave+2*n));
-  } /* rffti */
-
-#ifdef __cplusplus
-}
-#endif
-- 
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