zschur.c

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#include <../../nrnconf.h>
 
/**************************************************************************
**
** Copyright (C) 1993 David E. Steward & Zbigniew Leyk, all rights reserved.
**
**               Meschach Library
** 
** This Meschach Library is provided "as is" without any express 
** or implied warranty of any kind with respect to this software. 
** In particular the authors shall not be liable for any direct, 
** indirect, special, incidental or consequential damages arising 
** in any way from use of the software.
** 
** Everyone is granted permission to copy, modify and redistribute this
** Meschach Library, provided:
**  1.  All copies contain this copyright notice.
**  2.  All modified copies shall carry a notice stating who
**      made the last modification and the date of such modification.
**  3.  No charge is made for this software or works derived from it.  
**      This clause shall not be construed as constraining other software
**      distributed on the same medium as this software, nor is a
**      distribution fee considered a charge.
**
***************************************************************************/
 
/*  
    File containing routines for computing the Schur decomposition
    of a complex non-symmetric matrix
    See also: hessen.c
    Complex version
*/
 
 
#include    <stdio.h>
#include    "zmatrix.h"
#include        "zmatrix2.h"
#include    <math.h>
 
static char rcsid[] = "zschur.c,v 1.1 1997/12/04 17:56:16 hines Exp";
 
#define is_zero(z)  ((z).re == 0.0 && (z).im == 0.0)
#define b2s(t_or_f) ((t_or_f) ? "TRUE" : "FALSE")
 
 
/* zschur -- computes the Schur decomposition of the matrix A in situ
    -- optionally, gives Q matrix such that Q^*.A.Q is upper triangular
    -- returns upper triangular Schur matrix */
ZMAT    *zschur(A,Q)
ZMAT    *A, *Q;
{
    int     i, j, iter, k, k_min, k_max, k_tmp, n, split;
    Real    c;
    complex det, discrim, lambda, lambda0, lambda1, s, sum, ztmp;
    complex x, y;   /* for chasing algorithm */
    complex **A_me;
    static  ZVEC    *diag=ZVNULL;
    
    if ( ! A )
    error(E_NULL,"zschur");
    if ( A->m != A->n || ( Q && Q->m != Q->n ) )
    error(E_SQUARE,"zschur");
    if ( Q != ZMNULL && Q->m != A->m )
    error(E_SIZES,"zschur");
    n = A->n;
    diag = zv_resize(diag,A->n);
    MEM_STAT_REG(diag,TYPE_ZVEC);
    /* compute Hessenberg form */
    zHfactor(A,diag);
    
    /* save Q if necessary, and make A explicitly Hessenberg */
    zHQunpack(A,diag,Q,A);
 
    k_min = 0;  A_me = A->me;
 
    while ( k_min < n )
    {
    /* find k_max to suit:
       submatrix k_min..k_max should be irreducible */
    k_max = n-1;
    for ( k = k_min; k < k_max; k++ )
        if ( is_zero(A_me[k+1][k]) )
        {   k_max = k;  break;  }
 
    if ( k_max <= k_min )
    {
        k_min = k_max + 1;
        continue;       /* outer loop */
    }
 
    /* now have r x r block with r >= 2:
       apply Francis QR step until block splits */
    split = FALSE;      iter = 0;
    while ( ! split )
    {
        complex a00, a01, a10, a11;
        iter++;
        
        /* set up Wilkinson/Francis complex shift */
        /* use the smallest eigenvalue of the bottom 2 x 2 submatrix */
        k_tmp = k_max - 1;
 
        a00 = A_me[k_tmp][k_tmp];
        a01 = A_me[k_tmp][k_max];
        a10 = A_me[k_max][k_tmp];
        a11 = A_me[k_max][k_max];
        ztmp.re = 0.5*(a00.re - a11.re);
        ztmp.im = 0.5*(a00.im - a11.im);
        discrim = zsqrt(zadd(zmlt(ztmp,ztmp),zmlt(a01,a10)));
        sum.re  = 0.5*(a00.re + a11.re);
        sum.im  = 0.5*(a00.im + a11.im);
        lambda0 = zadd(sum,discrim);
        lambda1 = zsub(sum,discrim);
        det = zsub(zmlt(a00,a11),zmlt(a01,a10)); 
        
        if ( is_zero(lambda0) && is_zero(lambda1) )
          {                                                          
        lambda.re = lambda.im = 0.0;
          } 
        else if ( zabs(lambda0) > zabs(lambda1) )
        lambda = zdiv(det,lambda0);
        else
        lambda = zdiv(det,lambda1);
 
        /* perturb shift if convergence is slow */
        if ( (iter % 10) == 0 )
        {
        lambda.re += iter*0.02;
        lambda.im += iter*0.02;
        }
 
        /* set up Householder transformations */
        k_tmp = k_min + 1;
 
        x = zsub(A->me[k_min][k_min],lambda);
        y = A->me[k_min+1][k_min];
 
        /* use Givens' rotations to "chase" off-Hessenberg entry */
        for ( k = k_min; k <= k_max-1; k++ )
        {
        zgivens(x,y,&c,&s);
        zrot_cols(A,k,k+1,c,s,A);
        zrot_rows(A,k,k+1,c,s,A);
        if ( Q != ZMNULL )
            zrot_cols(Q,k,k+1,c,s,Q);
 
        /* zero things that should be zero */
        if ( k > k_min )
            A->me[k+1][k-1].re = A->me[k+1][k-1].im = 0.0;
 
        /* get next entry to chase along sub-diagonal */
        x = A->me[k+1][k];
        if ( k <= k_max - 2 )
            y = A->me[k+2][k];
        else
            y.re = y.im = 0.0;
        }
 
        for ( k = k_min; k <= k_max-2; k++ )
        {
        /* zero appropriate sub-diagonals */
        A->me[k+2][k].re = A->me[k+2][k].im = 0.0;
        }
 
        /* test to see if matrix should split */
        for ( k = k_min; k < k_max; k++ )
        if ( zabs(A_me[k+1][k]) < MACHEPS*
            (zabs(A_me[k][k])+zabs(A_me[k+1][k+1])) )
        {
            A_me[k+1][k].re = A_me[k+1][k].im = 0.0;
            split = TRUE;
        }
 
    }
    }
    
    /* polish up A by zeroing strictly lower triangular elements
       and small sub-diagonal elements */
    for ( i = 0; i < A->m; i++ )
    for ( j = 0; j < i-1; j++ )
        A_me[i][j].re = A_me[i][j].im = 0.0;
    for ( i = 0; i < A->m - 1; i++ )
    if ( zabs(A_me[i+1][i]) < MACHEPS*
        (zabs(A_me[i][i])+zabs(A_me[i+1][i+1])) )
        A_me[i+1][i].re = A_me[i+1][i].im = 0.0;
 
    return A;
}
 
 
#if 0
/* schur_vecs -- returns eigenvectors computed from the real Schur
        decomposition of a matrix
    -- T is the block upper triangular Schur matrix
    -- Q is the orthognal matrix where A = Q.T.Q^T
    -- if Q is null, the eigenvectors of T are returned
    -- X_re is the real part of the matrix of eigenvectors,
        and X_im is the imaginary part of the matrix.
    -- X_re is returned */
MAT *schur_vecs(T,Q,X_re,X_im)
MAT *T, *Q, *X_re, *X_im;
{
    int i, j, limit;
    Real    t11_re, t11_im, t12, t21, t22_re, t22_im;
    Real    l_re, l_im, det_re, det_im, invdet_re, invdet_im,
        val1_re, val1_im, val2_re, val2_im,
        tmp_val1_re, tmp_val1_im, tmp_val2_re, tmp_val2_im, **T_me;
    Real    sum, diff, discrim, magdet, norm, scale;
    static VEC  *tmp1_re=VNULL, *tmp1_im=VNULL,
            *tmp2_re=VNULL, *tmp2_im=VNULL;
 
    if ( ! T || ! X_re )
        error(E_NULL,"schur_vecs");
    if ( T->m != T->n || X_re->m != X_re->n ||
        ( Q != MNULL && Q->m != Q->n ) ||
        ( X_im != MNULL && X_im->m != X_im->n ) )
        error(E_SQUARE,"schur_vecs");
    if ( T->m != X_re->m ||
        ( Q != MNULL && T->m != Q->m ) ||
        ( X_im != MNULL && T->m != X_im->m ) )
        error(E_SIZES,"schur_vecs");
 
    tmp1_re = v_resize(tmp1_re,T->m);
    tmp1_im = v_resize(tmp1_im,T->m);
    tmp2_re = v_resize(tmp2_re,T->m);
    tmp2_im = v_resize(tmp2_im,T->m);
    MEM_STAT_REG(tmp1_re,TYPE_VEC);
    MEM_STAT_REG(tmp1_im,TYPE_VEC);
    MEM_STAT_REG(tmp2_re,TYPE_VEC);
    MEM_STAT_REG(tmp2_im,TYPE_VEC);
 
    T_me = T->me;
    i = 0;
    while ( i < T->m )
    {
        if ( i+1 < T->m && T->me[i+1][i] != 0.0 )
        {   /* complex eigenvalue */
        sum  = 0.5*(T_me[i][i]+T_me[i+1][i+1]);
        diff = 0.5*(T_me[i][i]-T_me[i+1][i+1]);
        discrim = diff*diff + T_me[i][i+1]*T_me[i+1][i];
        l_re = l_im = 0.0;
        if ( discrim < 0.0 )
        {   /* yes -- complex e-vals */
            l_re = sum;
            l_im = sqrt(-discrim);
        }
        else /* not correct Real Schur form */
            error(E_RANGE,"schur_vecs");
        }
        else
        {
        l_re = T_me[i][i];
        l_im = 0.0;
        }
 
        v_zero(tmp1_im);
        v_rand(tmp1_re);
        sv_mlt(MACHEPS,tmp1_re,tmp1_re);
 
        /* solve (T-l.I)x = tmp1 */
        limit = ( l_im != 0.0 ) ? i+1 : i;
        /* printf("limit = %d\n",limit); */
        for ( j = limit+1; j < T->m; j++ )
        tmp1_re->ve[j] = 0.0;
        j = limit;
        while ( j >= 0 )
        {
        if ( j > 0 && T->me[j][j-1] != 0.0 )
        {   /* 2 x 2 diagonal block */
            /* printf("checkpoint A\n"); */
            val1_re = tmp1_re->ve[j-1] -
              __ip__(&(tmp1_re->ve[j+1]),&(T->me[j-1][j+1]),limit-j);
            /* printf("checkpoint B\n"); */
            val1_im = tmp1_im->ve[j-1] -
              __ip__(&(tmp1_im->ve[j+1]),&(T->me[j-1][j+1]),limit-j);
            /* printf("checkpoint C\n"); */
            val2_re = tmp1_re->ve[j] -
              __ip__(&(tmp1_re->ve[j+1]),&(T->me[j][j+1]),limit-j);
            /* printf("checkpoint D\n"); */
            val2_im = tmp1_im->ve[j] -
              __ip__(&(tmp1_im->ve[j+1]),&(T->me[j][j+1]),limit-j);
            /* printf("checkpoint E\n"); */
            
            t11_re = T_me[j-1][j-1] - l_re;
            t11_im = - l_im;
            t22_re = T_me[j][j] - l_re;
            t22_im = - l_im;
            t12 = T_me[j-1][j];
            t21 = T_me[j][j-1];
 
            scale =  fabs(T_me[j-1][j-1]) + fabs(T_me[j][j]) +
            fabs(t12) + fabs(t21) + fabs(l_re) + fabs(l_im);
 
            det_re = t11_re*t22_re - t11_im*t22_im - t12*t21;
            det_im = t11_re*t22_im + t11_im*t22_re;
            magdet = det_re*det_re+det_im*det_im;
            if ( sqrt(magdet) < MACHEPS*scale )
            {
                det_re = MACHEPS*scale;
            magdet = det_re*det_re+det_im*det_im;
            }
            invdet_re =   det_re/magdet;
            invdet_im = - det_im/magdet;
            tmp_val1_re = t22_re*val1_re-t22_im*val1_im-t12*val2_re;
            tmp_val1_im = t22_im*val1_re+t22_re*val1_im-t12*val2_im;
            tmp_val2_re = t11_re*val2_re-t11_im*val2_im-t21*val1_re;
            tmp_val2_im = t11_im*val2_re+t11_re*val2_im-t21*val1_im;
            tmp1_re->ve[j-1] = invdet_re*tmp_val1_re -
                    invdet_im*tmp_val1_im;
            tmp1_im->ve[j-1] = invdet_im*tmp_val1_re +
                    invdet_re*tmp_val1_im;
            tmp1_re->ve[j]   = invdet_re*tmp_val2_re -
                    invdet_im*tmp_val2_im;
            tmp1_im->ve[j]   = invdet_im*tmp_val2_re +
                    invdet_re*tmp_val2_im;
            j -= 2;
            }
            else
        {
            t11_re = T_me[j][j] - l_re;
            t11_im = - l_im;
            magdet = t11_re*t11_re + t11_im*t11_im;
            scale = fabs(T_me[j][j]) + fabs(l_re);
            if ( sqrt(magdet) < MACHEPS*scale )
            {
                t11_re = MACHEPS*scale;
            magdet = t11_re*t11_re + t11_im*t11_im;
            }
            invdet_re =   t11_re/magdet;
            invdet_im = - t11_im/magdet;
            /* printf("checkpoint F\n"); */
            val1_re = tmp1_re->ve[j] -
              __ip__(&(tmp1_re->ve[j+1]),&(T->me[j][j+1]),limit-j);
            /* printf("checkpoint G\n"); */
            val1_im = tmp1_im->ve[j] -
              __ip__(&(tmp1_im->ve[j+1]),&(T->me[j][j+1]),limit-j);
            /* printf("checkpoint H\n"); */
            tmp1_re->ve[j] = invdet_re*val1_re - invdet_im*val1_im;
            tmp1_im->ve[j] = invdet_im*val1_re + invdet_re*val1_im;
            j -= 1;
        }
        }
 
        norm = v_norm_inf(tmp1_re) + v_norm_inf(tmp1_im);
        sv_mlt(1/norm,tmp1_re,tmp1_re);
        if ( l_im != 0.0 )
        sv_mlt(1/norm,tmp1_im,tmp1_im);
        mv_mlt(Q,tmp1_re,tmp2_re);
        if ( l_im != 0.0 )
        mv_mlt(Q,tmp1_im,tmp2_im);
        if ( l_im != 0.0 )
        norm = sqrt(in_prod(tmp2_re,tmp2_re)+in_prod(tmp2_im,tmp2_im));
        else
        norm = v_norm2(tmp2_re);
        sv_mlt(1/norm,tmp2_re,tmp2_re);
        if ( l_im != 0.0 )
        sv_mlt(1/norm,tmp2_im,tmp2_im);
 
        if ( l_im != 0.0 )
        {
        if ( ! X_im )
        error(E_NULL,"schur_vecs");
        set_col(X_re,i,tmp2_re);
        set_col(X_im,i,tmp2_im);
        sv_mlt(-1.0,tmp2_im,tmp2_im);
        set_col(X_re,i+1,tmp2_re);
        set_col(X_im,i+1,tmp2_im);
        i += 2;
        }
        else
        {
        set_col(X_re,i,tmp2_re);
        if ( X_im != MNULL )
            set_col(X_im,i,tmp1_im);    /* zero vector */
        i += 1;
        }
    }
 
    return X_re;
}
 
#endif