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eigen.c

/*
  Teem: Tools to process and visualize scientific data and images              
  Copyright (C) 2008, 2007, 2006, 2005  Gordon Kindlmann
  Copyright (C) 2004, 2003, 2002, 2001, 2000, 1999, 1998  University of Utah

  This library is free software; you can redistribute it and/or
  modify it under the terms of the GNU Lesser General Public License
  (LGPL) as published by the Free Software Foundation; either
  version 2.1 of the License, or (at your option) any later version.
  The terms of redistributing and/or modifying this software also
  include exceptions to the LGPL that facilitate static linking.

  This library is distributed in the hope that it will be useful,
  but WITHOUT ANY WARRANTY; without even the implied warranty of
  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
  Lesser General Public License for more details.

  You should have received a copy of the GNU Lesser General Public License
  along with this library; if not, write to Free Software Foundation, Inc.,
  51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
*/


#include "ell.h"

/* lop A
  fprintf(stderr, "_ellAlign3: ----------\n");
  fprintf(stderr, "_ellAlign3: v0 = %g %g %g\n", (v+0)[0], (v+0)[1], (v+0)[2]);
  fprintf(stderr, "_ellAlign3: v3 = %g %g %g\n", (v+3)[0], (v+3)[1], (v+3)[2]);
  fprintf(stderr, "_ellAlign3: v6 = %g %g %g\n", (v+6)[0], (v+6)[1], (v+6)[2]);
  fprintf(stderr, "_ellAlign3: d = %g %g %g -> %d %d %d\n",
          d0, d1, d2, Mi, ai, bi);
  fprintf(stderr, "_ellAlign3:  pre dot signs (03, 06, 36): %d %d %d\n",
          airSgn(ELL_3V_DOT(v+0, v+3)),
          airSgn(ELL_3V_DOT(v+0, v+6)),
          airSgn(ELL_3V_DOT(v+3, v+6)));
  */

/* lop B
  fprintf(stderr, "_ellAlign3: v0 = %g %g %g\n", (v+0)[0], (v+0)[1], (v+0)[2]);
  fprintf(stderr, "_ellAlign3: v3 = %g %g %g\n", (v+3)[0], (v+3)[1], (v+3)[2]);
  fprintf(stderr, "_ellAlign3: v6 = %g %g %g\n", (v+6)[0], (v+6)[1], (v+6)[2]);
  fprintf(stderr, "_ellAlign3:  post dot signs %d %d %d\n",
          airSgn(ELL_3V_DOT(v+0, v+3)),
          airSgn(ELL_3V_DOT(v+0, v+6)),
          airSgn(ELL_3V_DOT(v+3, v+6)));
  if (airSgn(ELL_3V_DOT(v+0, v+3)) < 0
      || airSgn(ELL_3V_DOT(v+0, v+6)) < 0
      || airSgn(ELL_3V_DOT(v+3, v+6)) < 0) {
    exit(1);
  }
  */

void
_ell_align3_d(double v[9]) {
  double d0, d1, d2;
  int Mi, ai, bi;
  
  d0 = ELL_3V_DOT(v+0, v+0);
  d1 = ELL_3V_DOT(v+3, v+3);
  d2 = ELL_3V_DOT(v+6, v+6);
  Mi = ELL_MAX3_IDX(d0, d1, d2);
  ai = (Mi + 1) % 3;
  bi = (Mi + 2) % 3;
  /* lop A */
  if (ELL_3V_DOT(v+3*Mi, v+3*ai) < 0) {
    ELL_3V_SCALE(v+3*ai, -1, v+3*ai);
  }
  if (ELL_3V_DOT(v+3*Mi, v+3*bi) < 0) {
    ELL_3V_SCALE(v+3*bi, -1, v+3*bi);
  }
  /* lob B */
  /* we can't guarantee that dot(v+3*ai,v+3*bi) > 0 ... */
}

/*
** leaves v+3*0 untouched, but makes sure that v+3*0, v+3*1, and v+3*2
** are mutually orthogonal.  Also leaves the magnitudes of all
** vectors unchanged.
*/
void
_ell_3m_enforce_orthogonality(double v[9]) {
  double d00, d10, d11, d20, d21, d22, scl, tv[3];

  d00 = ELL_3V_DOT(v+3*0, v+3*0);
  d10 = ELL_3V_DOT(v+3*1, v+3*0);
  d11 = ELL_3V_DOT(v+3*1, v+3*1);
  ELL_3V_SCALE_ADD2(tv, 1, v+3*1, -d10/d00, v+3*0);
  scl = sqrt(d11/ELL_3V_DOT(tv, tv));
  ELL_3V_SCALE(v+3*1, scl, tv);
  d20 = ELL_3V_DOT(v+3*2, v+3*0);
  d21 = ELL_3V_DOT(v+3*2, v+3*1);
  d22 = ELL_3V_DOT(v+3*2, v+3*2);
  ELL_3V_SCALE_ADD3(tv, 1, v+3*2, -d20/d00, v+3*0, -d21/d00, v+3*1);
  scl = sqrt(d22/ELL_3V_DOT(tv, tv));
  ELL_3V_SCALE(v+3*2, scl, tv);
  return;
}

/*
** makes sure that v+3*2 has a positive dot product with
** cross product of v+3*0 and v+3*1
*/
void
_ell_3m_make_right_handed_d(double v[9]) {
  double x[3];
  
  ELL_3V_CROSS(x, v+3*0, v+3*1);
  if (0 > ELL_3V_DOT(x, v+3*2)) {
    ELL_3V_SCALE(v+3*2, -1, v+3*2);
  }
}

/* lop A
  fprintf(stderr, "===  pre ===\n");
  fprintf(stderr, "crosses:  %g %g %g\n", (t+0)[0], (t+0)[1], (t+0)[2]);
  fprintf(stderr, "          %g %g %g\n", (t+3)[0], (t+3)[1], (t+3)[2]);
  fprintf(stderr, "          %g %g %g\n", (t+6)[0], (t+6)[1], (t+6)[2]);
  fprintf(stderr, "cross dots:  %g %g %g\n",
          ELL_3V_DOT(t+0, t+3), ELL_3V_DOT(t+0, t+6), ELL_3V_DOT(t+3, t+6));
*/

/* lop B
  fprintf(stderr, "=== post ===\n");
  fprintf(stderr, "crosses:  %g %g %g\n", (t+0)[0], (t+0)[1], (t+0)[2]);
  fprintf(stderr, "          %g %g %g\n", (t+3)[0], (t+3)[1], (t+3)[2]);
  fprintf(stderr, "          %g %g %g\n", (t+6)[0], (t+6)[1], (t+6)[2]);
  fprintf(stderr, "cross dots:  %g %g %g\n",
          ELL_3V_DOT(t+0, t+3), ELL_3V_DOT(t+0, t+6), ELL_3V_DOT(t+3, t+6));
*/

/*
******** ell_3m_1d_nullspace_d()
**
** the given matrix is assumed to have a nullspace of dimension one.
** A normalized vector which spans the nullspace is put into ans.
**
** The given nullspace matrix is NOT modified.  
**
** This does NOT use biff
*/
void
ell_3m_1d_nullspace_d(double ans[3], const double _n[9]) {
  double t[9], n[9], norm;
  
  ELL_3M_TRANSPOSE(n, _n);
  /* find the three cross-products of pairs of column vectors of n */
  ELL_3V_CROSS(t+0, n+0, n+3);
  ELL_3V_CROSS(t+3, n+0, n+6);
  ELL_3V_CROSS(t+6, n+3, n+6);
  /* lop A */
  _ell_align3_d(t);
  /* lop B */
  /* add them up (longer, hence more accurate, should dominate) */
  ELL_3V_ADD3(ans, t+0, t+3, t+6);

  /* normalize */
  ELL_3V_NORM(ans, ans, norm);

  return;
}

/*
******** ell_3m_2d_nullspace_d()
**
** the given matrix is assumed to have a nullspace of dimension two.
**
** The given nullspace matrix is NOT modified 
**
** This does NOT use biff
*/
void
ell_3m_2d_nullspace_d(double ans0[3], double ans1[3], const double _n[9]) {
  double n[9], tmp[3], norm;

  ELL_3M_TRANSPOSE(n, _n);
  _ell_align3_d(n);
  ELL_3V_ADD3(tmp, n+0, n+3, n+6);
  ELL_3V_NORM(tmp, tmp, norm);
  
  /* any two vectors which are perpendicular to the (supposedly 1D)
     span of the column vectors span the nullspace */
  ell_3v_perp_d(ans0, tmp);
  ELL_3V_NORM(ans0, ans0, norm);
  ELL_3V_CROSS(ans1, tmp, ans0);

  return;
}

/*
******** ell_3m_eigenvalues_d()
**
** finds eigenvalues of given matrix.
**
** returns information about the roots according to ellCubeRoot enum,
** see header for ellCubic for details.
**
** given matrix is NOT modified
**
** This does NOT use biff
**
** Doing the frobenius normalization proved successfull in avoiding the
** the creating of NaN eigenvalues when the coefficients of the matrix
** were really large (> 50000).  Also, when the matrix norm was really
** small, the comparison to "epsilon" in ell_cubic mistook three seperate
** roots for a single and a double, with this matrix in particular:
**  1.7421892  0.0137642  0.0152975
**  0.0137642  1.7565432 -0.0062296
**  0.0152975 -0.0062296  1.7700019
** (actually, this is prior to tenEigensolve's isotropic removal)
*/
int
ell_3m_eigenvalues_d(double _eval[3], const double _m[9], const int newton) {
  double A, B, C, scale, frob, m[9], eval[3];
  int roots;

  frob = ELL_3M_FROB(_m);
  scale = frob ? 1.0/frob : 1.0;
  ELL_3M_SCALE(m, scale, _m);
  /* 
  ** from gordon with mathematica; these are the coefficients of the
  ** cubic polynomial in x: det(x*I - M).  The full cubic is
  ** x^3 + A*x^2 + B*x + C.
  */
  A = -m[0] - m[4] - m[8];
  B = m[0]*m[4] - m[3]*m[1] 
    + m[0]*m[8] - m[6]*m[2] 
    + m[4]*m[8] - m[7]*m[5];
  C = (m[6]*m[4] - m[3]*m[7])*m[2]
    + (m[0]*m[7] - m[6]*m[1])*m[5]
    + (m[3]*m[1] - m[0]*m[4])*m[8];
  roots = ell_cubic(eval, A, B, C, newton);
  /* no longer need to sort here */
  ELL_3V_SCALE(_eval, 1.0/scale, eval);
  return roots;
}

void
_ell_3m_evecs_d(double evec[9], double eval[3], int roots, 
                const double m[9]) {
  double n[9], e0, e1, e2, t /* , tmpv[3] */ ;

  ELL_3V_GET(e0, e1, e2, eval);
  /* if (ell_debug) {
    printf("ell_3m_evecs_d: numroots = %d\n", numroots);
    } */

  /* we form m - lambda*I by doing a memcpy from m, and then
     (repeatedly) over-writing the diagonal elements */
  ELL_3M_COPY(n, m);
  switch (roots) {
  case ell_cubic_root_three:
    /* if (ell_debug) {
      printf("ell_3m_evecs_d: evals: %20.15f %20.15f %20.15f\n", 
             eval[0], eval[1], eval[2]);
             } */
    ELL_3M_DIAG_SET(n, m[0]-e0, m[4]-e0, m[8]-e0);
    ell_3m_1d_nullspace_d(evec+0, n);
    ELL_3M_DIAG_SET(n, m[0]-e1, m[4]-e1, m[8]-e1);
    ell_3m_1d_nullspace_d(evec+3, n);
    ELL_3M_DIAG_SET(n, m[0]-e2, m[4]-e2, m[8]-e2);
    ell_3m_1d_nullspace_d(evec+6, n);
    _ell_3m_enforce_orthogonality(evec);
    _ell_3m_make_right_handed_d(evec);
    ELL_3V_SET(eval, e0, e1, e2);
    break;
  case ell_cubic_root_single_double:
    ELL_SORT3(e0, e1, e2, t);
    if (e0 > e1) {
      /* one big (e0) , two small (e1, e2) : more like a cigar */
      ELL_3M_DIAG_SET(n, m[0]-e0, m[4]-e0, m[8]-e0);
      ell_3m_1d_nullspace_d(evec+0, n);
      ELL_3M_DIAG_SET(n, m[0]-e1, m[4]-e1, m[8]-e1);
      ell_3m_2d_nullspace_d(evec+3, evec+6, n);
    }
    else {
      /* two big (e0, e1), one small (e2): more like a pancake */
      ELL_3M_DIAG_SET(n, m[0]-e0, m[4]-e0, m[8]-e0);
      ell_3m_2d_nullspace_d(evec+0, evec+3, n);
      ELL_3M_DIAG_SET(n, m[0]-e2, m[4]-e2, m[8]-e2);
      ell_3m_1d_nullspace_d(evec+6, n);
    }
    _ell_3m_enforce_orthogonality(evec);
    _ell_3m_make_right_handed_d(evec);
    ELL_3V_SET(eval, e0, e1, e2);
    break;
  case ell_cubic_root_triple:
    /* one triple root; use any basis as the eigenvectors */
    ELL_3V_SET(evec+0, 1, 0, 0);
    ELL_3V_SET(evec+3, 0, 1, 0);
    ELL_3V_SET(evec+6, 0, 0, 1);
    ELL_3V_SET(eval, e0, e1, e2);
    break;
  case ell_cubic_root_single:
    /* only one real root */
    ELL_3M_DIAG_SET(n, m[0]-e0, m[4]-e0, m[8]-e0);
    ell_3m_1d_nullspace_d(evec+0, n);
    ELL_3V_SET(evec+3, AIR_NAN, AIR_NAN, AIR_NAN);
    ELL_3V_SET(evec+6, AIR_NAN, AIR_NAN, AIR_NAN);
    ELL_3V_SET(eval, e0, AIR_NAN, AIR_NAN);
    break;
  }
  /* if (ell_debug) {
    printf("ell_3m_evecs_d (numroots = %d): evecs: \n", numroots);
    ELL_3MV_MUL(tmpv, m, evec[0]);
    printf(" (%g:%g): %20.15f %20.15f %20.15f\n", 
           eval[0], ELL_3V_DOT(evec[0], tmpv), 
           evec[0][0], evec[0][1], evec[0][2]);
    ELL_3MV_MUL(tmpv, m, evec[1]);
    printf(" (%g:%g): %20.15f %20.15f %20.15f\n", 
           eval[1], ELL_3V_DOT(evec[1], tmpv), 
           evec[1][0], evec[1][1], evec[1][2]);
    ELL_3MV_MUL(tmpv, m, evec[2]);
    printf(" (%g:%g): %20.15f %20.15f %20.15f\n", 
           eval[2], ELL_3V_DOT(evec[2], tmpv), 
           evec[2][0], evec[2][1], evec[2][2]);
           } */
  return;
}

/*
******** ell_3m_eigensolve_d()
**
** finds eigenvalues and eigenvectors of given matrix m
**
** returns information about the roots according to ellCubeRoot enum,
** see header for ellCubic for details.  When eval[i] is set, evec+3*i
** is set to a corresponding eigenvector.  The eigenvectors are
** (evec+0)[], (evec+3)[], and (evec+6)[]
**
** NOTE: Even in the post-Teem-1.7 switch from column-major to 
** row-major- its still the case that the eigenvectors are at
** evec+0, evec+3, evec+6: this means that they USED to be the
** "columns" of the matrix, and NOW they're the rows.  
**
** The eigenvalues (and associated eigenvectors) are sorted in
** descending order.
**
** This does NOT use biff
*/
int
ell_3m_eigensolve_d(double eval[3], double evec[9],
                    const double m[9], const int newton) {
  int roots;

  /* if (ell_debug) {
    printf("ell_3m_eigensolve_d: input matrix:\n");
    printf("{{%20.15f,\t%20.15f,\t%20.15f},\n", m[0], m[1], m[2]);
    printf(" {%20.15f,\t%20.15f,\t%20.15f},\n", m[3], m[4], m[5]);
    printf(" {%20.15f,\t%20.15f,\t%20.15f}};\n",m[6], m[7], m[8]);
    } */
  
  roots = ell_3m_eigenvalues_d(eval, m, newton);
  _ell_3m_evecs_d(evec, eval, roots, m);

  return roots;
}

/*
******** ell_3m_svd_d
**
** singular value decomposition:
** mat = uu * diag(sval) * vv
**
** singular values are square roots of eigenvalues of mat * mat^T
** columns of uu are eigenvectors of mat * mat^T
** rows of vv are eigenvectors of mat^T * mat
**
** returns info about singular values according to ellCubeRoot enum
**
** HEY: I think this does the wrong thing when given a symmetric
** matrix with negative eigenvalues...
*/
int
ell_3m_svd_d(double uu[9], double sval[3], double vv[9], 
             const double mat[9], const int newton) {
  double trn[9], msqr[9], eval[3], evec[9];
  int roots;
  
  ELL_3M_TRANSPOSE(trn, mat);
  ELL_3M_MUL(msqr, mat, trn);
  roots = ell_3m_eigensolve_d(eval, evec, msqr, newton);
  sval[0] = sqrt(eval[0]);
  sval[1] = sqrt(eval[1]);
  sval[2] = sqrt(eval[2]);
  ELL_3M_TRANSPOSE(uu, evec);
  ELL_3M_MUL(msqr, trn, mat);
  _ell_3m_evecs_d(vv, eval, roots, msqr);

  return roots;
}

/*
** NOTE: profiling showed that about a quarter of the execution time of
** ell_6ms_eigensolve_d() is spent here; so reconsider its need and
** implementation... (fabs vs. AIR_ABS() made no difference)
*/
static void
_maxI_sum_find(unsigned int maxI[2], double *sumon, double *sumoff,
               double mat[6][6]) {
  double maxm, tmp;
  unsigned int rrI, ccI;

  /* we hope that all these loops are unrolled by the optimizer */
  *sumon = *sumoff = 0.0;
  for (rrI=0; rrI<6; rrI++) {
    *sumon += AIR_ABS(mat[rrI][rrI]);
  }
  maxm = -1;
  for (rrI=0; rrI<5; rrI++) {
    for (ccI=rrI+1; ccI<6; ccI++) {
      tmp = AIR_ABS(mat[rrI][ccI]);
      *sumoff += tmp;
      if (tmp > maxm) {
        maxm = tmp;
        maxI[0] = rrI;
        maxI[1] = ccI;
      }
    }
  }

  /*
  if (1) {
    double nrm, trc;
    nrm = trc = 0;
    for (rrI=0; rrI<6; rrI++) {
      trc += mat[rrI][rrI];
      nrm += mat[rrI][rrI]*mat[rrI][rrI];
    }
    for (rrI=0; rrI<5; rrI++) {
      for (ccI=rrI+1; ccI<6; ccI++) {
        nrm += 2*mat[rrI][ccI]*mat[rrI][ccI];
      }
    }
    fprintf(stderr, "---------------- invars = %g %g\n", trc, nrm);
  }
  */

  return;
}

static int
_compar(const void *A_void, const void *B_void) {
  double *A, *B;
  A = AIR_CAST(double *, A_void);
  B = AIR_CAST(double *, B_void);
  return (A[0] < B[0] ? 1 : (A[0] > B[0] ? -1 : 0));
}

/*
******* ell_6ms_eigensolve_d
**
** uses Jacobi iterations to find eigensystem of 6x6 symmetric matrix,
** given in sym[21].  Puts eigenvalues, in descending order, in eval[6],
** and corresponding eigenvectors in _evec+0, _evec+6, ..., _evec+30.
** NOTE: you can pass a NULL _evec if eigenvectors aren't needed.
** 
** does NOT use biff
*/
int
ell_6ms_eigensolve_d(double eval[6], double _evec[36],
                     const double sym[21], const double eps) {
  /* char me[]="ell_6ms_eigensolve_d"; */
  double mat[2][6][6], evec[2][6][6], sumon, sumoff, evtmp[12];
  unsigned int cur, rrI, ccI, maxI[2], iter;

  if (!( eval && sym && eps >= 0 )) {
    return 1;
  }
  /* copy symmetric matrix sym[] into upper tris of mat[0][][] & mat[1][][] */
  mat[0][0][0] = sym[ 0];
  mat[0][0][1] = sym[ 1];
  mat[0][0][2] = sym[ 2];
  mat[0][0][3] = sym[ 3];
  mat[0][0][4] = sym[ 4];
  mat[0][0][5] = sym[ 5];
  mat[0][1][1] = sym[ 6];
  mat[0][1][2] = sym[ 7];
  mat[0][1][3] = sym[ 8];
  mat[0][1][4] = sym[ 9];
  mat[0][1][5] = sym[10];
  mat[0][2][2] = sym[11];
  mat[0][2][3] = sym[12];
  mat[0][2][4] = sym[13];
  mat[0][2][5] = sym[14];
  mat[0][3][3] = sym[15];
  mat[0][3][4] = sym[16];
  mat[0][3][5] = sym[17];
  mat[0][4][4] = sym[18];
  mat[0][4][5] = sym[19];
  mat[0][5][5] = sym[20];
  if (_evec) {
    /* initialize evec[0]; */
    for (rrI=0; rrI<6; rrI++) {
      for (ccI=0; ccI<6; ccI++) {
        evec[0][ccI][rrI] = (rrI == ccI);
      }
    }
  }
  /*
  fprintf(stderr, "!%s(INIT): m = [", me);
  for (rrI=0; rrI<6; rrI++) {
    for (ccI=0; ccI<6; ccI++) {
      fprintf(stderr, "%f%s", 
              (rrI <= ccI ? mat[0][rrI][ccI] : mat[0][ccI][rrI]), 
              ccI<5 ? "," : (rrI<5 ? ";" : "]"));
    }
    fprintf(stderr, "\n");
  }
  */
  _maxI_sum_find(maxI, &sumon, &sumoff, mat[0]);
  /*
  maxI[0] = 0;
  maxI[1] = 3;
  */
  cur = 1;         /* fake out anticipating first line of loop */
  iter = 0;
  while (sumoff/sumon > eps) {
    double th, tt, cc, ss; 
    unsigned int P, Q;
    /*
    fprintf(stderr, "!%s(%u): sumoff/sumon = %g/%g = %g > %g\n", me, iter,
            sumoff, sumon, sumoff/sumon, eps);
    */
    cur = 1 - cur;

    P = maxI[0];
    Q = maxI[1];
    th = (mat[cur][Q][Q] - mat[cur][P][P])/(2*mat[cur][P][Q]);
    tt = (th > 0 ? +1 : -1)/(AIR_ABS(th) + sqrt(th*th + 1));
    cc = 1/sqrt(tt*tt + 1);
    ss = cc*tt;
    /*
    fprintf(stderr, "!%s(%u): maxI = (P,Q) = (%u,%u) --> ss=%f, cc=%f\n",
            me, iter, P, Q, ss, cc);
    fprintf(stderr, "     r = [");
    for (rrI=0; rrI<6; rrI++) {
      for (ccI=0; ccI<6; ccI++) {
        fprintf(stderr, "%g%s",
                (rrI == ccI
                 ? (rrI == P || rrI == Q ? cc : 1.0)
                 : (rrI == P && ccI == Q
                    ? ss
                    : (rrI == Q && ccI == P
                       ? -ss
                       : 0))),
                ccI<5 ? "," : (rrI<5 ? ";" : "]"));
      }
      fprintf(stderr, "\n");
    }
    */
    /* initialize by copying whole matrix */
    for (rrI=0; rrI<6; rrI++) {
      for (ccI=rrI; ccI<6; ccI++) {
        mat[1-cur][rrI][ccI] = mat[cur][rrI][ccI];
      }
    }
    /* perform Jacobi rotation */
    for (rrI=0; rrI<P; rrI++) {
      mat[1-cur][rrI][P] = cc*mat[cur][rrI][P] - ss*mat[cur][rrI][Q];
    }
    for (ccI=P+1; ccI<6; ccI++) {
      mat[1-cur][P][ccI] = cc*mat[cur][P][ccI] - ss*(Q <= ccI 
                                                     ? mat[cur][Q][ccI] 
                                                     : mat[cur][ccI][Q]);
    }
    for (rrI=0; rrI<Q; rrI++) {
      mat[1-cur][rrI][Q] = ss*(rrI <= P 
                               ? mat[cur][rrI][P]
                               : mat[cur][P][rrI]) + cc*mat[cur][rrI][Q];
    }
    for (ccI=Q+1; ccI<6; ccI++) {
      mat[1-cur][Q][ccI] = ss*mat[cur][P][ccI] + cc*mat[cur][Q][ccI];
    }
    /* set special entries */
    mat[1-cur][P][P] = mat[cur][P][P] - tt*mat[cur][P][Q];
    mat[1-cur][Q][Q] = mat[cur][Q][Q] + tt*mat[cur][P][Q];
    mat[1-cur][P][Q] = 0.0;
    if (_evec) {
      /* NOTE: the eigenvectors use transpose of indexing of mat */
      /* start by copying all */
      for (rrI=0; rrI<6; rrI++) {
        for (ccI=0; ccI<6; ccI++) {
          evec[1-cur][ccI][rrI] = evec[cur][ccI][rrI];
        }
      }
      for (rrI=0; rrI<6; rrI++) {
        evec[1-cur][P][rrI] = cc*evec[cur][P][rrI] - ss*evec[cur][Q][rrI];
        evec[1-cur][Q][rrI] = ss*evec[cur][P][rrI] + cc*evec[cur][Q][rrI];
      }
    }

    _maxI_sum_find(maxI, &sumon, &sumoff, mat[1-cur]);

    /*
    fprintf(stderr, "!%s(%u): m = [", me, iter);
    for (rrI=0; rrI<6; rrI++) {
      for (ccI=0; ccI<6; ccI++) {
        fprintf(stderr, "%f%s", 
                (rrI <= ccI ? mat[1-cur][rrI][ccI] : mat[1-cur][ccI][rrI]), 
                ccI<5 ? "," : (rrI<5 ? ";" : "]"));
      }
      fprintf(stderr, "\n");
    }
    */
    iter++;
  }
  /* 1-cur is index of final solution */

  /* sort evals */
  for (ccI=0; ccI<6; ccI++) {
    evtmp[0 + 2*ccI] = mat[1-cur][ccI][ccI];
    evtmp[1 + 2*ccI] = ccI;
  }
  qsort(evtmp, 6, 2*sizeof(double), _compar);

  /* copy out solution */
  for (ccI=0; ccI<6; ccI++) {
    eval[ccI] = evtmp[0 + 2*ccI];
    if (_evec) {
      unsigned eeI;
      for (rrI=0; rrI<6; rrI++) {
        eeI = AIR_CAST(unsigned int, evtmp[1 + 2*ccI]);
        _evec[rrI + 6*ccI] = evec[1-cur][eeI][rrI];
      }
    }
  }
  
  return 0;
}

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