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465 lines
16 KiB
C++
465 lines
16 KiB
C++
// This file is part of Eigen, a lightweight C++ template library
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// for linear algebra.
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//
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// Copyright (C) 2011 Jitse Niesen <jitse@maths.leeds.ac.uk>
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//
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// This Source Code Form is subject to the terms of the Mozilla
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// Public License v. 2.0. If a copy of the MPL was not distributed
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// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
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#ifndef EIGEN_MATRIX_SQUARE_ROOT
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#define EIGEN_MATRIX_SQUARE_ROOT
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namespace Eigen {
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/** \ingroup MatrixFunctions_Module
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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>
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class MatrixSquareRootQuasiTriangular : internal::noncopyable
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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 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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explicit 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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* 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> 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 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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};
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template <typename MatrixType>
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template <typename ResultType>
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void MatrixSquareRootQuasiTriangular<MatrixType>::compute(ResultType &result)
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{
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result.resize(m_A.rows(), m_A.cols());
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computeDiagonalPartOfSqrt(result, m_A);
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computeOffDiagonalPartOfSqrt(result, m_A);
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}
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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 MatrixSquareRootQuasiTriangular<MatrixType>::computeDiagonalPartOfSqrt(MatrixType& sqrtT,
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const MatrixType& T)
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{
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using std::sqrt;
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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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if (i == size - 1 || T.coeff(i+1, i) == 0) {
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eigen_assert(T(i,i) >= 0);
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sqrtT.coeffRef(i,i) = sqrt(T.coeff(i,i));
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}
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else {
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compute2x2diagonalBlock(sqrtT, T, i);
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++i;
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}
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}
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}
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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 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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if (T.coeff(j, j-1) != 0) // if T(j-1:j, j-1:j) is a 2-by-2 block
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continue;
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for (Index i = j-1; i >= 0; i--) {
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if (i > 0 && T.coeff(i, i-1) != 0) // if T(i-1:i, i-1:i) is a 2-by-2 block
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continue;
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bool iBlockIs2x2 = (i < size - 1) && (T.coeff(i+1, i) != 0);
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bool jBlockIs2x2 = (j < size - 1) && (T.coeff(j+1, j) != 0);
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if (iBlockIs2x2 && jBlockIs2x2)
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compute2x2offDiagonalBlock(sqrtT, T, i, j);
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else if (iBlockIs2x2 && !jBlockIs2x2)
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compute2x1offDiagonalBlock(sqrtT, T, i, j);
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else if (!iBlockIs2x2 && jBlockIs2x2)
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compute1x2offDiagonalBlock(sqrtT, T, i, j);
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else if (!iBlockIs2x2 && !jBlockIs2x2)
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compute1x1offDiagonalBlock(sqrtT, T, i, j);
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}
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}
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}
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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 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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Matrix<Scalar,2,2> block = T.template block<2,2>(i,i);
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EigenSolver<Matrix<Scalar,2,2> > es(block);
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sqrtT.template block<2,2>(i,i)
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= (es.eigenvectors() * es.eigenvalues().cwiseSqrt().asDiagonal() * es.eigenvectors().inverse()).real();
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}
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// pre: block structure of T is such that (i,j) is a 1x1 block,
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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 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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}
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// similar to compute1x1offDiagonalBlock()
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template <typename MatrixType>
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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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rhs -= sqrtT.block(i, i+1, 1, j-i-1) * sqrtT.block(i+1, j, j-i-1, 2);
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Matrix<Scalar,2,2> A = sqrtT.coeff(i,i) * Matrix<Scalar,2,2>::Identity();
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A += sqrtT.template block<2,2>(j,j).transpose();
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sqrtT.template block<1,2>(i,j).transpose() = A.fullPivLu().solve(rhs.transpose());
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}
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// similar to compute1x1offDiagonalBlock()
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template <typename MatrixType>
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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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rhs -= sqrtT.block(i, i+2, 2, j-i-2) * sqrtT.block(i+2, j, j-i-2, 1);
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Matrix<Scalar,2,2> A = sqrtT.coeff(j,j) * Matrix<Scalar,2,2>::Identity();
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A += sqrtT.template block<2,2>(i,i);
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sqrtT.template block<2,1>(i,j) = A.fullPivLu().solve(rhs);
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}
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// similar to compute1x1offDiagonalBlock()
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template <typename MatrixType>
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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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Matrix<Scalar,2,2> C = T.template block<2,2>(i,j);
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if (j-i > 2)
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C -= sqrtT.block(i, i+2, 2, j-i-2) * sqrtT.block(i+2, j, j-i-2, 2);
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Matrix<Scalar,2,2> X;
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solveAuxiliaryEquation(X, A, B, C);
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sqrtT.template block<2,2>(i,j) = X;
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}
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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 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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Matrix<Scalar,4,4> coeffMatrix = Matrix<Scalar,4,4>::Zero();
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coeffMatrix.coeffRef(0,0) = A.coeff(0,0) + B.coeff(0,0);
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coeffMatrix.coeffRef(1,1) = A.coeff(0,0) + B.coeff(1,1);
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coeffMatrix.coeffRef(2,2) = A.coeff(1,1) + B.coeff(0,0);
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coeffMatrix.coeffRef(3,3) = A.coeff(1,1) + B.coeff(1,1);
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coeffMatrix.coeffRef(0,1) = B.coeff(1,0);
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coeffMatrix.coeffRef(0,2) = A.coeff(0,1);
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coeffMatrix.coeffRef(1,0) = B.coeff(0,1);
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coeffMatrix.coeffRef(1,3) = A.coeff(0,1);
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coeffMatrix.coeffRef(2,0) = A.coeff(1,0);
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coeffMatrix.coeffRef(2,3) = B.coeff(1,0);
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coeffMatrix.coeffRef(3,1) = A.coeff(1,0);
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coeffMatrix.coeffRef(3,2) = B.coeff(0,1);
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Matrix<Scalar,4,1> rhs;
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rhs.coeffRef(0) = C.coeff(0,0);
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rhs.coeffRef(1) = C.coeff(0,1);
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rhs.coeffRef(2) = C.coeff(1,0);
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rhs.coeffRef(3) = C.coeff(1,1);
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Matrix<Scalar,4,1> result;
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result = coeffMatrix.fullPivLu().solve(rhs);
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X.coeffRef(0,0) = result.coeff(0);
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X.coeffRef(0,1) = result.coeff(1);
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X.coeffRef(1,0) = result.coeff(2);
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X.coeffRef(1,1) = result.coeff(3);
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}
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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 MatrixSquareRootTriangular : internal::noncopyable
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{
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public:
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explicit 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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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 MatrixSquareRootTriangular<MatrixType>::compute(ResultType &result)
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{
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using std::sqrt;
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// Compute square root of m_A and store it in upper triangular part of result
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// This uses that the square root of triangular matrices can be computed directly.
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result.resize(m_A.rows(), m_A.cols());
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typedef typename MatrixType::Index Index;
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for (Index i = 0; i < m_A.rows(); i++) {
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result.coeffRef(i,i) = sqrt(m_A.coeff(i,i));
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}
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for (Index j = 1; j < m_A.cols(); j++) {
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for (Index i = j-1; i >= 0; i--) {
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typedef typename MatrixType::Scalar Scalar;
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// if i = j-1, then segment has length 0 so tmp = 0
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Scalar tmp = (result.row(i).segment(i+1,j-i-1) * result.col(j).segment(i+1,j-i-1)).value();
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// denominator may be zero if original matrix is singular
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result.coeffRef(i,j) = (m_A.coeff(i,j) - tmp) / (result.coeff(i,i) + result.coeff(j,j));
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}
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}
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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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explicit 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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explicit 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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MatrixType sqrtT = MatrixType::Zero(m_A.rows(), m_A.cols());
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MatrixSquareRootQuasiTriangular<MatrixType>(T).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> : internal::noncopyable
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{
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public:
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explicit 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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MatrixType sqrtT;
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MatrixSquareRootTriangular<MatrixType>(T).compute(sqrtT);
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// Compute square root of m_A
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result = U * (sqrtT.template triangularView<Upper>() * 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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*
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* \tparam Derived Type of the argument to the matrix square root.
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*
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* This class holds the argument to the matrix square root until it
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* is assigned or evaluated for some other reason (so the argument
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* should not be changed in the meantime). It is the return type of
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* MatrixBase::sqrt() and most of the time this is the only way it is
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* used.
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*/
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template<typename Derived> class MatrixSquareRootReturnValue
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: public ReturnByValue<MatrixSquareRootReturnValue<Derived> >
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{
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typedef typename Derived::Index Index;
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public:
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/** \brief Constructor.
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*
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* \param[in] src %Matrix (expression) forming the argument of the
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* matrix square root.
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*/
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explicit MatrixSquareRootReturnValue(const Derived& src) : m_src(src) { }
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/** \brief Compute the matrix square root.
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*
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* \param[out] result the matrix square root of \p src in the
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* constructor.
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*/
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template <typename ResultType>
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inline void evalTo(ResultType& result) const
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{
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const typename Derived::PlainObject srcEvaluated = m_src.eval();
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MatrixSquareRoot<typename Derived::PlainObject> me(srcEvaluated);
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me.compute(result);
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}
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Index rows() const { return m_src.rows(); }
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Index cols() const { return m_src.cols(); }
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|
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protected:
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const Derived& m_src;
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|
};
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|
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namespace internal {
|
|
template<typename Derived>
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|
struct traits<MatrixSquareRootReturnValue<Derived> >
|
|
{
|
|
typedef typename Derived::PlainObject ReturnType;
|
|
};
|
|
}
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|
|
|
template <typename Derived>
|
|
const MatrixSquareRootReturnValue<Derived> MatrixBase<Derived>::sqrt() const
|
|
{
|
|
eigen_assert(rows() == cols());
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|
return MatrixSquareRootReturnValue<Derived>(derived());
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|
}
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|
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} // end namespace Eigen
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|
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#endif // EIGEN_MATRIX_FUNCTION
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