Tools/Math/LA_Eigenvalues.cpp

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//------------------------------------------------------------------------------
#include <math.h>
#include "LA_Eigenvalues.h"
//------------------------------------------------------------------------------
namespace LA
{
//------------------------------------------------------------------------------
Eigenvalues::Eigenvalues(const Matrix& A)
{
  int i, j;
  
  n = A.columns();

  V.create(n, n);
  d.create(n);
  e.create(n);

  issymmetric = 1;
  for (j = 0; (j < n) && issymmetric; j++)
  {
    for (i = 0; (i < n) && issymmetric; i++)
    {
      issymmetric = (A[i][j] == A[j][i]);
    }
  }

  if (issymmetric)
  {
    for (i = 0; i < n; i++)
    {
      for (j = 0; j < n; j++)
      {
        V[i][j] = A[i][j];
      }
    }
   
    // Tridiagonalize.
    tred2();
   
    // Diagonalize.
    tql2();
  }
  else
  {
    H.create(n, n);
    ort.create(n);

    for (j = 0; j < n; j++)
    {
      for (i = 0; i < n; i++)
      {
        H[i][j] = A[i][j];
      }
    }
   
    // Reduce to Hessenberg form.
    orthes();
   
    // Reduce Hessenberg to real Schur form.
    hqr2();
  }
}
//------------------------------------------------------------------------------
const Matrix& Eigenvalues::getV() const
{
  return V;
}
//------------------------------------------------------------------------------
const Vector& Eigenvalues::getRealEigenvalues() const
{
  return d;
}
//------------------------------------------------------------------------------
const Vector& Eigenvalues::getImagEigenvalues() const
{
  return e;
}
//------------------------------------------------------------------------------
Matrix Eigenvalues::getD() const
{
  Matrix D(n, n);
  for (int i = 0; i < n; i++)
  {
    for (int j = 0; j < n; j++)
    {
      D[i][j] = 0.0;
    }
    D[i][i] = d[i];
    if (e[i] > 0)
    {
      D[i][i+1] = e[i];
    }
    else if (e[i] < 0)
    {
      D[i][i-1] = e[i];
    }
  }
  return D;
}
//------------------------------------------------------------------------------
void Eigenvalues::tred2()
{
  //  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.

  int i, j, k;
  
  for (j = 0; j < n; j++)
  {
    d[j] = V[n-1][j];
  }

  // Householder reduction to tridiagonal form.
   
  for (i = n-1; i > 0; i--)
  {
    // Scale to avoid under/overflow.
    double scale = 0.0;
    double h = 0.0;
    for (k = 0; k < i; k++)
    {
      scale = scale + fabs(d[k]);
    }
    if (scale == 0.0)
    {
      e[i] = d[i-1];
      for (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 (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 (j = 0; j < i; j++)
      {
        e[j] = 0.0;
      }

      // Apply similarity transformation to remaining columns.
      for (j = 0; j < i; j++)
      {
        f = d[j];
        V[j][i] = f;
        g = e[j] + V[j][j] * f;
        for (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 (j = 0; j < i; j++)
      {
        e[j] /= h;
        f += e[j] * d[j];
      }
      double hh = f / (h + h);
      for (j = 0; j < i; j++)
      {
        e[j] -= hh * d[j];
      }
      for (j = 0; j < i; j++)
      {
        f = d[j];
        g = e[j];
        for (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 (i = 0; i < n-1; i++)
  {
    V[n-1][i] = V[i][i];
    V[i][i] = 1.0;
    double h = d[i+1];
    if (h != 0.0)
    {
      for (k = 0; k <= i; k++)
      {
        d[k] = V[k][i+1] / h;
      }
      for (j = 0; j <= i; j++)
      {
        double g = 0.0;
        for (k = 0; k <= i; k++)
        {
          g += V[k][i+1] * V[k][j];
        }
        for (k = 0; k <= i; k++)
        {
          V[k][j] -= g * d[k];
        }
      }
    }
    for (k = 0; k <= i; k++)
    {
      V[k][i+1] = 0.0;
    }
  }
  for (j = 0; j < n; j++)
  {
    d[j] = V[n-1][j];
    V[n-1][j] = 0.0;
  }
  V[n-1][n-1] = 1.0;
  e[0] = 0.0;
}
//------------------------------------------------------------------------------
void Eigenvalues::tql2()
{
  //  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.

  int i, j, k, l;

  for (i = 1; i < n; i++)
  {
    e[i-1] = e[i];
  }
  e[n-1] = 0.0;
   
  double f = 0.0;
  double tst1 = 0.0;
  double eps = pow(2.0,-52.0);
  for (l = 0; l < n; l++)
  {
    // Find small subdiagonal element
    double tst2 = fabs(d[l]) + fabs(e[l]);
    tst1 = (tst1 > tst2)? tst1: tst2;
    int m = l;

    // Original while-loop from Java code
    while (m < n)
    {
      if (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]);
#ifdef _WIN32
        double r = _hypot(p,1.0);
#else
        double r = hypot(p,1.0);
#endif
        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 (i = l+2; i < n; 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 (i = m-1; i >= l; i--)
        {
          c3 = c2;
          c2 = c;
          s2 = s;
          g = c * e[i];
          h = c * p;
#ifdef _WIN32
          r = _hypot(p,e[i]);
#else
          r = hypot(p,e[i]);
#endif
          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 (k = 0; k < n; 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 (fabs(e[l]) > eps*tst1);
    }
    d[l] = d[l] + f;
    e[l] = 0.0;
  }
     
  // Sort eigenvalues and corresponding vectors.
  for (i = 0; i < n-1; i++)
  {
    k = i;
    double p = d[i];
    for (j = i+1; j < n; j++)
    {
      if (d[j] < p)
      {
        k = j;
        p = d[j];
      }
    }
    if (k != i)
    {
      d[k] = d[i];
      d[i] = p;
      for (j = 0; j < n; j++)
      {
        p = V[j][i];
        V[j][i] = V[j][k];
        V[j][k] = p;
      }
    }
  }
}
//------------------------------------------------------------------------------
void Eigenvalues::orthes()
{
  //  This is derived from the Algol procedures orthes and ortran,
  //  by Martin and Wilkinson, Handbook for Auto. Comp.,
  //  Vol.ii-Linear Algebra, and the corresponding
  //  Fortran subroutines in EISPACK.
   
  int low = 0;
  int high = n-1;

  int i, j, m;
   
  for (m = low+1; m <= high-1; m++)
  {
    // Scale column.
    double scale = 0.0;
    for (i = m; i <= high; i++)
    {
      scale = scale + fabs(H[i][m-1]);
    }
    if (scale != 0.0)
    {
      // Compute Householder transformation.
      double h = 0.0;
      for (i = high; i >= m; i--)
      {
        ort[i] = H[i][m-1]/scale;
        h += ort[i] * ort[i];
      }
      double g = sqrt(h);
      if (ort[m] > 0)
      {
        g = -g;
      }
      h = h - ort[m] * g;
      ort[m] = ort[m] - g;
   
      // Apply Householder similarity transformation
      // H = (I-u*u'/h)*H*(I-u*u')/h)
   
      for (j = m; j < n; j++)
      {
        double f = 0.0;
        for (i = high; i >= m; i--)
        {
          f += ort[i]*H[i][j];
        }
        f = f/h;
        for (i = m; i <= high; i++)
        {
          H[i][j] -= f*ort[i];
        }
      }
   
      for (i = 0; i <= high; i++)
      {
        double f = 0.0;
        for (j = high; j >= m; j--)
        {
          f += ort[j]*H[i][j];
        }
        f = f/h;
        for (j = m; j <= high; j++)
        {
          H[i][j] -= f*ort[j];
        }
      }
      ort[m] = scale*ort[m];
      H[m][m-1] = scale*g;
    }
  }
   
  // Accumulate transformations (Algol's ortran).
  for (i = 0; i < n; i++)
  {
    for (j = 0; j < n; j++)
    {
      V[i][j] = (i == j ? 1.0 : 0.0);
    }
  }

  for (m = high-1; m >= low+1; m--)
  {
    if (H[m][m-1] != 0.0)
    {
      for (i = m+1; i <= high; i++)
      {
        ort[i] = H[i][m-1];
      }
      for (j = m; j <= high; j++)
      {
        double g = 0.0;
        for (i = m; i <= high; i++)
        {
          g += ort[i] * V[i][j];
        }
        // Double division avoids possible underflow
        g = (g / ort[m]) / H[m][m-1];
        for (i = m; i <= high; i++)
        {
          V[i][j] += g * ort[i];
        }
      }
    }
  }
}
//------------------------------------------------------------------------------
void Eigenvalues::cdiv(double xr, double xi, double yr, double yi)
{
  double r, d;
  if (fabs(yr) > fabs(yi))
  {
    r = yi/yr;
    d = yr + r*yi;
    cdivr = (xr + r*xi)/d;
    cdivi = (xi - r*xr)/d;
  }
  else
  {
    r = yr/yi;
    d = yi + r*yr;
    cdivr = (r*xr + xi)/d;
    cdivi = (r*xi - xr)/d;
  }
}
//------------------------------------------------------------------------------
void Eigenvalues::hqr2()
{
  //  This is derived from the Algol procedure hqr2,
  //  by Martin and Wilkinson, Handbook for Auto. Comp.,
  //  Vol.ii-Linear Algebra, and the corresponding
  //  Fortran subroutine in EISPACK.

  int i, j, k;
   
  // Initialize
  int nn = this->n;
  int n = nn-1;
  int low = 0;
  int high = nn-1;
  double eps = pow(2.0,-52.0);
  double exshift = 0.0;
  double p=0,q=0,r=0,s=0,z=0,t,w,x,y;
   
  // Store roots isolated by balanc and compute matrix norm
  double norm = 0.0;
  for (i = 0; i < nn; i++)
  {
    if ((i < low) || (i > high))
    {
      d[i] = H[i][i];
      e[i] = 0.0;
    }
    for (j = ((i-1)>0)?i-1:0; j < nn; j++)
    {
      norm = norm + fabs(H[i][j]);
    }
  }
   
  // Outer loop over eigenvalue index
  int iter = 0;
  while (n >= low)
  {
    // Look for single small sub-diagonal element
    int l = n;
    while (l > low)
    {
      s = fabs(H[l-1][l-1]) + fabs(H[l][l]);
      if (s == 0.0)
      {
        s = norm;
      }
      if (fabs(H[l][l-1]) < eps * s)
      {
        break;
      }
      l--;
    }
       
    // Check for convergence
    // One root found
    if (l == n)
    {
      H[n][n] = H[n][n] + exshift;
      d[n] = H[n][n];
      e[n] = 0.0;
      n--;
      iter = 0;
   
      // Two roots found
    }
    else if (l == n-1)
    {
      w = H[n][n-1] * H[n-1][n];
      p = (H[n-1][n-1] - H[n][n]) / 2.0;
      q = p * p + w;
      z = sqrt(fabs(q));
      H[n][n] = H[n][n] + exshift;
      H[n-1][n-1] = H[n-1][n-1] + exshift;
      x = H[n][n];
  
      // Real pair
      if (q >= 0)
      {
        if (p >= 0)
        {
          z = p + z;
        }
        else
        {
          z = p - z;
        }
        d[n-1] = x + z;
        d[n] = d[n-1];
        if (z != 0.0)
        {
          d[n] = x - w / z;
        }
        e[n-1] = 0.0;
        e[n] = 0.0;
        x = H[n][n-1];
        s = fabs(x) + fabs(z);
        p = x / s;
        q = z / s;
        r = sqrt(p * p+q * q);
        p = p / r;
        q = q / r;
  
        // Row modification
        for (j = n-1; j < nn; j++)
        {
          z = H[n-1][j];
          H[n-1][j] = q * z + p * H[n][j];
          H[n][j] = q * H[n][j] - p * z;
        }
   
        // Column modification
        for (i = 0; i <= n; i++)
        {
          z = H[i][n-1];
          H[i][n-1] = q * z + p * H[i][n];
          H[i][n] = q * H[i][n] - p * z;
        }
   
        // Accumulate transformations
        for (i = low; i <= high; i++)
        {
          z = V[i][n-1];
          V[i][n-1] = q * z + p * V[i][n];
          V[i][n] = q * V[i][n] - p * z;
        }
   
        // Complex pair
      }
      else
      {
        d[n-1] = x + p;
        d[n] = x + p;
        e[n-1] = z;
        e[n] = -z;
      }
      n = n - 2;
      iter = 0;
   
      // No convergence yet
    }
    else
    {
      // Form shift
      x = H[n][n];
      y = 0.0;
      w = 0.0;
      if (l < n)
      {
        y = H[n-1][n-1];
        w = H[n][n-1] * H[n-1][n];
      }
   
      // Wilkinson's original ad hoc shift
      if (iter == 10)
      {
        exshift += x;
        for (i = low; i <= n; i++)
        {
          H[i][i] -= x;
        }
        s = fabs(H[n][n-1]) + fabs(H[n-1][n-2]);
        x = y = 0.75 * s;
        w = -0.4375 * s * s;
      }

      // MATLAB's new ad hoc shift
      if (iter == 30)
      {
        s = (y - x) / 2.0;
        s = s * s + w;
        if (s > 0)
        {
          s = sqrt(s);
          if (y < x)
          {
            s = -s;
          }
          s = x - w / ((y - x) / 2.0 + s);
          for (i = low; i <= n; i++)
          {
            H[i][i] -= s;
          }
          exshift += s;
          x = y = w = 0.964;
        }
      }
      iter = iter + 1;   // (Could check iteration count here.)
   
      // Look for two consecutive small sub-diagonal elements
      int m = n-2;
      while (m >= l)
      {
        z = H[m][m];
        r = x - z;
        s = y - z;
        p = (r * s - w) / H[m+1][m] + H[m][m+1];
        q = H[m+1][m+1] - z - r - s;
        r = H[m+2][m+1];
        s = fabs(p) + fabs(q) + fabs(r);
        p = p / s;
        q = q / s;
        r = r / s;
        if (m == l)
        {
          break;
        }
        if (fabs(H[m][m-1]) * (fabs(q) + fabs(r)) <
            eps * (fabs(p) * (fabs(H[m-1][m-1]) + fabs(z) +
              fabs(H[m+1][m+1]))))
        {
          break;
        }
        m--;
      }
   
      for (i = m+2; i <= n; i++)
      {
        H[i][i-2] = 0.0;
        if (i > m+2)
        {
          H[i][i-3] = 0.0;
        }
      }
   
      // Double QR step involving rows l:n and columns m:n
      for (k = m; k <= n-1; k++)
      {
        int notlast = (k != n-1);
        if (k != m)
        {
          p = H[k][k-1];
          q = H[k+1][k-1];
          r = (notlast ? H[k+2][k-1] : 0.0);
          x = fabs(p) + fabs(q) + fabs(r);
          if (x != 0.0)
          {
            p = p / x;
            q = q / x;
            r = r / x;
          }
        }
        if (x == 0.0)
        {
          break;
        }
        s = sqrt(p * p + q * q + r * r);
        if (p < 0)
        {
          s = -s;
        }
        if (s != 0)
        {
          if (k != m)
          {
            H[k][k-1] = -s * x;
          }
          else if (l != m)
          {
            H[k][k-1] = -H[k][k-1];
          }
          p = p + s;
          x = p / s;
          y = q / s;
          z = r / s;
          q = q / p;
          r = r / p;
  
          // Row modification
          for (j = k; j < nn; j++)
          {
            p = H[k][j] + q * H[k+1][j];
            if (notlast)
            {
              p = p + r * H[k+2][j];
              H[k+2][j] = H[k+2][j] - p * z;
            }
            H[k][j] = H[k][j] - p * x;
            H[k+1][j] = H[k+1][j] - p * y;
          }
   
          // Column modification
          int limit = (n<(k+3))?n:k+3;
          for (i = 0; i <= limit; i++)
          {
            p = x * H[i][k] + y * H[i][k+1];
            if (notlast)
            {
              p = p + z * H[i][k+2];
              H[i][k+2] = H[i][k+2] - p * r;
            }
            H[i][k] = H[i][k] - p;
            H[i][k+1] = H[i][k+1] - p * q;
          }
   
          // Accumulate transformations
          for (i = low; i <= high; i++)
          {
            p = x * V[i][k] + y * V[i][k+1];
            if (notlast)
            {
              p = p + z * V[i][k+2];
              V[i][k+2] = V[i][k+2] - p * r;
            }
            V[i][k] = V[i][k] - p;
            V[i][k+1] = V[i][k+1] - p * q;
          }
        }  // (s != 0)
      }  // k loop
    }  // check convergence
  }  // while (n >= low)
      
  // Backsubstitute to find vectors of upper triangular form
  if (norm == 0.0)
  {
    return;
  }
   
  for (n = nn-1; n >= 0; n--)
  {
    p = d[n];
    q = e[n];
   
    // Real vector
    if (q == 0)
    {
      int l = n;
      H[n][n] = 1.0;
      for (i = n-1; i >= 0; i--)
      {
        w = H[i][i] - p;
        r = 0.0;
        for (j = l; j <= n; j++)
        {
          r = r + H[i][j] * H[j][n];
        }
        if (e[i] < 0.0)
        {
          z = w;
          s = r;
        }
        else
        {
          l = i;
          if (e[i] == 0.0)
          {
            if (w != 0.0)
            {
              H[i][n] = -r / w;
            }
            else
            {
              H[i][n] = -r / (eps * norm);
            }
   
            // Solve real equations
          }
          else
          {
            x = H[i][i+1];
            y = H[i+1][i];
            q = (d[i] - p) * (d[i] - p) + e[i] * e[i];
            t = (x * s - z * r) / q;
            H[i][n] = t;
            if (fabs(x) > fabs(z))
            {
              H[i+1][n] = (-r - w * t) / x;
            }
            else
            {
              H[i+1][n] = (-s - y * t) / z;
            }
          }
   
          // Overflow control
          t = fabs(H[i][n]);
          if ((eps * t) * t > 1)
          {
            for (j = i; j <= n; j++)
            {
              H[j][n] = H[j][n] / t;
            }
          }
        }
      }
   
      // Complex vector
    }
    else if (q < 0)
    {
      int l = n-1;

      // Last vector component imaginary so matrix is triangular
      if (fabs(H[n][n-1]) > fabs(H[n-1][n]))
      {
        H[n-1][n-1] = q / H[n][n-1];
        H[n-1][n] = -(H[n][n] - p) / H[n][n-1];
      }
      else
      {
        cdiv(0.0,-H[n-1][n],H[n-1][n-1]-p,q);
        H[n-1][n-1] = cdivr;
        H[n-1][n] = cdivi;
      }
      H[n][n-1] = 0.0;
      H[n][n] = 1.0;
      for (i = n-2; i >= 0; i--)
      {
        double ra,sa,vr,vi;
        ra = 0.0;
        sa = 0.0;
        for (j = l; j <= n; j++)
        {
          ra = ra + H[i][j] * H[j][n-1];
          sa = sa + H[i][j] * H[j][n];
        }
        w = H[i][i] - p;
   
        if (e[i] < 0.0)
        {
          z = w;
          r = ra;
          s = sa;
        }
        else
        {
          l = i;
          if (e[i] == 0)
          {
            cdiv(-ra,-sa,w,q);
            H[i][n-1] = cdivr;
            H[i][n] = cdivi;
          }
          else
          {
            // Solve complex equations
            x = H[i][i+1];
            y = H[i+1][i];
            vr = (d[i] - p) * (d[i] - p) + e[i] * e[i] - q * q;
            vi = (d[i] - p) * 2.0 * q;
            if ((vr == 0.0) && (vi == 0.0))
            {
              vr = eps * norm * (fabs(w) + fabs(q) +
                        fabs(x) + fabs(y) + fabs(z));
            }
            cdiv(x*r-z*ra+q*sa,x*s-z*sa-q*ra,vr,vi);
            H[i][n-1] = cdivr;
            H[i][n] = cdivi;
            if (fabs(x) > (fabs(z) + fabs(q)))
            {
              H[i+1][n-1] = (-ra - w * H[i][n-1] + q * H[i][n]) / x;
              H[i+1][n] = (-sa - w * H[i][n] - q * H[i][n-1]) / x;
            }
            else
            {
              cdiv(-r-y*H[i][n-1],-s-y*H[i][n],z,q);
              H[i+1][n-1] = cdivr;
              H[i+1][n] = cdivi;
            }
          }
   
          // Overflow control
          double t1 = fabs(H[i][n-1]);
          double t2 = fabs(H[i][n]);
          t = (t1>t2)?t1:t2;
          if ((eps * t) * t > 1)
          {
            for (j = i; j <= n; j++)
            {
              H[j][n-1] = H[j][n-1] / t;
              H[j][n] = H[j][n] / t;
            }
          }
        }
      }
    }
  }
   
  // Vectors of isolated roots
  for (i = 0; i < nn; i++)
  {
    if (i < low || i > high)
    {
      for (j = i; j < nn; j++)
      {
        V[i][j] = H[i][j];
      }
    }
  }
   
  // Back transformation to get eigenvectors of original matrix
  for (j = nn-1; j >= low; j--)
  {
    for (i = low; i <= high; i++)
    {
      z = 0.0;
      int limit = (j<high)?j:high;
      for (k = low; k <= limit; k++)
      {
        z = z + V[i][k] * H[k][j];
      }
      V[i][j] = z;
    }
  }
}
//------------------------------------------------------------------------------
}
//------------------------------------------------------------------------------

---