source: branches/profile/GDE/PHYML/eigen.c

/***********************************************************
*  This eigen() routine works for eigenvalue/vector analysis
*         for real general square matrix A
*         A will be destroyed
*         rr,ri are vectors containing eigenvalues
*         vr,vi are matrices containing (right) eigenvectors
*
*              A*[vr+vi*i] = [vr+vi*i] * diag{rr+ri*i}
*
*  Algorithm: Handbook for Automatic Computation, vol 2
*             by Wilkinson and Reinsch, 1971
*             most of source codes were taken from a public domain
*             solftware called MATCALC.
*  Credits:   to the authors of MATCALC
*
*  return     -1 not converged
*              0 no complex eigenvalues/vectors
*              1 complex eigenvalues/vectors
*  Tianlin Wang at University of Illinois
*  Thu May  6 15:22:31 CDT 1993
***************************************************************/

#include "utilities.h"
#include "eigen.h"

#define BASE        2    /* base of floating point arithmetic */
#define DIGITS     40    /* no. of digits to the base BASE in the fraction */
/*
#define DIGITS     53
*/
#define MAXITER    30    /* max2. no. of iterations to converge */

#define pos(i,j,n)      ((i)*(n)+(j))


/* rr/vr : real parts of eigen values/vectors */
/* ri/vi : imaginary part s of eigen values/vectors */


int eigen(int job, double *A, int n, double *rr, double *ri, 
          double *vr, double *vi, double *work)
{    
/* job=0: eigen values only
       1: both eigen values and eigen vectors 
   double w[n*2]: work space
*/
    int low,hi,i,j,k, it, istate=0;
    double tiny=sqrt(pow((double)BASE,(double)(1-DIGITS))), t; 

/*     printf("EIGEN\n"); */

    balance(A,n,&low,&hi,work);
    elemhess(job,A,n,low,hi,vr,vi, (int*)(work+n));
    if (-1 == realeig(job,A,n,low,hi,rr,ri,vr,vi)) return (-1);
    if (job) unbalance(n,vr,vi,low,hi,work);

/* sort, added by Z. Yang */
   for (i=0; i<n; i++) {
       for (j=i+1,it=i,t=rr[i]; j<n; j++)
           if (t<rr[j]) { t=rr[j]; it=j; }
       rr[it]=rr[i];   rr[i]=t;
       t=ri[it];       ri[it]=ri[i];  ri[i]=t;
       for (k=0; k<n; k++) {
          t=vr[k*n+it];  vr[k*n+it]=vr[k*n+i];  vr[k*n+i]=t;
          t=vi[k*n+it];  vi[k*n+it]=vi[k*n+i];  vi[k*n+i]=t;
       }
       if (fabs(ri[i])>tiny) istate=1;
   }

    return (istate) ;
}

/* complex funcctions
*/

complex compl (double re,double im)
{
    complex r;

    r.re = re;
    r.im = im;
    return(r);
}

complex _conj (complex a)
{
    a.im = -a.im;
    return(a);
}

#define csize(a) (fabs(a.re)+fabs(a.im))

complex cplus (complex a, complex b)
{
   complex c;
   c.re = a.re+b.re;  
   c.im = a.im+b.im;   
   return (c);
}

complex cminus (complex a, complex b)
{
   complex c;
   c.re = a.re-b.re;  
   c.im = a.im-b.im;   
   return (c);
}

complex cby (complex a, complex b)
{
   complex c;
   c.re = a.re*b.re-a.im*b.im ;
   c.im = a.re*b.im+a.im*b.re ;
   return (c);
}

complex cdiv (complex a,complex b)
{
    double ratio, den;
    complex c;

    if (fabs(b.re) <= fabs(b.im)) {
        ratio = b.re / b.im;
        den = b.im * (1 + ratio * ratio);
        c.re = (a.re * ratio + a.im) / den;
        c.im = (a.im * ratio - a.re) / den;
    }
    else {
        ratio = b.im / b.re;
        den = b.re * (1 + ratio * ratio);
        c.re = (a.re + a.im * ratio) / den;
        c.im = (a.im - a.re * ratio) / den;
    }
    return(c);
}

/* complex local_cexp (complex a) */
/* { */
/*    complex c; */
/*    c.re = exp(a.re); */
/*    if (fabs(a.im)==0) c.im = 0;  */
/*    else  { c.im = c.re*sin(a.im); c.re*=cos(a.im); } */
/*    return (c); */
/* } */

complex cfactor (complex x, double a)
{
   complex c;
   c.re = a*x.re; 
   c.im = a*x.im;
   return (c);
}

int cxtoy (complex *x, complex *y, int n)
{
   int i;
   For (i,n) y[i]=x[i];
   return (0);
}

int cmatby (complex *a, complex *b, complex *c, int n,int m,int k)
/* a[n*m], b[m*k], c[n*k]  ......  c = a*b
*/
{
   int i,j,i1;
   complex t;

   For (i,n)  For(j,k) {
       for (i1=0,t=compl(0,0); i1<m; i1++)  
           t = cplus (t, cby(a[i*m+i1],b[i1*k+j]));
       c[i*k+j] = t;
   }
   return (0);
}

int cmatinv( complex *x, int n, int m, double *space)
{
/* x[n*m]  ... m>=n
*/
   int i,j,k, *irow=(int*) space;
   double xmaxsize, ee=1e-20;
   complex xmax, t,t1;

   For(i,n)  {
       xmaxsize = 0.;
       for (j=i; j<n; j++) {
          if ( xmaxsize < csize (x[j*m+i]))  {
               xmaxsize = csize (x[j*m+i]);
               xmax = x[j*m+i];
               irow[i] = j;
          }
       }
       if (xmaxsize < ee)   {
           printf("\nDet goes to zero at %8d!\t\n", i+1);
           return(-1);
       }
       if (irow[i] != i) {
           For(j,m) {
                t = x[i*m+j];
                x[i*m+j] = x[irow[i]*m+j];
                x[ irow[i]*m+j] = t;
           }
       }
       t = cdiv (compl(1,0), x[i*m+i]);
       For(j,n) {
           if (j == i) continue;
           t1 = cby (t,x[j*m+i]);
           For(k,m)  x[j*m+k] = cminus (x[j*m+k], cby(t1,x[i*m+k]));
           x[j*m+i] = cfactor (t1, -1);
       }
       For(j,m)   x[i*m+j] = cby (x[i*m+j], t);
       x[i*m+i] = t;
   }                         
   for (i=n-1; i>=0; i--) {
        if (irow[i] == i) continue;
        For(j,n)  {
           t = x[j*m+i];
           x[j*m+i] = x[j*m+irow[i]];
           x[ j*m+irow[i]] = t;
        }
   }
   return (0);
}


void balance(double *mat, int n,int *low, int *hi, double *scale)
{
/* Balance a matrix for calculation of eigenvalues and eigenvectors
*/
    double c,f,g,r,s;
    int i,j,k,l,done;
        /* search for rows isolating an eigenvalue and push them down */
    for (k = n - 1; k >= 0; k--) {
        for (j = k; j >= 0; j--) {
            for (i = 0; i <= k; i++) {
                if (i != j && fabs(mat[pos(j,i,n)]) != 0) break;
            }

            if (i > k) {
                scale[k] = j;

                if (j != k) {
                    for (i = 0; i <= k; i++) {
                       c = mat[pos(i,j,n)];
                       mat[pos(i,j,n)] = mat[pos(i,k,n)];
                       mat[pos(i,k,n)] = c;
                    }

                    for (i = 0; i < n; i++) {
                       c = mat[pos(j,i,n)];
                       mat[pos(j,i,n)] = mat[pos(k,i,n)];
                       mat[pos(k,i,n)] = c;
                    }
                }
                break;
            }
        }
        if (j < 0) break;
    }

    /* search for columns isolating an eigenvalue and push them left */

    for (l = 0; l <= k; l++) {
        for (j = l; j <= k; j++) {
            for (i = l; i <= k; i++) {
                if (i != j && fabs(mat[pos(i,j,n)]) != 0) break;
            }
            if (i > k) {
                scale[l] = j;
                if (j != l) {
                    for (i = 0; i <= k; i++) {
                       c = mat[pos(i,j,n)];
                       mat[pos(i,j,n)] = mat[pos(i,l,n)];
                       mat[pos(i,l,n)] = c;
                    }

                    for (i = l; i < n; i++) {
                       c = mat[pos(j,i,n)];
                       mat[pos(j,i,n)] = mat[pos(l,i,n)];
                       mat[pos(l,i,n)] = c;
                    }
                }

                break;
            }
        }

        if (j > k) break;
    }

    *hi = k;
    *low = l;

    /* balance the submatrix in rows l through k */

    for (i = l; i <= k; i++) {
        scale[i] = 1;
    }

    do {
        for (done = 1,i = l; i <= k; i++) {
            for (c = 0,r = 0,j = l; j <= k; j++) {
                if (j != i) {
                    c += fabs(mat[pos(j,i,n)]);
                    r += fabs(mat[pos(i,j,n)]);
                }
            }

            if (c != 0 && r != 0) {
                g = r / BASE;
                f = 1;
                s = c + r;

                while (c < g) {
                    f *= BASE;
                    c *= BASE * BASE;
                }

                g = r * BASE;

                while (c >= g) {
                    f /= BASE;
                    c /= BASE * BASE;
                }

                if ((c + r) / f < 0.95 * s) {
                    done = 0;
                    g = 1 / f;
                    scale[i] *= f;

                    for (j = l; j < n; j++) {
                        mat[pos(i,j,n)] *= g;
                    }

                    for (j = 0; j <= k; j++) {
                        mat[pos(j,i,n)] *= f;
                    }
                }
            }
        }
    } while (!done);
}


/*
 * Transform back eigenvectors of a balanced matrix
 * into the eigenvectors of the original matrix
 */
void unbalance(int n,double *vr,double *vi, int low, int hi, double *scale)
{
    int i,j,k;
    double tmp;

    for (i = low; i <= hi; i++) {
        for (j = 0; j < n; j++) {
            vr[pos(i,j,n)] *= scale[i];
            vi[pos(i,j,n)] *= scale[i];
        }
    }

    for (i = low - 1; i >= 0; i--) {
        if ((k = (int)scale[i]) != i) {
            for (j = 0; j < n; j++) {
                tmp = vr[pos(i,j,n)];
                vr[pos(i,j,n)] = vr[pos(k,j,n)];
                vr[pos(k,j,n)] = tmp;

                tmp = vi[pos(i,j,n)];
                vi[pos(i,j,n)] = vi[pos(k,j,n)];
                vi[pos(k,j,n)] = tmp;        
            }
        }
    }

    for (i = hi + 1; i < n; i++) {
        if ((k = (int)scale[i]) != i) {
            for (j = 0; j < n; j++) {
                tmp = vr[pos(i,j,n)];
                vr[pos(i,j,n)] = vr[pos(k,j,n)];
                vr[pos(k,j,n)] = tmp;

                tmp = vi[pos(i,j,n)];
                vi[pos(i,j,n)] = vi[pos(k,j,n)];
                vi[pos(k,j,n)] = tmp;        
            }
        }
    }
}

/*
 * Reduce the submatrix in rows and columns low through hi of real matrix mat to
 * Hessenberg form by elementary similarity transformations
 */
void elemhess(int job,double *mat,int n,int low,int hi, double *vr,
              double *vi, int *work)
{
/* work[n] */
    int i,j,m;
    double x,y;

    for (m = low + 1; m < hi; m++) {
        for (x = 0,i = m,j = m; j <= hi; j++) {
            if (fabs(mat[pos(j,m-1,n)]) > fabs(x)) {
                x = mat[pos(j,m-1,n)];
                i = j;
            }
        }

        if ((work[m] = i) != m) {
            for (j = m - 1; j < n; j++) {
               y = mat[pos(i,j,n)];
               mat[pos(i,j,n)] = mat[pos(m,j,n)];
               mat[pos(m,j,n)] = y;
            }

            for (j = 0; j <= hi; j++) {
               y = mat[pos(j,i,n)];
               mat[pos(j,i,n)] = mat[pos(j,m,n)];
               mat[pos(j,m,n)] = y;
            }
        }

        if (x != 0) {
            for (i = m + 1; i <= hi; i++) {
                if ((y = mat[pos(i,m-1,n)]) != 0) {
                    y = mat[pos(i,m-1,n)] = y / x;

                    for (j = m; j < n; j++) {
                        mat[pos(i,j,n)] -= y * mat[pos(m,j,n)];
                    }

                    for (j = 0; j <= hi; j++) {
                        mat[pos(j,m,n)] += y * mat[pos(j,i,n)];
                    }
                }
            }
        }
    }
    if (job) {
       for (i=0; i<n; i++) {
          for (j=0; j<n; j++) {
             vr[pos(i,j,n)] = 0.0; vi[pos(i,j,n)] = 0.0;
          }
          vr[pos(i,i,n)] = 1.0;
       }

       for (m = hi - 1; m > low; m--) {
          for (i = m + 1; i <= hi; i++) {
             vr[pos(i,m,n)] = mat[pos(i,m-1,n)];
          }

         if ((i = work[m]) != m) {
            for (j = m; j <= hi; j++) {
               vr[pos(m,j,n)] = vr[pos(i,j,n)];
               vr[pos(i,j,n)] = 0.0;
            }
            vr[pos(i,m,n)] = 1.0;
         }
      }
   }
}

/*
 * Calculate eigenvalues and eigenvectors of a real upper Hessenberg matrix
 * Return 1 if converges successfully and 0 otherwise
 */
 
int realeig(int job,double *mat,int n,int low, int hi, double *valr,
      double *vali, double *vr,double *vi)
{
   complex v;
   double p=0,q=0,r=0,s=0,t,w,x,y,z=0,ra,sa,norm,eps;
   int niter,en,i,j,k,l,m;
   double precision  = pow((double)BASE,(double)(1-DIGITS));

   eps = precision;
   for (i=0; i<n; i++) {
      valr[i]=0.0;
      vali[i]=0.0;
   }
      /* store isolated roots and calculate norm */
   for (norm = 0,i = 0; i < n; i++) {
      for (j = MAX(0,i-1); j < n; j++) {
         norm += fabs(mat[pos(i,j,n)]);
      }
      if (i < low || i > hi) valr[i] = mat[pos(i,i,n)];
   }
   t = 0;
   en = hi;

   while (en >= low) {
      niter = 0;
      for (;;) {

       /* look for single small subdiagonal element */

         for (l = en; l > low; l--) {
            s = fabs(mat[pos(l-1,l-1,n)]) + fabs(mat[pos(l,l,n)]);
            if (s == 0) s = norm;
            if (fabs(mat[pos(l,l-1,n)]) <= eps * s) break;
         }

         /* form shift */

         x = mat[pos(en,en,n)];

         if (l == en) {             /* one root found */
            valr[en] = x + t;
            if (job) mat[pos(en,en,n)] = x + t;
            en--;
            break;
         }

         y = mat[pos(en-1,en-1,n)];
         w = mat[pos(en,en-1,n)] * mat[pos(en-1,en,n)];

         if (l == en - 1) {                /* two roots found */
            p = (y - x) / 2;
            q = p * p + w;
            z = sqrt(fabs(q));
            x += t;
            if (job) {
               mat[pos(en,en,n)] = x;
               mat[pos(en-1,en-1,n)] = y + t;
            }
            if (q < 0) {                /* complex pair */
               valr[en-1] = x+p;
               vali[en-1] = z;
               valr[en] = x+p;
               vali[en] = -z;
            }
            else {                      /* real pair */
               z = (p < 0) ? p - z : p + z;
               valr[en-1] = x + z;
               valr[en] = (z == 0) ? x + z : x - w / z;
               if (job) {
                  x = mat[pos(en,en-1,n)];
                  s = fabs(x) + fabs(z);
                  p = x / s;
                  q = z / s;
                  r = sqrt(p*p+q*q);
                  p /= r;
                  q /= r;
                  for (j = en - 1; j < n; j++) {
                     z = mat[pos(en-1,j,n)];
                     mat[pos(en-1,j,n)] = q * z + p *
                     mat[pos(en,j,n)];
                     mat[pos(en,j,n)] = q * mat[pos(en,j,n)] - p*z;
                  }
                  for (i = 0; i <= en; i++) {
                     z = mat[pos(i,en-1,n)];
                     mat[pos(i,en-1,n)] = q * z + p * mat[pos(i,en,n)];
                     mat[pos(i,en,n)] = q * mat[pos(i,en,n)] - p*z;
                  }
                  for (i = low; i <= hi; i++) {
                     z = vr[pos(i,en-1,n)];
                     vr[pos(i,en-1,n)] = q*z + p*vr[pos(i,en,n)];
                     vr[pos(i,en,n)] = q*vr[pos(i,en,n)] - p*z;
                  }
               }
            }
            en -= 2;
            break;
         }
         if (niter == MAXITER) return(-1);
         if (niter != 0 && niter % 10 == 0) {
            t += x;
            for (i = low; i <= en; i++) mat[pos(i,i,n)] -= x;
            s = fabs(mat[pos(en,en-1,n)]) + fabs(mat[pos(en-1,en-2,n)]);
            x = y = 0.75 * s;
            w = -0.4375 * s * s;
         }
         niter++;
           /* look for two consecutive small subdiagonal elements */
         for (m = en - 2; m >= l; m--) {
            z = mat[pos(m,m,n)];
            r = x - z;
            s = y - z;
            p = (r * s - w) / mat[pos(m+1,m,n)] + mat[pos(m,m+1,n)];
            q = mat[pos(m+1,m+1,n)] - z - r - s;
            r = mat[pos(m+2,m+1,n)];
            s = fabs(p) + fabs(q) + fabs(r);
            p /= s;
            q /= s;
            r /= s;
            if (m == l || fabs(mat[pos(m,m-1,n)]) * (fabs(q)+fabs(r)) <=
                eps * (fabs(mat[pos(m-1,m-1,n)]) + fabs(z) +
                fabs(mat[pos(m+1,m+1,n)])) * fabs(p)) break;
         }
         for (i = m + 2; i <= en; i++) mat[pos(i,i-2,n)] = 0;
         for (i = m + 3; i <= en; i++) mat[pos(i,i-3,n)] = 0;
             /* double QR step involving rows l to en and columns m to en */
         for (k = m; k < en; k++) {
            if (k != m) {
               p = mat[pos(k,k-1,n)];
               q = mat[pos(k+1,k-1,n)];
               r = (k == en - 1) ? 0 : mat[pos(k+2,k-1,n)];
               if ((x = fabs(p) + fabs(q) + fabs(r)) == 0) continue;
               p /= x;
               q /= x;
               r /= x;
            }
            s = sqrt(p*p+q*q+r*r);
            if (p < 0) s = -s;
            if (k != m) {
               mat[pos(k,k-1,n)] = -s * x;
            }
            else if (l != m) {
               mat[pos(k,k-1,n)] = -mat[pos(k,k-1,n)];
            }
            p += s;
            x = p / s;
            y = q / s;
            z = r / s;
            q /= p;
            r /= p;
                /* row modification */
            for (j = k; j <= (!job ? en : n-1); j++){
               p = mat[pos(k,j,n)] + q * mat[pos(k+1,j,n)];
               if (k != en - 1) {
                  p += r * mat[pos(k+2,j,n)];
                  mat[pos(k+2,j,n)] -= p * z;
               }
               mat[pos(k+1,j,n)] -= p * y;
               mat[pos(k,j,n)] -= p * x;
            }
            j = MIN(en,k+3);
              /* column modification */
            for (i = (!job ? l : 0); i <= j; i++) {
               p = x * mat[pos(i,k,n)] + y * mat[pos(i,k+1,n)];
               if (k != en - 1) {
                  p += z * mat[pos(i,k+2,n)];
                  mat[pos(i,k+2,n)] -= p*r;
               }
               mat[pos(i,k+1,n)] -= p*q;
               mat[pos(i,k,n)] -= p;
            }
            if (job) {             /* accumulate transformations */
               for (i = low; i <= hi; i++) {
                  p = x * vr[pos(i,k,n)] + y * vr[pos(i,k+1,n)];
                  if (k != en - 1) {
                     p += z * vr[pos(i,k+2,n)];
                     vr[pos(i,k+2,n)] -= p*r;
                  }
                  vr[pos(i,k+1,n)] -= p*q;
                  vr[pos(i,k,n)] -= p;
               }
            }
         }
      }
   }

   if (!job) return(0);
   if (norm != 0) {
       /* back substitute to find vectors of upper triangular form */
      for (en = n-1; en >= 0; en--) {
         p = valr[en];
         if ((q = vali[en]) < 0) {            /* complex vector */
            m = en - 1;
            if (fabs(mat[pos(en,en-1,n)]) > fabs(mat[pos(en-1,en,n)])) {
               mat[pos(en-1,en-1,n)] = q / mat[pos(en,en-1,n)];
               mat[pos(en-1,en,n)] = (p - mat[pos(en,en,n)]) /
                     mat[pos(en,en-1,n)];
            }
            else {
               v = cdiv(compl(0.0,-mat[pos(en-1,en,n)]),
                    compl(mat[pos(en-1,en-1,n)]-p,q));
               mat[pos(en-1,en-1,n)] = v.re;
               mat[pos(en-1,en,n)] = v.im;
            }
            mat[pos(en,en-1,n)] = 0;
            mat[pos(en,en,n)] = 1;
            for (i = en - 2; i >= 0; i--) {
               w = mat[pos(i,i,n)] - p;
               ra = 0;
               sa = mat[pos(i,en,n)];
               for (j = m; j < en; j++) {
                  ra += mat[pos(i,j,n)] * mat[pos(j,en-1,n)];
                  sa += mat[pos(i,j,n)] * mat[pos(j,en,n)];
               }
               if (vali[i] < 0) {
                  z = w;
                  r = ra;
                  s = sa;
               }
               else {
                  m = i;
                  if (vali[i] == 0) {
                     v = cdiv(compl(-ra,-sa),compl(w,q));
                     mat[pos(i,en-1,n)] = v.re;
                     mat[pos(i,en,n)] = v.im;
                  }
                  else {                      /* solve complex equations */
                     x = mat[pos(i,i+1,n)];
                     y = mat[pos(i+1,i,n)];
                     v.re = (valr[i]- p)*(valr[i]-p) + vali[i]*vali[i] - q*q;
                     v.im = (valr[i] - p)*2*q;
                     if ((fabs(v.re) + fabs(v.im)) == 0) {
                        v.re = eps * norm * (fabs(w) +
                                fabs(q) + fabs(x) + fabs(y) + fabs(z));
                     }
                     v = cdiv(compl(x*r-z*ra+q*sa,x*s-z*sa-q*ra),v);
                     mat[pos(i,en-1,n)] = v.re;
                     mat[pos(i,en,n)] = v.im;
                     if (fabs(x) > fabs(z) + fabs(q)) {
                        mat[pos(i+1,en-1,n)] = 
                             (-ra - w * mat[pos(i,en-1,n)] +
                             q * mat[pos(i,en,n)]) / x;
                        mat[pos(i+1,en,n)] = (-sa - w * mat[pos(i,en,n)] -
                             q * mat[pos(i,en-1,n)]) / x;
                     }
                     else {
                        v = cdiv(compl(-r-y*mat[pos(i,en-1,n)],
                             -s-y*mat[pos(i,en,n)]),compl(z,q));
                        mat[pos(i+1,en-1,n)] = v.re;
                        mat[pos(i+1,en,n)] = v.im;
                     }
                  }
               }
            }
         }
         else if (q == 0) {                             /* real vector */
            m = en;
            mat[pos(en,en,n)] = 1;
            for (i = en - 1; i >= 0; i--) {
               w = mat[pos(i,i,n)] - p;
               r = mat[pos(i,en,n)];
               for (j = m; j < en; j++) {
                  r += mat[pos(i,j,n)] * mat[pos(j,en,n)];
               }
               if (vali[i] < 0) {
                  z = w;
                  s = r;
               }
               else {
                  m = i;
                  if (vali[i] == 0) {
                     if ((t = w) == 0) t = eps * norm;
                     mat[pos(i,en,n)] = -r / t;
                  }
                  else {            /* solve real equations */
                     x = mat[pos(i,i+1,n)];
                     y = mat[pos(i+1,i,n)];
                     q = (valr[i] - p) * (valr[i] - p) + vali[i]*vali[i];
                     t = (x * s - z * r) / q;
                     mat[pos(i,en,n)] = t;
                     if (fabs(x) <= fabs(z)) {
                        mat[pos(i+1,en,n)] = (-s - y * t) / z;
                     }
                     else {
                        mat[pos(i+1,en,n)] = (-r - w * t) / x;
                     }
                  }
               }
            }
         }
      }
             /* vectors of isolated roots */
      for (i = 0; i < n; i++) {
         if (i < low || i > hi) {
            for (j = i; j < n; j++) {
               vr[pos(i,j,n)] = mat[pos(i,j,n)];
            }
         }
      }
       /* multiply by transformation matrix */

      for (j = n-1; j >= low; j--) {
         m = MIN(j,hi);
         for (i = low; i <= hi; i++) {
            for (z = 0,k = low; k <= m; k++) {
               z += vr[pos(i,k,n)] * mat[pos(k,j,n)];
            }
            vr[pos(i,j,n)] = z;
         }
      }
   }
    /* rearrange complex eigenvectors */
   for (j = 0; j < n; j++) {
      if (vali[j] != 0) {
         for (i = 0; i < n; i++) {
            vi[pos(i,j,n)] = vr[pos(i,j+1,n)];
            vr[pos(i,j+1,n)] = vr[pos(i,j,n)];
            vi[pos(i,j+1,n)] = -vi[pos(i,j,n)];
         }
         j++;
      }
   }
   return(0);
}