Jungfraujoch/gemmi_gph/eig3.cpp

// Based on public domain code from Connelly Barnes:
// http://barnesc.blogspot.com/2007/02/eigenvectors-of-3x3-symmetric-matrix.html
// which in turn is based on the public domain Java Matrix library JAMA.

#include <gemmi/eig3.hpp>
#include <cmath>

// Symmetric Householder reduction to tridiagonal form.
//  This is derived from the Algol procedures tred2 by
//  Bowdler, Martin, Reinsch, and Wilkinson, Handbook for
//  Auto. Comp., Vol.ii-Linear Algebra, and the corresponding
//  Fortran subroutine in EISPACK.
static void tred2(gemmi::Mat33& V, double d[3], double e[3]) {
  for (int j = 0; j < 3; j++) {
    d[j] = V[3-1][j];
  }

  // Householder reduction to tridiagonal form.
  for (int i = 3-1; i > 0; i--) {
    // Scale to avoid under/overflow.
    double scale = 0.0;
    double h = 0.0;
    for (int k = 0; k < i; k++) {
      scale = scale + std::fabs(d[k]);
    }
    if (scale == 0.0) {
      e[i] = d[i-1];
      for (int j = 0; j < i; j++) {
        d[j] = V[i-1][j];
        V[i][j] = 0.0;
        V[j][i] = 0.0;
      }
    } else {
      // Generate Householder vector.
      for (int k = 0; k < i; k++) {
        d[k] /= scale;
        h += d[k] * d[k];
      }
      double f = d[i-1];
      double g = sqrt(h);
      if (f > 0) {
        g = -g;
      }
      e[i] = scale * g;
      h = h - f * g;
      d[i-1] = f - g;
      for (int j = 0; j < i; j++) {
        e[j] = 0.0;
      }

      // Apply similarity transformation to remaining columns.
      for (int j = 0; j < i; j++) {
        f = d[j];
        V[j][i] = f;
        g = e[j] + V[j][j] * f;
        for (int k = j+1; k <= i-1; k++) {
          g += V[k][j] * d[k];
          e[k] += V[k][j] * f;
        }
        e[j] = g;
      }
      f = 0.0;
      for (int j = 0; j < i; j++) {
        e[j] /= h;
        f += e[j] * d[j];
      }
      double hh = f / (h + h);
      for (int j = 0; j < i; j++) {
        e[j] -= hh * d[j];
      }
      for (int j = 0; j < i; j++) {
        f = d[j];
        g = e[j];
        for (int k = j; k <= i-1; k++) {
          V[k][j] -= (f * e[k] + g * d[k]);
        }
        d[j] = V[i-1][j];
        V[i][j] = 0.0;
      }
    }
    d[i] = h;
  }

  // Accumulate transformations.
  for (int i = 0; i < 3-1; i++) {
    V[3-1][i] = V[i][i];
    V[i][i] = 1.0;
    double h = d[i+1];
    if (h != 0.0) {
      for (int k = 0; k <= i; k++) {
        d[k] = V[k][i+1] / h;
      }
      for (int j = 0; j <= i; j++) {
        double g = 0.0;
        for (int k = 0; k <= i; k++) {
          g += V[k][i+1] * V[k][j];
        }
        for (int k = 0; k <= i; k++) {
          V[k][j] -= g * d[k];
        }
      }
    }
    for (int k = 0; k <= i; k++) {
      V[k][i+1] = 0.0;
    }
  }
  for (int j = 0; j < 3; j++) {
    d[j] = V[3-1][j];
    V[3-1][j] = 0.0;
  }
  V[3-1][3-1] = 1.0;
  e[0] = 0.0;
}

// Symmetric tridiagonal QL algorithm.
//  This is derived from the Algol procedures tql2, by
//  Bowdler, Martin, Reinsch, and Wilkinson, Handbook for
//  Auto. Comp., Vol.ii-Linear Algebra, and the corresponding
//  Fortran subroutine in EISPACK.
static void tql2(gemmi::Mat33& V, double d[3], double e[3]) {
  for (int i = 1; i < 3; i++) {
    e[i-1] = e[i];
  }
  e[3-1] = 0.0;

  double f = 0.0;
  double tst1 = 0.0;
  double eps = std::pow(2.0,-52.0);
  for (int l = 0; l < 3; l++) {

    // Find small subdiagonal element
    tst1 = std::max(tst1, std::fabs(d[l]) + std::fabs(e[l]));
    int m = l;
    // MW: I changed "m < 3" to "m < 2", because if m==2 is followed by m++,
    // we access out of bounds d[3] in "p = d[m];" and later on e[3], getting
    // *** stack smashing detected ***.
    while (m < 2) {
      if (std::fabs(e[m]) <= eps*tst1) {
        break;
      }
      m++;
    }

    // If m == l, d[l] is an eigenvalue,
    // otherwise, iterate.
    if (m > l) {
      int iter = 0;
      do {
        iter = iter + 1;  // (Could check iteration count here.)

        // Compute implicit shift
        double g = d[l];
        double p = (d[l+1] - g) / (2.0 * e[l]);
        double r = std::sqrt(p*p + 1.0);
        if (p < 0) {
          r = -r;
        }
        d[l] = e[l] / (p + r);
        d[l+1] = e[l] * (p + r);
        double dl1 = d[l+1];
        double h = g - d[l];
        for (int i = l+2; i < 3; i++) {
          d[i] -= h;
        }
        f = f + h;

        // Implicit QL transformation.
        p = d[m];
        double c = 1.0;
        double c2 = c;
        double c3 = c;
        double el1 = e[l+1];
        double s = 0.0;
        double s2 = 0.0;
        for (int i = m-1; i >= l; i--) {
          c3 = c2;
          c2 = c;
          s2 = s;
          g = c * e[i];
          h = c * p;
          r = std::sqrt(p*p + e[i]*e[i]);
          e[i+1] = s * r;
          s = e[i] / r;
          c = p / r;
          p = c * d[i] - s * g;
          d[i+1] = h + s * (c * g + s * d[i]);

          // Accumulate transformation.
          for (int k = 0; k < 3; k++) {
            h = V[k][i+1];
            V[k][i+1] = s * V[k][i] + c * h;
            V[k][i] = c * V[k][i] - s * h;
          }
        }
        p = -s * s2 * c3 * el1 * e[l] / dl1;
        e[l] = s * p;
        d[l] = c * p;

        // Check for convergence.
      } while (std::fabs(e[l]) > eps*tst1);
    }
    d[l] = d[l] + f;
    e[l] = 0.0;
  }

  // Sort eigenvalues and corresponding vectors.
  for (int i = 0; i < 3-1; i++) {
    int k = i;
    double p = d[i];
    for (int j = i+1; j < 3; j++) {
      if (d[j] < p) {
        k = j;
        p = d[j];
      }
    }
    if (k != i) {
      d[k] = d[i];
      d[i] = p;
      for (int j = 0; j < 3; j++) {
        p = V[j][i];
        V[j][i] = V[j][k];
        V[j][k] = p;
      }
    }
  }
}

namespace gemmi {

Mat33 eigen_decomposition(const SMat33<double>& A, double (&d)[3]) {
  double e[3];
  Mat33 V = A.as_mat33();
  tred2(V, d, e);
  tql2(V, d, e);
  return V;
}

} // namespace gemmi