Split code for (quasi)triangular matrices from MatrixSquareRoot.

This way, (quasi)triangular matrices can avoid the costly Schur decomposition.

unsupported/Eigen/src/MatrixFunctions/MatrixSquareRoot.h

@@ -26,76 +26,71 @@
 #define EIGEN_MATRIX_SQUARE_ROOT
 
 /** \ingroup MatrixFunctions_Module
-  * \brief Class for computing matrix square roots.
+  * \brief Class for computing matrix square roots of upper quasi-triangular matrices.
-  * \tparam MatrixType type of the argument of the matrix square root,
+  * \tparam  MatrixType  type of the argument of the matrix square root,
-  * expected to be an instantiation of the Matrix class template.
+  *                      expected to be an instantiation of the Matrix class template.
+  *
+  * This class computes the square root of the upper quasi-triangular
+  * matrix stored in the upper Hessenberg part of the matrix passed to
+  * the constructor.
+  *
+  * \sa MatrixSquareRoot, MatrixSquareRootTriangular
   */
-template <typename MatrixType, int IsComplex = NumTraits<typename internal::traits<MatrixType>::Scalar>::IsComplex>
+template <typename MatrixType>
-class MatrixSquareRoot
+class MatrixSquareRootQuasiTriangular
-{  
+{
   public:
 
     /** \brief Constructor. 
       *
-      * \param[in]  A  matrix whose square root is to be computed.
+      * \param[in]  A  upper quasi-triangular matrix whose square root 
+      *                is to be computed.
       *
       * The class stores a reference to \p A, so it should not be
       * changed (or destroyed) before compute() is called.
       */
-    MatrixSquareRoot(const MatrixType& A);
+    MatrixSquareRootQuasiTriangular(const MatrixType& A) 
-
+      : m_A(A) 
+    {
+      eigen_assert(A.rows() == A.cols());
+    }
+    
     /** \brief Compute the matrix square root
       *
       * \param[out] result  square root of \p A, as specified in the constructor.
       *
-      * See MatrixBase::sqrt() for details on how this computation
+      * Only the upper Hessenberg part of \p result is updated, the
-      * is implemented.
+      * rest is not touched.  See MatrixBase::sqrt() for details on
+      * how this computation is implemented.
       */
-    template <typename ResultType> 
+    template <typename ResultType> void compute(ResultType &result);    
-    void compute(ResultType &result);    
+    
-};
+  private:
-
+    typedef typename MatrixType::Index Index;
-
+    typedef typename MatrixType::Scalar Scalar;
-// ********** Partial specialization for real matrices **********
+    
-
+    void computeDiagonalPartOfSqrt(MatrixType& sqrtT, const MatrixType& T);
-template <typename MatrixType>
+    void computeOffDiagonalPartOfSqrt(MatrixType& sqrtT, const MatrixType& T);
-class MatrixSquareRoot<MatrixType, 0>
+    void compute2x2diagonalBlock(MatrixType& sqrtT, const MatrixType& T, typename MatrixType::Index i);
-{
+    void compute1x1offDiagonalBlock(MatrixType& sqrtT, const MatrixType& T, 
-public:
+  				  typename MatrixType::Index i, typename MatrixType::Index j);
-  MatrixSquareRoot(const MatrixType& A) 
+    void compute1x2offDiagonalBlock(MatrixType& sqrtT, const MatrixType& T, 
-    : m_A(A) 
+  				  typename MatrixType::Index i, typename MatrixType::Index j);
-  {
+    void compute2x1offDiagonalBlock(MatrixType& sqrtT, const MatrixType& T, 
-    eigen_assert(A.rows() == A.cols());
+  				  typename MatrixType::Index i, typename MatrixType::Index j);
-  }
+    void compute2x2offDiagonalBlock(MatrixType& sqrtT, const MatrixType& T, 
+  				  typename MatrixType::Index i, typename MatrixType::Index j);
   
-  template <typename ResultType> void compute(ResultType &result);    
+    template <typename SmallMatrixType>
+    static void solveAuxiliaryEquation(SmallMatrixType& X, const SmallMatrixType& A, 
+  				     const SmallMatrixType& B, const SmallMatrixType& C);
   
-private:
+    const MatrixType& m_A;
-  typedef typename MatrixType::Index Index;
-  typedef typename MatrixType::Scalar Scalar;
-  
-  void computeDiagonalPartOfSqrt(MatrixType& sqrtT, const MatrixType& T);
-  void computeOffDiagonalPartOfSqrt(MatrixType& sqrtT, const MatrixType& T);
-  void compute2x2diagonalBlock(MatrixType& sqrtT, const MatrixType& T, typename MatrixType::Index i);
-  void compute1x1offDiagonalBlock(MatrixType& sqrtT, const MatrixType& T, 
-				  typename MatrixType::Index i, typename MatrixType::Index j);
-  void compute1x2offDiagonalBlock(MatrixType& sqrtT, const MatrixType& T, 
-				  typename MatrixType::Index i, typename MatrixType::Index j);
-  void compute2x1offDiagonalBlock(MatrixType& sqrtT, const MatrixType& T, 
-				  typename MatrixType::Index i, typename MatrixType::Index j);
-  void compute2x2offDiagonalBlock(MatrixType& sqrtT, const MatrixType& T, 
-				  typename MatrixType::Index i, typename MatrixType::Index j);
-
-  template <typename SmallMatrixType>
-  static void solveAuxiliaryEquation(SmallMatrixType& X, const SmallMatrixType& A, 
-				     const SmallMatrixType& B, const SmallMatrixType& C);
-
-  const MatrixType& m_A;
 };
 
 template <typename MatrixType>
 template <typename ResultType> 
-void MatrixSquareRoot<MatrixType, 0>::compute(ResultType &result)
+void MatrixSquareRootQuasiTriangular<MatrixType>::compute(ResultType &result)
 {
   // Compute Schur decomposition of m_A
   const RealSchur<MatrixType> schurOfA(m_A);  
@@ -114,7 +109,8 @@ void MatrixSquareRoot<MatrixType, 0>::compute(ResultType &result)
 // pre:  T is quasi-upper-triangular and sqrtT is a zero matrix of the same size
 // post: the diagonal blocks of sqrtT are the square roots of the diagonal blocks of T
 template <typename MatrixType>
-void MatrixSquareRoot<MatrixType, 0>::computeDiagonalPartOfSqrt(MatrixType& sqrtT, const MatrixType& T)
+void MatrixSquareRootQuasiTriangular<MatrixType>::computeDiagonalPartOfSqrt(MatrixType& sqrtT, 
+									  const MatrixType& T)
 {
   const Index size = m_A.rows();
   for (Index i = 0; i < size; i++) {
@@ -132,7 +128,8 @@ void MatrixSquareRoot<MatrixType, 0>::computeDiagonalPartOfSqrt(MatrixType& sqrt
 // pre:  T is quasi-upper-triangular and diagonal blocks of sqrtT are square root of diagonal blocks of T.
 // post: sqrtT is the square root of T.
 template <typename MatrixType>
-void MatrixSquareRoot<MatrixType, 0>::computeOffDiagonalPartOfSqrt(MatrixType& sqrtT, const MatrixType& T)
+void MatrixSquareRootQuasiTriangular<MatrixType>::computeOffDiagonalPartOfSqrt(MatrixType& sqrtT, 
+									     const MatrixType& T)
 {
   const Index size = m_A.rows();
   for (Index j = 1; j < size; j++) {
@@ -158,9 +155,8 @@ void MatrixSquareRoot<MatrixType, 0>::computeOffDiagonalPartOfSqrt(MatrixType& s
 // pre:  T.block(i,i,2,2) has complex conjugate eigenvalues
 // post: sqrtT.block(i,i,2,2) is square root of T.block(i,i,2,2)
 template <typename MatrixType>
-void MatrixSquareRoot<MatrixType, 0>::compute2x2diagonalBlock(MatrixType& sqrtT, 
+void MatrixSquareRootQuasiTriangular<MatrixType>
-							      const MatrixType& T, 
+     ::compute2x2diagonalBlock(MatrixType& sqrtT, const MatrixType& T, typename MatrixType::Index i)
-							      typename MatrixType::Index i)
 {
   // TODO: This case (2-by-2 blocks with complex conjugate eigenvalues) is probably hidden somewhere
   //       in EigenSolver. If we expose it, we could call it directly from here.
@@ -174,10 +170,9 @@ void MatrixSquareRoot<MatrixType, 0>::compute2x2diagonalBlock(MatrixType& sqrtT,
 //       all blocks of sqrtT to left of and below (i,j) are correct
 // post: sqrtT(i,j) has the correct value
 template <typename MatrixType>
-void MatrixSquareRoot<MatrixType, 0>::compute1x1offDiagonalBlock(MatrixType& sqrtT, 
+void MatrixSquareRootQuasiTriangular<MatrixType>
-								 const MatrixType& T, 
+     ::compute1x1offDiagonalBlock(MatrixType& sqrtT, const MatrixType& T, 
-								 typename MatrixType::Index i,
+				  typename MatrixType::Index i, typename MatrixType::Index j)
-								 typename MatrixType::Index j)
 {
   Scalar tmp = (sqrtT.row(i).segment(i+1,j-i-1) * sqrtT.col(j).segment(i+1,j-i-1)).value();
   sqrtT.coeffRef(i,j) = (T.coeff(i,j) - tmp) / (sqrtT.coeff(i,i) + sqrtT.coeff(j,j));
@@ -185,10 +180,9 @@ void MatrixSquareRoot<MatrixType, 0>::compute1x1offDiagonalBlock(MatrixType& sqr
 
 // similar to compute1x1offDiagonalBlock()
 template <typename MatrixType>
-void MatrixSquareRoot<MatrixType, 0>::compute1x2offDiagonalBlock(MatrixType& sqrtT, 
+void MatrixSquareRootQuasiTriangular<MatrixType>
-								 const MatrixType& T, 
+     ::compute1x2offDiagonalBlock(MatrixType& sqrtT, const MatrixType& T, 
-								 typename MatrixType::Index i,
+				  typename MatrixType::Index i, typename MatrixType::Index j)
-								 typename MatrixType::Index j)
 {
   Matrix<Scalar,1,2> rhs = T.template block<1,2>(i,j);
   if (j-i > 1)
@@ -200,10 +194,9 @@ void MatrixSquareRoot<MatrixType, 0>::compute1x2offDiagonalBlock(MatrixType& sqr
 
 // similar to compute1x1offDiagonalBlock()
 template <typename MatrixType>
-void MatrixSquareRoot<MatrixType, 0>::compute2x1offDiagonalBlock(MatrixType& sqrtT, 
+void MatrixSquareRootQuasiTriangular<MatrixType>
-								 const MatrixType& T, 
+     ::compute2x1offDiagonalBlock(MatrixType& sqrtT, const MatrixType& T, 
-								 typename MatrixType::Index i,
+				  typename MatrixType::Index i, typename MatrixType::Index j)
-								 typename MatrixType::Index j)
 {
   Matrix<Scalar,2,1> rhs = T.template block<2,1>(i,j);
   if (j-i > 2)
@@ -215,10 +208,9 @@ void MatrixSquareRoot<MatrixType, 0>::compute2x1offDiagonalBlock(MatrixType& sqr
 
 // similar to compute1x1offDiagonalBlock()
 template <typename MatrixType>
-void MatrixSquareRoot<MatrixType, 0>::compute2x2offDiagonalBlock(MatrixType& sqrtT, 
+void MatrixSquareRootQuasiTriangular<MatrixType>
-								 const MatrixType& T, 
+     ::compute2x2offDiagonalBlock(MatrixType& sqrtT, const MatrixType& T, 
-								 typename MatrixType::Index i,
+				  typename MatrixType::Index i, typename MatrixType::Index j)
-								 typename MatrixType::Index j)
 {
   Matrix<Scalar,2,2> A = sqrtT.template block<2,2>(i,i);
   Matrix<Scalar,2,2> B = sqrtT.template block<2,2>(j,j);
@@ -233,10 +225,9 @@ void MatrixSquareRoot<MatrixType, 0>::compute2x2offDiagonalBlock(MatrixType& sqr
 // solves the equation A X + X B = C where all matrices are 2-by-2
 template <typename MatrixType>
 template <typename SmallMatrixType>
-void MatrixSquareRoot<MatrixType, 0>::solveAuxiliaryEquation(SmallMatrixType& X,
+void MatrixSquareRootQuasiTriangular<MatrixType>
-							     const SmallMatrixType& A,
+     ::solveAuxiliaryEquation(SmallMatrixType& X, const SmallMatrixType& A,
-							     const SmallMatrixType& B,
+			      const SmallMatrixType& B, const SmallMatrixType& C)
-							     const SmallMatrixType& C)
 {
   EIGEN_STATIC_ASSERT((internal::is_same<SmallMatrixType, Matrix<Scalar,2,2> >::value),
 		      EIGEN_INTERNAL_ERROR_PLEASE_FILE_A_BUG_REPORT);
@@ -270,18 +261,37 @@ void MatrixSquareRoot<MatrixType, 0>::solveAuxiliaryEquation(SmallMatrixType& X,
   X.coeffRef(1,1) = result.coeff(3);
 }
 
-// ********** Partial specialization for complex matrices **********
 
+/** \ingroup MatrixFunctions_Module
+  * \brief Class for computing matrix square roots of upper triangular matrices.
+  * \tparam  MatrixType  type of the argument of the matrix square root,
+  *                      expected to be an instantiation of the Matrix class template.
+  *
+  * This class computes the square root of the upper triangular matrix
+  * stored in the upper triangular part (including the diagonal) of
+  * the matrix passed to the constructor.
+  *
+  * \sa MatrixSquareRoot, MatrixSquareRootQuasiTriangular
+  */
 template <typename MatrixType>
-class MatrixSquareRoot<MatrixType, 1>
+class MatrixSquareRootTriangular
 {
   public:
-    MatrixSquareRoot(const MatrixType& A) 
+    MatrixSquareRootTriangular(const MatrixType& A) 
       : m_A(A) 
     {
       eigen_assert(A.rows() == A.cols());
     }
 
+    /** \brief Compute the matrix square root
+      *
+      * \param[out] result  square root of \p A, as specified in the constructor.
+      *
+      * Only the upper triangular part (including the diagonal) of 
+      * \p result is updated, the rest is not touched.  See
+      * MatrixBase::sqrt() for details on how this computation is
+      * implemented.
+      */
     template <typename ResultType> void compute(ResultType &result);    
 
  private:
@@ -290,7 +300,7 @@ class MatrixSquareRoot<MatrixType, 1>
 
 template <typename MatrixType>
 template <typename ResultType> 
-void MatrixSquareRoot<MatrixType, 1>::compute(ResultType &result)
+void MatrixSquareRootTriangular<MatrixType>::compute(ResultType &result)
 {
   // Compute Schur decomposition of m_A
   const ComplexSchur<MatrixType> schurOfA(m_A);  
@@ -320,6 +330,107 @@ void MatrixSquareRoot<MatrixType, 1>::compute(ResultType &result)
   result.noalias() = tmp * U.adjoint();
 }
 
+
+/** \ingroup MatrixFunctions_Module
+  * \brief Class for computing matrix square roots of general matrices.
+  * \tparam  MatrixType  type of the argument of the matrix square root,
+  *                      expected to be an instantiation of the Matrix class template.
+  *
+  * \sa MatrixSquareRootTriangular, MatrixSquareRootQuasiTriangular, MatrixBase::sqrt()
+  */
+template <typename MatrixType, int IsComplex = NumTraits<typename internal::traits<MatrixType>::Scalar>::IsComplex>
+class MatrixSquareRoot
+{
+  public:
+
+    /** \brief Constructor. 
+      *
+      * \param[in]  A  matrix whose square root is to be computed.
+      *
+      * The class stores a reference to \p A, so it should not be
+      * changed (or destroyed) before compute() is called.
+      */
+    MatrixSquareRoot(const MatrixType& A); 
+    
+    /** \brief Compute the matrix square root
+      *
+      * \param[out] result  square root of \p A, as specified in the constructor.
+      *
+      * See MatrixBase::sqrt() for details on how this computation is
+      * implemented.
+      */
+    template <typename ResultType> void compute(ResultType &result);    
+};
+
+
+// ********** Partial specialization for real matrices **********
+
+template <typename MatrixType>
+class MatrixSquareRoot<MatrixType, 0>
+{
+  public:
+
+    MatrixSquareRoot(const MatrixType& A) 
+      : m_A(A) 
+    {  
+      eigen_assert(A.rows() == A.cols());
+    }
+  
+    template <typename ResultType> void compute(ResultType &result)
+    {
+      // Compute Schur decomposition of m_A
+      const RealSchur<MatrixType> schurOfA(m_A);  
+      const MatrixType& T = schurOfA.matrixT();
+      const MatrixType& U = schurOfA.matrixU();
+    
+      // Compute square root of T
+      MatrixSquareRootQuasiTriangular<MatrixType> tmp(T);
+      MatrixType sqrtT = MatrixType::Zero(m_A.rows(), m_A.rows());
+      tmp.compute(sqrtT);
+    
+      // Compute square root of m_A
+      result = U * sqrtT * U.adjoint();
+    }
+    
+  private:
+    const MatrixType& m_A;
+};
+
+
+// ********** Partial specialization for complex matrices **********
+
+template <typename MatrixType>
+class MatrixSquareRoot<MatrixType, 1>
+{
+  public:
+
+    MatrixSquareRoot(const MatrixType& A) 
+      : m_A(A) 
+    {  
+      eigen_assert(A.rows() == A.cols());
+    }
+  
+    template <typename ResultType> void compute(ResultType &result)
+    {
+      // Compute Schur decomposition of m_A
+      const ComplexSchur<MatrixType> schurOfA(m_A);  
+      const MatrixType& T = schurOfA.matrixT();
+      const MatrixType& U = schurOfA.matrixU();
+    
+      // Compute square root of T
+      MatrixSquareRootTriangular<MatrixType> tmp(T);
+      MatrixType sqrtT = MatrixType::Zero(m_A.rows(), m_A.rows());
+      tmp.compute(sqrtT);
+    
+      // Compute square root of m_A
+      result = U * sqrtT * U.adjoint();
+    }
+    
+  private:
+    const MatrixType& m_A;
+};
+
+
 /** \ingroup MatrixFunctions_Module
   *
   * \brief Proxy for the matrix square root of some matrix (expression).