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https://gitlab.com/libeigen/eigen.git
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Split code for (quasi)triangular matrices from MatrixSquareRoot.
This way, (quasi)triangular matrices can avoid the costly Schur decomposition.
This commit is contained in:
Jitse Niesen
parent
8ddd1e390b
commit
c01ed935dd
@ -26,76 +26,71 @@
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#define EIGEN_MATRIX_SQUARE_ROOT
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/** \ingroup MatrixFunctions_Module
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* \brief Class for computing matrix square roots.
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* \tparam MatrixType type of the argument of the matrix square root,
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* expected to be an instantiation of the Matrix class template.
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* \brief Class for computing matrix square roots of upper quasi-triangular matrices.
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* \tparam MatrixType type of the argument of the matrix square root,
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* expected to be an instantiation of the Matrix class template.
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*
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* This class computes the square root of the upper quasi-triangular
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* matrix stored in the upper Hessenberg part of the matrix passed to
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* the constructor.
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*
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* \sa MatrixSquareRoot, MatrixSquareRootTriangular
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*/
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template <typename MatrixType, int IsComplex = NumTraits<typename internal::traits<MatrixType>::Scalar>::IsComplex>
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class MatrixSquareRoot
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{
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template <typename MatrixType>
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class MatrixSquareRootQuasiTriangular
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{
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public:
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/** \brief Constructor.
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*
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* \param[in] A matrix whose square root is to be computed.
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* \param[in] A upper quasi-triangular matrix whose square root
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* is to be computed.
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*
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* The class stores a reference to \p A, so it should not be
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* changed (or destroyed) before compute() is called.
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*/
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MatrixSquareRoot(const MatrixType& A);
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MatrixSquareRootQuasiTriangular(const MatrixType& A)
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: m_A(A)
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{
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eigen_assert(A.rows() == A.cols());
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}
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/** \brief Compute the matrix square root
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*
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* \param[out] result square root of \p A, as specified in the constructor.
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*
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* See MatrixBase::sqrt() for details on how this computation
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* is implemented.
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* Only the upper Hessenberg part of \p result is updated, the
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* rest is not touched. See MatrixBase::sqrt() for details on
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* how this computation is implemented.
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*/
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template <typename ResultType>
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void compute(ResultType &result);
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};
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// ********** Partial specialization for real matrices **********
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template <typename MatrixType>
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class MatrixSquareRoot<MatrixType, 0>
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{
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public:
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MatrixSquareRoot(const MatrixType& A)
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: m_A(A)
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{
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eigen_assert(A.rows() == A.cols());
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}
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template <typename ResultType> void compute(ResultType &result);
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private:
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typedef typename MatrixType::Index Index;
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typedef typename MatrixType::Scalar Scalar;
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void computeDiagonalPartOfSqrt(MatrixType& sqrtT, const MatrixType& T);
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void computeOffDiagonalPartOfSqrt(MatrixType& sqrtT, const MatrixType& T);
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void compute2x2diagonalBlock(MatrixType& sqrtT, const MatrixType& T, typename MatrixType::Index i);
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void compute1x1offDiagonalBlock(MatrixType& sqrtT, const MatrixType& T,
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typename MatrixType::Index i, typename MatrixType::Index j);
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void compute1x2offDiagonalBlock(MatrixType& sqrtT, const MatrixType& T,
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typename MatrixType::Index i, typename MatrixType::Index j);
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void compute2x1offDiagonalBlock(MatrixType& sqrtT, const MatrixType& T,
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typename MatrixType::Index i, typename MatrixType::Index j);
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void compute2x2offDiagonalBlock(MatrixType& sqrtT, const MatrixType& T,
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typename MatrixType::Index i, typename MatrixType::Index j);
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template <typename ResultType> void compute(ResultType &result);
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template <typename SmallMatrixType>
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static void solveAuxiliaryEquation(SmallMatrixType& X, const SmallMatrixType& A,
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const SmallMatrixType& B, const SmallMatrixType& C);
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private:
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typedef typename MatrixType::Index Index;
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typedef typename MatrixType::Scalar Scalar;
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void computeDiagonalPartOfSqrt(MatrixType& sqrtT, const MatrixType& T);
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void computeOffDiagonalPartOfSqrt(MatrixType& sqrtT, const MatrixType& T);
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void compute2x2diagonalBlock(MatrixType& sqrtT, const MatrixType& T, typename MatrixType::Index i);
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void compute1x1offDiagonalBlock(MatrixType& sqrtT, const MatrixType& T,
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typename MatrixType::Index i, typename MatrixType::Index j);
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void compute1x2offDiagonalBlock(MatrixType& sqrtT, const MatrixType& T,
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typename MatrixType::Index i, typename MatrixType::Index j);
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void compute2x1offDiagonalBlock(MatrixType& sqrtT, const MatrixType& T,
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typename MatrixType::Index i, typename MatrixType::Index j);
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void compute2x2offDiagonalBlock(MatrixType& sqrtT, const MatrixType& T,
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typename MatrixType::Index i, typename MatrixType::Index j);
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template <typename SmallMatrixType>
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static void solveAuxiliaryEquation(SmallMatrixType& X, const SmallMatrixType& A,
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const SmallMatrixType& B, const SmallMatrixType& C);
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const MatrixType& m_A;
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const MatrixType& m_A;
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};
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template <typename MatrixType>
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template <typename ResultType>
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void MatrixSquareRoot<MatrixType, 0>::compute(ResultType &result)
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void MatrixSquareRootQuasiTriangular<MatrixType>::compute(ResultType &result)
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{
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// Compute Schur decomposition of m_A
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const RealSchur<MatrixType> schurOfA(m_A);
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@ -114,7 +109,8 @@ void MatrixSquareRoot<MatrixType, 0>::compute(ResultType &result)
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// pre: T is quasi-upper-triangular and sqrtT is a zero matrix of the same size
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// post: the diagonal blocks of sqrtT are the square roots of the diagonal blocks of T
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template <typename MatrixType>
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void MatrixSquareRoot<MatrixType, 0>::computeDiagonalPartOfSqrt(MatrixType& sqrtT, const MatrixType& T)
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void MatrixSquareRootQuasiTriangular<MatrixType>::computeDiagonalPartOfSqrt(MatrixType& sqrtT,
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const MatrixType& T)
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{
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const Index size = m_A.rows();
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for (Index i = 0; i < size; i++) {
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@ -132,7 +128,8 @@ void MatrixSquareRoot<MatrixType, 0>::computeDiagonalPartOfSqrt(MatrixType& sqrt
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// pre: T is quasi-upper-triangular and diagonal blocks of sqrtT are square root of diagonal blocks of T.
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// post: sqrtT is the square root of T.
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template <typename MatrixType>
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void MatrixSquareRoot<MatrixType, 0>::computeOffDiagonalPartOfSqrt(MatrixType& sqrtT, const MatrixType& T)
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void MatrixSquareRootQuasiTriangular<MatrixType>::computeOffDiagonalPartOfSqrt(MatrixType& sqrtT,
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const MatrixType& T)
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{
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const Index size = m_A.rows();
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for (Index j = 1; j < size; j++) {
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@ -158,9 +155,8 @@ void MatrixSquareRoot<MatrixType, 0>::computeOffDiagonalPartOfSqrt(MatrixType& s
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// pre: T.block(i,i,2,2) has complex conjugate eigenvalues
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// post: sqrtT.block(i,i,2,2) is square root of T.block(i,i,2,2)
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template <typename MatrixType>
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void MatrixSquareRoot<MatrixType, 0>::compute2x2diagonalBlock(MatrixType& sqrtT,
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const MatrixType& T,
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typename MatrixType::Index i)
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void MatrixSquareRootQuasiTriangular<MatrixType>
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::compute2x2diagonalBlock(MatrixType& sqrtT, const MatrixType& T, typename MatrixType::Index i)
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{
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// TODO: This case (2-by-2 blocks with complex conjugate eigenvalues) is probably hidden somewhere
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// in EigenSolver. If we expose it, we could call it directly from here.
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@ -174,10 +170,9 @@ void MatrixSquareRoot<MatrixType, 0>::compute2x2diagonalBlock(MatrixType& sqrtT,
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// all blocks of sqrtT to left of and below (i,j) are correct
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// post: sqrtT(i,j) has the correct value
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template <typename MatrixType>
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void MatrixSquareRoot<MatrixType, 0>::compute1x1offDiagonalBlock(MatrixType& sqrtT,
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const MatrixType& T,
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typename MatrixType::Index i,
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typename MatrixType::Index j)
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void MatrixSquareRootQuasiTriangular<MatrixType>
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::compute1x1offDiagonalBlock(MatrixType& sqrtT, const MatrixType& T,
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typename MatrixType::Index i, typename MatrixType::Index j)
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{
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Scalar tmp = (sqrtT.row(i).segment(i+1,j-i-1) * sqrtT.col(j).segment(i+1,j-i-1)).value();
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sqrtT.coeffRef(i,j) = (T.coeff(i,j) - tmp) / (sqrtT.coeff(i,i) + sqrtT.coeff(j,j));
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@ -185,10 +180,9 @@ void MatrixSquareRoot<MatrixType, 0>::compute1x1offDiagonalBlock(MatrixType& sqr
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// similar to compute1x1offDiagonalBlock()
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template <typename MatrixType>
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void MatrixSquareRoot<MatrixType, 0>::compute1x2offDiagonalBlock(MatrixType& sqrtT,
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const MatrixType& T,
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typename MatrixType::Index i,
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typename MatrixType::Index j)
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void MatrixSquareRootQuasiTriangular<MatrixType>
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::compute1x2offDiagonalBlock(MatrixType& sqrtT, const MatrixType& T,
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typename MatrixType::Index i, typename MatrixType::Index j)
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{
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Matrix<Scalar,1,2> rhs = T.template block<1,2>(i,j);
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if (j-i > 1)
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@ -200,10 +194,9 @@ void MatrixSquareRoot<MatrixType, 0>::compute1x2offDiagonalBlock(MatrixType& sqr
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// similar to compute1x1offDiagonalBlock()
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template <typename MatrixType>
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void MatrixSquareRoot<MatrixType, 0>::compute2x1offDiagonalBlock(MatrixType& sqrtT,
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const MatrixType& T,
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typename MatrixType::Index i,
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typename MatrixType::Index j)
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void MatrixSquareRootQuasiTriangular<MatrixType>
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::compute2x1offDiagonalBlock(MatrixType& sqrtT, const MatrixType& T,
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typename MatrixType::Index i, typename MatrixType::Index j)
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{
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Matrix<Scalar,2,1> rhs = T.template block<2,1>(i,j);
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if (j-i > 2)
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@ -215,10 +208,9 @@ void MatrixSquareRoot<MatrixType, 0>::compute2x1offDiagonalBlock(MatrixType& sqr
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// similar to compute1x1offDiagonalBlock()
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template <typename MatrixType>
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void MatrixSquareRoot<MatrixType, 0>::compute2x2offDiagonalBlock(MatrixType& sqrtT,
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const MatrixType& T,
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typename MatrixType::Index i,
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typename MatrixType::Index j)
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void MatrixSquareRootQuasiTriangular<MatrixType>
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::compute2x2offDiagonalBlock(MatrixType& sqrtT, const MatrixType& T,
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typename MatrixType::Index i, typename MatrixType::Index j)
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{
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Matrix<Scalar,2,2> A = sqrtT.template block<2,2>(i,i);
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Matrix<Scalar,2,2> B = sqrtT.template block<2,2>(j,j);
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@ -233,10 +225,9 @@ void MatrixSquareRoot<MatrixType, 0>::compute2x2offDiagonalBlock(MatrixType& sqr
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// solves the equation A X + X B = C where all matrices are 2-by-2
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template <typename MatrixType>
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template <typename SmallMatrixType>
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void MatrixSquareRoot<MatrixType, 0>::solveAuxiliaryEquation(SmallMatrixType& X,
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const SmallMatrixType& A,
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const SmallMatrixType& B,
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const SmallMatrixType& C)
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void MatrixSquareRootQuasiTriangular<MatrixType>
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::solveAuxiliaryEquation(SmallMatrixType& X, const SmallMatrixType& A,
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const SmallMatrixType& B, const SmallMatrixType& C)
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{
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EIGEN_STATIC_ASSERT((internal::is_same<SmallMatrixType, Matrix<Scalar,2,2> >::value),
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EIGEN_INTERNAL_ERROR_PLEASE_FILE_A_BUG_REPORT);
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@ -270,18 +261,37 @@ void MatrixSquareRoot<MatrixType, 0>::solveAuxiliaryEquation(SmallMatrixType& X,
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X.coeffRef(1,1) = result.coeff(3);
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}
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// ********** Partial specialization for complex matrices **********
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/** \ingroup MatrixFunctions_Module
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* \brief Class for computing matrix square roots of upper triangular matrices.
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* \tparam MatrixType type of the argument of the matrix square root,
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* expected to be an instantiation of the Matrix class template.
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*
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* This class computes the square root of the upper triangular matrix
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* stored in the upper triangular part (including the diagonal) of
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* the matrix passed to the constructor.
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*
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* \sa MatrixSquareRoot, MatrixSquareRootQuasiTriangular
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*/
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template <typename MatrixType>
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class MatrixSquareRoot<MatrixType, 1>
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class MatrixSquareRootTriangular
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{
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public:
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MatrixSquareRoot(const MatrixType& A)
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MatrixSquareRootTriangular(const MatrixType& A)
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: m_A(A)
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{
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eigen_assert(A.rows() == A.cols());
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}
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/** \brief Compute the matrix square root
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*
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* \param[out] result square root of \p A, as specified in the constructor.
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*
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* Only the upper triangular part (including the diagonal) of
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* \p result is updated, the rest is not touched. See
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* MatrixBase::sqrt() for details on how this computation is
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* implemented.
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*/
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template <typename ResultType> void compute(ResultType &result);
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private:
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@ -290,7 +300,7 @@ class MatrixSquareRoot<MatrixType, 1>
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template <typename MatrixType>
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template <typename ResultType>
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void MatrixSquareRoot<MatrixType, 1>::compute(ResultType &result)
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void MatrixSquareRootTriangular<MatrixType>::compute(ResultType &result)
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{
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// Compute Schur decomposition of m_A
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const ComplexSchur<MatrixType> schurOfA(m_A);
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@ -320,6 +330,107 @@ void MatrixSquareRoot<MatrixType, 1>::compute(ResultType &result)
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result.noalias() = tmp * U.adjoint();
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}
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/** \ingroup MatrixFunctions_Module
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* \brief Class for computing matrix square roots of general matrices.
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* \tparam MatrixType type of the argument of the matrix square root,
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* expected to be an instantiation of the Matrix class template.
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*
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* \sa MatrixSquareRootTriangular, MatrixSquareRootQuasiTriangular, MatrixBase::sqrt()
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*/
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template <typename MatrixType, int IsComplex = NumTraits<typename internal::traits<MatrixType>::Scalar>::IsComplex>
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class MatrixSquareRoot
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{
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public:
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/** \brief Constructor.
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*
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* \param[in] A matrix whose square root is to be computed.
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*
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* The class stores a reference to \p A, so it should not be
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* changed (or destroyed) before compute() is called.
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*/
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MatrixSquareRoot(const MatrixType& A);
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/** \brief Compute the matrix square root
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*
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* \param[out] result square root of \p A, as specified in the constructor.
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*
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* See MatrixBase::sqrt() for details on how this computation is
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* implemented.
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*/
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template <typename ResultType> void compute(ResultType &result);
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};
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// ********** Partial specialization for real matrices **********
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template <typename MatrixType>
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class MatrixSquareRoot<MatrixType, 0>
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{
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public:
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MatrixSquareRoot(const MatrixType& A)
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: m_A(A)
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{
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eigen_assert(A.rows() == A.cols());
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}
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template <typename ResultType> void compute(ResultType &result)
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{
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// Compute Schur decomposition of m_A
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const RealSchur<MatrixType> schurOfA(m_A);
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const MatrixType& T = schurOfA.matrixT();
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const MatrixType& U = schurOfA.matrixU();
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// Compute square root of T
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MatrixSquareRootQuasiTriangular<MatrixType> tmp(T);
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MatrixType sqrtT = MatrixType::Zero(m_A.rows(), m_A.rows());
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tmp.compute(sqrtT);
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// Compute square root of m_A
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result = U * sqrtT * U.adjoint();
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}
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private:
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const MatrixType& m_A;
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};
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// ********** Partial specialization for complex matrices **********
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template <typename MatrixType>
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class MatrixSquareRoot<MatrixType, 1>
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{
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public:
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MatrixSquareRoot(const MatrixType& A)
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: m_A(A)
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{
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eigen_assert(A.rows() == A.cols());
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}
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template <typename ResultType> void compute(ResultType &result)
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{
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// Compute Schur decomposition of m_A
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const ComplexSchur<MatrixType> schurOfA(m_A);
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const MatrixType& T = schurOfA.matrixT();
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const MatrixType& U = schurOfA.matrixU();
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// Compute square root of T
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MatrixSquareRootTriangular<MatrixType> tmp(T);
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MatrixType sqrtT = MatrixType::Zero(m_A.rows(), m_A.rows());
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tmp.compute(sqrtT);
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// Compute square root of m_A
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result = U * sqrtT * U.adjoint();
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}
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private:
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const MatrixType& m_A;
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};
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/** \ingroup MatrixFunctions_Module
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*
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* \brief Proxy for the matrix square root of some matrix (expression).
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