zheevq3.h

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// ----------------------------------------------------------------------------
// Numerical diagonalization of 3x3 matrcies
// Copyright (C) 2006  Joachim Kopp
// ----------------------------------------------------------------------------
// This library is free software; you can redistribute it and/or
// modify it under the terms of the GNU Lesser General Public
// License as published by the Free Software Foundation; either
// version 2.1 of the License, or (at your option) any later version.
//
// 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 the Free Software
// Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
// ----------------------------------------------------------------------------
#ifndef __ZHEEVQ3_H
#define __ZHEEVQ3_H

#include <complex>

#include <stdio.h>
#include <math.h>
#include <complex>
#include "zhetrd3.h"

// ----------------------------------------------------------------------------
template <typename T>
int zheevq3(std::complex<T> A[3][3], std::complex<T> Q[3][3], T w[3])
// ----------------------------------------------------------------------------
// Calculates the eigenvalues and normalized eigenvectors of a hermitian 3x3
// matrix A using the QL algorithm with implicit shifts, preceded by a
// Householder reduction to real tridiagonal form.
// The function accesses only the diagonal and upper triangular parts of A.
// The access is read-only.
// ----------------------------------------------------------------------------
// Parameters:
//   A: The hermitian input matrix
//   Q: Storage buffer for eigenvectors
//   w: Storage buffer for eigenvalues
// ----------------------------------------------------------------------------
// Return value:
//   0: Success
//  -1: Error (no convergence)
// ----------------------------------------------------------------------------
// Dependencies:
//   zhetrd3()
// ----------------------------------------------------------------------------
{
  const int n = 3;
  T e[3];                 // The third element is used only as temporary workspace
  T g, r, p, f, b, s, c;  // Intermediate storage
  std::complex<T> t(0,0);
  int nIter;
  int m;

  // Transform A to real tridiagonal form by the Householder method
  zhetrd3(A, Q, w, e);

  // Calculate eigensystem of the remaining real symmetric tridiagonal matrix
  // with the QL method
  //
  // Loop over all off-diagonal elements
  for (int l=0; l < n-1; l++)
  {
    nIter = 0;
    while (1)
    {
      // Check for convergence and exit iteration loop if off-diagonal
      // element e(l) is zero
      for (m=l; m <= n-2; m++)
      {
        g = fabs(w[m])+fabs(w[m+1]);
        if (fabs(e[m]) + g == g)
          break;
      }
      if (m == l)
        break;

      if (nIter++ >= 30)
        return -1;

      // Calculate g = d_m - k
      g = (w[l+1] - w[l]) / (e[l] + e[l]);
      r = sqrt(SQR(g) + 1.0);
      if (g > 0)
        g = w[m] - w[l] + e[l]/(g + r);
      else
        g = w[m] - w[l] + e[l]/(g - r);

      s = c = 1.0;
      p = 0.0;
      for (int i=m-1; i >= l; i--)
      {
        f = s * e[i];
        b = c * e[i];
        if (fabs(f) > fabs(g))
        {
          c      = g / f;
          r      = sqrt(SQR(c) + 1.0);
          e[i+1] = f * r;
          c     *= (s = 1.0/r);
        }
        else
        {
          s      = f / g;
          r      = sqrt(SQR(s) + 1.0);
          e[i+1] = g * r;
          s     *= (c = 1.0/r);
        }

        g = w[i+1] - p;
        r = (w[i] - g)*s + 2.0*c*b;
        p = s * r;
        w[i+1] = g + p;
        g = c*r - b;

        // Form eigenvectors
#ifndef EVALS_ONLY
        for (int k=0; k < n; k++)
        {
          t = Q[k][i+1];
          Q[k][i+1] = s*Q[k][i] + c*t;
          Q[k][i]   = c*Q[k][i] - s*t;
        }
#endif
      }
      w[l] -= p;
      e[l]  = g;
      e[m]  = 0.0;
    }
  }

  return 0;
}

#endif