Split computation of real Schur form in EigenSolver to its own class.

This is done with the minimal amount of work, so the result is very rough.

Eigen/src/Eigenvalues/EigenSolver.h

@@ -2,6 +2,7 @@
 // for linear algebra.
 //
 // Copyright (C) 2008 Gael Guennebaud <g.gael@free.fr>
+// Copyright (C) 2010 Jitse Niesen <jitse@maths.leeds.ac.uk>
 //
 // Eigen is free software; you can redistribute it and/or
 // modify it under the terms of the GNU Lesser General Public
@@ -25,7 +26,7 @@
 #ifndef EIGEN_EIGENSOLVER_H
 #define EIGEN_EIGENSOLVER_H
 
-#include "./HessenbergDecomposition.h"
+#include "./RealSchur.h"
 
 /** \eigenvalues_module \ingroup Eigenvalues_Module
   * \nonstableyet
@@ -246,10 +247,7 @@ template<typename _MatrixType> class EigenSolver
     EigenSolver& compute(const MatrixType& matrix);
 
   private:
-
-    typedef Matrix<RealScalar, ColsAtCompileTime, 1, Options, MaxColsAtCompileTime, 1> RealVectorType;
-    void orthes(MatrixType& matH, RealVectorType& ort);
-    void hqr2(MatrixType& matH);
+    void hqr2_step2(MatrixType& matH);
 
   protected:
     MatrixType m_eivec;
@@ -310,16 +308,15 @@ template<typename MatrixType>
 EigenSolver<MatrixType>& EigenSolver<MatrixType>::compute(const MatrixType& matrix)
 {
   assert(matrix.cols() == matrix.rows());
-  int n = matrix.cols();
-  m_eivalues.resize(n,1);
 
-  // Reduce to Hessenberg form.
-  HessenbergDecomposition<MatrixType> hd(matrix);
-  MatrixType matH = hd.matrixH();
-  m_eivec = hd.matrixQ();
+  // Reduce to real Schur form.
+  RealSchur<MatrixType> rs(matrix);
+  MatrixType matH = rs.matrixT();
+  m_eivec = rs.matrixU();
+  m_eivalues = rs.eigenvalues();
 
-  // Reduce Hessenberg to real Schur form.
-  hqr2(matH);
+  // Compute eigenvectors.
+  hqr2_step2(matH);
 
   m_isInitialized = true;
   return *this;
@@ -345,289 +342,29 @@ std::complex<Scalar> cdiv(Scalar xr, Scalar xi, Scalar yr, Scalar yi)
 }
 
 
-// Nonsymmetric reduction from Hessenberg to real Schur form.
 template<typename MatrixType>
-void EigenSolver<MatrixType>::hqr2(MatrixType& matH)
+void EigenSolver<MatrixType>::hqr2_step2(MatrixType& matH)
 {
-  //  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.
+  const int nn = m_eivec.cols();
+  const int low = 0;
+  const int high = nn-1;
+  const Scalar eps = ei_pow(Scalar(2),ei_is_same_type<Scalar,float>::ret ? Scalar(-23) : Scalar(-52));
+  Scalar p, q, r=0, s=0, t, w, x, y, z=0;
 
-  // Initialize
-  int nn = m_eivec.cols();
-  int n = nn-1;
-  int low = 0;
-  int high = nn-1;
-  Scalar eps = ei_pow(Scalar(2),ei_is_same_type<Scalar,float>::ret ? Scalar(-23) : Scalar(-52));
-  Scalar exshift = 0.0;
-  Scalar p=0,q=0,r=0,s=0,z=0,t,w,x,y;
-
-  // Store roots isolated by balanc and compute matrix norm
-  // FIXME to be efficient the following would requires a triangular reduxion code
-  // Scalar norm = matH.upper().cwiseAbs().sum() + matH.corner(BottomLeft,n,n).diagonal().cwiseAbs().sum();
+  // inefficient! this is already computed in RealSchur
   Scalar norm = 0.0;
   for (int j = 0; j < nn; ++j)
   {
-    // FIXME what's the purpose of the following since the condition is always false
-    if ((j < low) || (j > high))
-    {
-      m_eivalues.coeffRef(j) = Complex(matH.coeff(j,j), 0.0);
-    }
     norm += matH.row(j).segment(std::max(j-1,0), nn-std::max(j-1,0)).cwiseAbs().sum();
   }
-
-  // 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 = ei_abs(matH.coeff(l-1,l-1)) + ei_abs(matH.coeff(l,l));
-      if (s == 0.0)
-          s = norm;
-      if (ei_abs(matH.coeff(l,l-1)) < eps * s)
-        break;
-      l--;
-    }
-
-    // Check for convergence
-    // One root found
-    if (l == n)
-    {
-      matH.coeffRef(n,n) = matH.coeff(n,n) + exshift;
-      m_eivalues.coeffRef(n) = Complex(matH.coeff(n,n), 0.0);
-      n--;
-      iter = 0;
-    }
-    else if (l == n-1) // Two roots found
-    {
-      w = matH.coeff(n,n-1) * matH.coeff(n-1,n);
-      p = (matH.coeff(n-1,n-1) - matH.coeff(n,n)) * Scalar(0.5);
-      q = p * p + w;
-      z = ei_sqrt(ei_abs(q));
-      matH.coeffRef(n,n) = matH.coeff(n,n) + exshift;
-      matH.coeffRef(n-1,n-1) = matH.coeff(n-1,n-1) + exshift;
-      x = matH.coeff(n,n);
-
-      // Scalar pair
-      if (q >= 0)
-      {
-        if (p >= 0)
-          z = p + z;
-        else
-          z = p - z;
-
-        m_eivalues.coeffRef(n-1) = Complex(x + z, 0.0);
-        m_eivalues.coeffRef(n) = Complex(z!=0.0 ? x - w / z : m_eivalues.coeff(n-1).real(), 0.0);
-
-        x = matH.coeff(n,n-1);
-        s = ei_abs(x) + ei_abs(z);
-        p = x / s;
-        q = z / s;
-        r = ei_sqrt(p * p+q * q);
-        p = p / r;
-        q = q / r;
-
-        // Row modification
-        for (int j = n-1; j < nn; ++j)
-        {
-          z = matH.coeff(n-1,j);
-          matH.coeffRef(n-1,j) = q * z + p * matH.coeff(n,j);
-          matH.coeffRef(n,j) = q * matH.coeff(n,j) - p * z;
-        }
-
-        // Column modification
-        for (int i = 0; i <= n; ++i)
-        {
-          z = matH.coeff(i,n-1);
-          matH.coeffRef(i,n-1) = q * z + p * matH.coeff(i,n);
-          matH.coeffRef(i,n) = q * matH.coeff(i,n) - p * z;
-        }
-
-        // Accumulate transformations
-        for (int i = low; i <= high; ++i)
-        {
-          z = m_eivec.coeff(i,n-1);
-          m_eivec.coeffRef(i,n-1) = q * z + p * m_eivec.coeff(i,n);
-          m_eivec.coeffRef(i,n) = q * m_eivec.coeff(i,n) - p * z;
-        }
-      }
-      else // Complex pair
-      {
-        m_eivalues.coeffRef(n-1) = Complex(x + p, z);
-        m_eivalues.coeffRef(n)   = Complex(x + p, -z);
-      }
-      n = n - 2;
-      iter = 0;
-    }
-    else // No convergence yet
-    {
-      // Form shift
-      x = matH.coeff(n,n);
-      y = 0.0;
-      w = 0.0;
-      if (l < n)
-      {
-          y = matH.coeff(n-1,n-1);
-          w = matH.coeff(n,n-1) * matH.coeff(n-1,n);
-      }
-
-      // Wilkinson's original ad hoc shift
-      if (iter == 10)
-      {
-        exshift += x;
-        for (int i = low; i <= n; ++i)
-          matH.coeffRef(i,i) -= x;
-        s = ei_abs(matH.coeff(n,n-1)) + ei_abs(matH.coeff(n-1,n-2));
-        x = y = Scalar(0.75) * s;
-        w = Scalar(-0.4375) * s * s;
-      }
-
-      // MATLAB's new ad hoc shift
-      if (iter == 30)
-      {
-        s = Scalar((y - x) / 2.0);
-        s = s * s + w;
-        if (s > 0)
-        {
-          s = ei_sqrt(s);
-          if (y < x)
-            s = -s;
-          s = Scalar(x - w / ((y - x) / 2.0 + s));
-          for (int i = low; i <= n; ++i)
-            matH.coeffRef(i,i) -= s;
-          exshift += s;
-          x = y = w = Scalar(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 = matH.coeff(m,m);
-        r = x - z;
-        s = y - z;
-        p = (r * s - w) / matH.coeff(m+1,m) + matH.coeff(m,m+1);
-        q = matH.coeff(m+1,m+1) - z - r - s;
-        r = matH.coeff(m+2,m+1);
-        s = ei_abs(p) + ei_abs(q) + ei_abs(r);
-        p = p / s;
-        q = q / s;
-        r = r / s;
-        if (m == l) {
-          break;
-        }
-        if (ei_abs(matH.coeff(m,m-1)) * (ei_abs(q) + ei_abs(r)) <
-          eps * (ei_abs(p) * (ei_abs(matH.coeff(m-1,m-1)) + ei_abs(z) +
-          ei_abs(matH.coeff(m+1,m+1)))))
-        {
-          break;
-        }
-        m--;
-      }
-
-      for (int i = m+2; i <= n; ++i)
-      {
-        matH.coeffRef(i,i-2) = 0.0;
-        if (i > m+2)
-          matH.coeffRef(i,i-3) = 0.0;
-      }
-
-      // Double QR step involving rows l:n and columns m:n
-      for (int k = m; k <= n-1; ++k)
-      {
-        int notlast = (k != n-1);
-        if (k != m) {
-          p = matH.coeff(k,k-1);
-          q = matH.coeff(k+1,k-1);
-          r = notlast ? matH.coeff(k+2,k-1) : Scalar(0);
-          x = ei_abs(p) + ei_abs(q) + ei_abs(r);
-          if (x != 0.0)
-          {
-            p = p / x;
-            q = q / x;
-            r = r / x;
-          }
-        }
-
-        if (x == 0.0)
-          break;
-
-        s = ei_sqrt(p * p + q * q + r * r);
-
-        if (p < 0)
-          s = -s;
-
-        if (s != 0)
-        {
-          if (k != m)
-            matH.coeffRef(k,k-1) = -s * x;
-          else if (l != m)
-            matH.coeffRef(k,k-1) = -matH.coeff(k,k-1);
-
-          p = p + s;
-          x = p / s;
-          y = q / s;
-          z = r / s;
-          q = q / p;
-          r = r / p;
-
-          // Row modification
-          for (int j = k; j < nn; ++j)
-          {
-            p = matH.coeff(k,j) + q * matH.coeff(k+1,j);
-            if (notlast)
-            {
-              p = p + r * matH.coeff(k+2,j);
-              matH.coeffRef(k+2,j) = matH.coeff(k+2,j) - p * z;
-            }
-            matH.coeffRef(k,j) = matH.coeff(k,j) - p * x;
-            matH.coeffRef(k+1,j) = matH.coeff(k+1,j) - p * y;
-          }
-
-          // Column modification
-          for (int i = 0; i <= std::min(n,k+3); ++i)
-          {
-            p = x * matH.coeff(i,k) + y * matH.coeff(i,k+1);
-            if (notlast)
-            {
-              p = p + z * matH.coeff(i,k+2);
-              matH.coeffRef(i,k+2) = matH.coeff(i,k+2) - p * r;
-            }
-            matH.coeffRef(i,k) = matH.coeff(i,k) - p;
-            matH.coeffRef(i,k+1) = matH.coeff(i,k+1) - p * q;
-          }
-
-          // Accumulate transformations
-          for (int i = low; i <= high; ++i)
-          {
-            p = x * m_eivec.coeff(i,k) + y * m_eivec.coeff(i,k+1);
-            if (notlast)
-            {
-              p = p + z * m_eivec.coeff(i,k+2);
-              m_eivec.coeffRef(i,k+2) = m_eivec.coeff(i,k+2) - p * r;
-            }
-            m_eivec.coeffRef(i,k) = m_eivec.coeff(i,k) - p;
-            m_eivec.coeffRef(i,k+1) = m_eivec.coeff(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--)
+  for (int n = nn-1; n >= 0; n--)
   {
     p = m_eivalues.coeff(n).real();
     q = m_eivalues.coeff(n).imag();