merge

3 files changed, 47 insertions, 28 deletions

diff --git a/Eigen/src/Eigenvalues/GeneralizedEigenSolver.h b/Eigen/src/Eigenvalues/GeneralizedEigenSolver.h
index 07a9ccf46..650617ca7 100644
--- a/Eigen/src/Eigenvalues/GeneralizedEigenSolver.h
+++ b/Eigen/src/Eigenvalues/GeneralizedEigenSolver.h
@@ -327,33 +327,22 @@ GeneralizedEigenSolver<MatrixType>::compute(const MatrixType& A, const MatrixTyp
       }
       else
       {
-        // We need to extract the generalized eigenvalues of the pair of a general 2x2 block S and a triangular 2x2 block T
-        // From the eigen decomposition of T = U * E * U^-1,
-        // we can extract the eigenvalues of (U^-1 * S * U) / E
-        // Here, we can take advantage that E = diag(T), and U = [ 1 T_01 ; 0 T_11-T_00], and U^-1 = [1 -T_11/(T_11-T_00) ; 0 1/(T_11-T_00)].
-        // Then taking beta=T_00*T_11*(T_11-T_00), we can avoid any division, and alpha is the eigenvalues of A = (U^-1 * S * U) * diag(T_11,T_00) * (T_11-T_00):
-
-        // T =  [a b ; 0 c]
-        // S =  [e f ; g h]
-        RealScalar a = m_realQZ.matrixT().coeff(i, i), b = m_realQZ.matrixT().coeff(i, i+1), c = m_realQZ.matrixT().coeff(i+1, i+1);
-        RealScalar e = m_matS.coeff(i, i), f = m_matS.coeff(i, i+1), g = m_matS.coeff(i+1, i), h = m_matS.coeff(i+1, i+1);
-        RealScalar d = c-a;
-        RealScalar gb = g*b;
-        Matrix<RealScalar,2,2> A;
-        A <<  (e*d-gb)*c,  ((e*b+f*d-h*b)*d-gb*b)*a,
-              g*c       ,  (gb+h*d)*a;
-
-        // NOTE, we could also compute the SVD of T's block during the QZ factorization so that the respective T block is guaranteed to be diagonal,
-        // and then we could directly apply the formula below (while taking care of scaling S columns by T11,T00):
-
-        Scalar p = Scalar(0.5) * (A.coeff(i, i) - A.coeff(i+1, i+1));
-        Scalar z = sqrt(abs(p * p + A.coeff(i+1, i) * A.coeff(i, i+1)));
-        m_alphas.coeffRef(i)   = ComplexScalar(A.coeff(i+1, i+1) + p, z);
-        m_alphas.coeffRef(i+1) = ComplexScalar(A.coeff(i+1, i+1) + p, -z);
+        // We need to extract the generalized eigenvalues of the pair of a general 2x2 block S and a positive diagonal 2x2 block T
+        // Then taking beta=T_00*T_11, we can avoid any division, and alpha is the eigenvalues of A = (U^-1 * S * U) * diag(T_11,T_00):
 
-        m_betas.coeffRef(i)   =
-        m_betas.coeffRef(i+1) = a*c*d;
+        // T =  [a 0]
+        //      [0 b]
+        RealScalar a = m_realQZ.matrixT().coeff(i, i), b = m_realQZ.matrixT().coeff(i+1, i+1);
+        Matrix<RealScalar,2,2> S2 = m_matS.template block<2,2>(i,i) * Matrix<Scalar,2,1>(b,a).asDiagonal();
+
+        Scalar p = Scalar(0.5) * (S2.coeff(0,0) - S2.coeff(1,1));
+        Scalar z = sqrt(abs(p * p + S2.coeff(1,0) * S2.coeff(0,1)));
+        m_alphas.coeffRef(i)   = ComplexScalar(S2.coeff(1,1) + p,  z);
+        m_alphas.coeffRef(i+1) = ComplexScalar(S2.coeff(1,1) + p, -z);
 
+        m_betas.coeffRef(i)   =
+        m_betas.coeffRef(i+1) = a*b;
+        
         i += 2;
       }
     }
diff --git a/Eigen/src/Eigenvalues/RealQZ.h b/Eigen/src/Eigenvalues/RealQZ.h
index a62071d42..b3a910dd9 100644
--- a/Eigen/src/Eigenvalues/RealQZ.h
+++ b/Eigen/src/Eigenvalues/RealQZ.h
@@ -552,7 +552,6 @@ namespace Eigen {
       m_T.coeffRef(l,l-1) = Scalar(0.0);
     }
 
-
   template<typename MatrixType>
     RealQZ<MatrixType>& RealQZ<MatrixType>::compute(const MatrixType& A_in, const MatrixType& B_in, bool computeQZ)
     {
@@ -616,6 +615,37 @@ namespace Eigen {
       }
       // check if we converged before reaching iterations limit
       m_info = (local_iter<m_maxIters) ? Success : NoConvergence;
+
+      // For each non triangular 2x2 diagonal block of S,
+      //    reduce the respective 2x2 diagonal block of T to positive diagonal form using 2x2 SVD.
+      // This step is not mandatory for QZ, but it does help further extraction of eigenvalues/eigenvectors,
+      // and is in par with Lapack/Matlab QZ.
+      if(m_info==Success)
+      {
+        for(Index i=0; i<dim-1; ++i)
+        {
+          if(m_S.coeff(i+1, i) != Scalar(0))
+          {
+            JacobiRotation<Scalar> j_left, j_right;
+            internal::real_2x2_jacobi_svd(m_T, i, i+1, &j_left, &j_right);
+
+            // Apply resulting Jacobi rotations
+            m_S.applyOnTheLeft(i,i+1,j_left);
+            m_S.applyOnTheRight(i,i+1,j_right);
+            m_T.applyOnTheLeft(i,i+1,j_left);
+            m_T.applyOnTheRight(i,i+1,j_right);
+            m_T(i+1,i) = m_T(i,i+1) = Scalar(0);
+
+            if(m_computeQZ) {
+              m_Q.applyOnTheRight(i,i+1,j_left.transpose());
+              m_Z.applyOnTheLeft(i,i+1,j_right.transpose());
+            }
+
+            i++;
+          }
+        }
+      }
+
       return *this;
     } // end compute
 
diff --git a/Eigen/src/Eigenvalues/Tridiagonalization.h b/Eigen/src/Eigenvalues/Tridiagonalization.h
index 2030b5be1..1d102c17b 100644
--- a/Eigen/src/Eigenvalues/Tridiagonalization.h
+++ b/Eigen/src/Eigenvalues/Tridiagonalization.h
@@ -367,10 +367,10 @@ void tridiagonalization_inplace(MatrixType& matA, CoeffVectorType& hCoeffs)
     hCoeffs.tail(n-i-1).noalias() = (matA.bottomRightCorner(remainingSize,remainingSize).template selfadjointView<Lower>()
                                   * (conj(h) * matA.col(i).tail(remainingSize)));
 
-    hCoeffs.tail(n-i-1) += (conj(h)*Scalar(-0.5)*(hCoeffs.tail(remainingSize).dot(matA.col(i).tail(remainingSize)))) * matA.col(i).tail(n-i-1);
+    hCoeffs.tail(n-i-1) += (conj(h)*RealScalar(-0.5)*(hCoeffs.tail(remainingSize).dot(matA.col(i).tail(remainingSize)))) * matA.col(i).tail(n-i-1);
 
     matA.bottomRightCorner(remainingSize, remainingSize).template selfadjointView<Lower>()
-      .rankUpdate(matA.col(i).tail(remainingSize), hCoeffs.tail(remainingSize), -1);
+      .rankUpdate(matA.col(i).tail(remainingSize), hCoeffs.tail(remainingSize), Scalar(-1));
 
     matA.col(i).coeffRef(i+1) = beta;
     hCoeffs.coeffRef(i) = h;