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253 lines
8.7 KiB
C++
253 lines
8.7 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) 2012 Chen-Pang He <jdh8@ms63.hinet.net>
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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_POWER_BASE
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#define EIGEN_MATRIX_POWER_BASE
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namespace Eigen {
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namespace internal {
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template<int IsComplex>
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struct recompose_complex_schur
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{
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template<typename ResultType, typename MatrixType>
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static inline void run(ResultType& res, const MatrixType& T, const MatrixType& U)
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{ res = U * (T.template triangularView<Upper>() * U.adjoint()); }
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};
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template<>
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struct recompose_complex_schur<0>
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{
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template<typename ResultType, typename MatrixType>
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static inline void run(ResultType& res, const MatrixType& T, const MatrixType& U)
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{ res = (U * (T.template triangularView<Upper>() * U.adjoint())).real(); }
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};
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template<typename Derived>
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struct traits<MatrixPowerProductBase<Derived> > : traits<Derived>
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{ };
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template<typename T>
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inline int binary_powering_cost(T p, int* squarings)
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{
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int applyings=0, tmp;
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if (frexp(p, squarings) != 0.5);
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--*squarings;
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while (std::frexp(p, &tmp), tmp > 0) {
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p -= std::ldexp(static_cast<T>(0.5), tmp);
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++applyings;
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}
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return applyings;
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}
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inline int matrix_power_get_pade_degree(float normIminusT)
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{
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const float maxNormForPade[] = { 2.8064004e-1f /* degree = 3 */ , 4.3386528e-1f };
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int degree = 3;
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for (; degree <= 4; ++degree)
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if (normIminusT <= maxNormForPade[degree - 3])
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break;
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return degree;
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}
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inline int matrix_power_get_pade_degree(double normIminusT)
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{
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const double maxNormForPade[] = { 1.884160592658218e-2 /* degree = 3 */ , 6.038881904059573e-2, 1.239917516308172e-1,
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1.999045567181744e-1, 2.789358995219730e-1 };
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int degree = 3;
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for (; degree <= 7; ++degree)
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if (normIminusT <= maxNormForPade[degree - 3])
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break;
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return degree;
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}
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inline int matrix_power_get_pade_degree(long double normIminusT)
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{
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#if LDBL_MANT_DIG == 53
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const int maxPadeDegree = 7;
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const double maxNormForPade[] = { 1.884160592658218e-2L /* degree = 3 */ , 6.038881904059573e-2L, 1.239917516308172e-1L,
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1.999045567181744e-1L, 2.789358995219730e-1L };
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#elif LDBL_MANT_DIG <= 64
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const int maxPadeDegree = 8;
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const double maxNormForPade[] = { 6.3854693117491799460e-3L /* degree = 3 */ , 2.6394893435456973676e-2L,
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6.4216043030404063729e-2L, 1.1701165502926694307e-1L, 1.7904284231268670284e-1L, 2.4471944416607995472e-1L };
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#elif LDBL_MANT_DIG <= 106
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const int maxPadeDegree = 10;
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const double maxNormForPade[] = { 1.0007161601787493236741409687186e-4L /* degree = 3 */ ,
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1.0007161601787493236741409687186e-3L, 4.7069769360887572939882574746264e-3L, 1.3220386624169159689406653101695e-2L,
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2.8063482381631737920612944054906e-2L, 4.9625993951953473052385361085058e-2L, 7.7367040706027886224557538328171e-2L,
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1.1016843812851143391275867258512e-1L };
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#else
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const int maxPadeDegree = 10;
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const double maxNormForPade[] = { 5.524506147036624377378713555116378e-5L /* degree = 3 */ ,
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6.640600568157479679823602193345995e-4L, 3.227716520106894279249709728084626e-3L,
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9.619593944683432960546978734646284e-3L, 2.134595382433742403911124458161147e-2L,
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3.908166513900489428442993794761185e-2L, 6.266780814639442865832535460550138e-2L,
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9.134603732914548552537150753385375e-2L };
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#endif
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int degree = 3;
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for (; degree <= maxPadeDegree; ++degree)
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if (normIminusT <= maxNormForPade[degree - 3])
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break;
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return degree;
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}
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} // namespace internal
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template<typename MatrixType, int UpLo = Upper> class MatrixPowerTriangularAtomic
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{
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private:
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typedef typename MatrixType::Scalar Scalar;
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typedef typename MatrixType::RealScalar RealScalar;
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typedef Array<Scalar,
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EIGEN_SIZE_MIN_PREFER_FIXED(MatrixType::RowsAtCompileTime,MatrixType::ColsAtCompileTime),
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1,ColMajor,
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EIGEN_SIZE_MIN_PREFER_FIXED(MatrixType::MaxRowsAtCompileTime,MatrixType::MaxColsAtCompileTime)> ArrayType;
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const MatrixType& m_T;
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void computePade(int degree, const MatrixType& IminusT, MatrixType& res, RealScalar p) const;
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void compute2x2(MatrixType& res, RealScalar p) const;
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void computeBig(MatrixType& res, RealScalar p) const;
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public:
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explicit MatrixPowerTriangularAtomic(const MatrixType& T);
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void compute(MatrixType& res, RealScalar p) const;
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};
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template<typename MatrixType, int UpLo>
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MatrixPowerTriangularAtomic<MatrixType,UpLo>::MatrixPowerTriangularAtomic(const MatrixType& T) :
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m_T(T)
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{ eigen_assert(T.rows() == T.cols()); }
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template<typename MatrixType, int UpLo>
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void MatrixPowerTriangularAtomic<MatrixType,UpLo>::compute(MatrixType& res, RealScalar p) const
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{
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switch (m_T.rows()) {
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case 0:
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break;
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case 1:
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res(0,0) = std::pow(m_T(0,0), p);
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break;
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case 2:
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compute2x2(res, p);
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break;
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default:
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computeBig(res, p);
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}
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}
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template<typename MatrixType, int UpLo>
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void MatrixPowerTriangularAtomic<MatrixType,UpLo>::computePade(int degree, const MatrixType& IminusT, MatrixType& res,
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RealScalar p) const
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{
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int i = degree<<1;
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res = (p-degree) / ((i-1)<<1) * IminusT;
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for (--i; i; --i) {
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res = (MatrixType::Identity(m_T.rows(), m_T.cols()) + res).template triangularView<UpLo>()
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.solve((i==1 ? -p : i&1 ? (-p-(i>>1))/(i<<1) : (p-(i>>1))/((i-1)<<1)) * IminusT).eval();
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}
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res += MatrixType::Identity(m_T.rows(), m_T.cols());
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}
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template<typename MatrixType, int UpLo>
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void MatrixPowerTriangularAtomic<MatrixType,UpLo>::compute2x2(MatrixType& res, RealScalar p) const
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{
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using std::abs;
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using std::pow;
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ArrayType logTdiag = m_T.diagonal().array().log();
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res(0,0) = pow(m_T(0,0), p);
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for (int i=1; i < m_T.cols(); ++i) {
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res(i,i) = pow(m_T(i,i), p);
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if (m_T(i-1,i-1) == m_T(i,i)) {
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res(i-1,i) = p * pow(m_T(i-1,i), p-1);
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}
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else if (2*abs(m_T(i-1,i-1)) < abs(m_T(i,i)) || 2*abs(m_T(i,i)) < abs(m_T(i-1,i-1))) {
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res(i-1,i) = m_T(i-1,i) * (res(i,i)-res(i-1,i-1)) / (m_T(i,i)-m_T(i-1,i-1));
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}
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else {
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// computation in previous branch is inaccurate if abs(m_T(i,i)) \approx abs(m_T(i-1,i-1))
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int unwindingNumber = std::ceil(((logTdiag[i]-logTdiag[i-1]).imag() - M_PI) / (2*M_PI));
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Scalar w = internal::atanh2(m_T(i,i)-m_T(i-1,i-1), m_T(i,i)+m_T(i-1,i-1)) + Scalar(0, M_PI*unwindingNumber);
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res(i-1,i) = m_T(i-1,i) * RealScalar(2) * std::exp(RealScalar(0.5) * p * (logTdiag[i]+logTdiag[i-1])) *
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std::sinh(p * w) / (m_T(i,i) - m_T(i-1,i-1));
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}
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}
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}
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template<typename MatrixType, int UpLo>
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void MatrixPowerTriangularAtomic<MatrixType,UpLo>::computeBig(MatrixType& res, RealScalar p) const
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{
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const int digits = std::numeric_limits<RealScalar>::digits;
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const RealScalar maxNormForPade = digits <= 24? 4.3386528e-1f: // sigle precision
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digits <= 53? 2.789358995219730e-1: // double precision
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digits <= 64? 2.4471944416607995472e-1L: // extended precision
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digits <= 106? 1.1016843812851143391275867258512e-01: // double-double
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9.134603732914548552537150753385375e-02; // quadruple precision
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MatrixType IminusT, sqrtT, T=m_T;
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RealScalar normIminusT;
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int degree, degree2, numberOfSquareRoots=0, numberOfExtraSquareRoots=0;
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while (true) {
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IminusT = MatrixType::Identity(m_T.rows(), m_T.cols()) - T;
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normIminusT = IminusT.cwiseAbs().colwise().sum().maxCoeff();
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if (normIminusT < maxNormForPade) {
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degree = internal::matrix_power_get_pade_degree(normIminusT);
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degree2 = internal::matrix_power_get_pade_degree(normIminusT/2);
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if (degree - degree2 <= 1 || numberOfExtraSquareRoots)
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break;
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++numberOfExtraSquareRoots;
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}
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MatrixSquareRootTriangular<MatrixType>(T).compute(sqrtT);
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T = sqrtT;
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++numberOfSquareRoots;
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}
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computePade(degree, IminusT, res, p);
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for (; numberOfSquareRoots; --numberOfSquareRoots) {
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compute2x2(res, std::ldexp(p,-numberOfSquareRoots));
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res *= res;
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}
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compute2x2(res, p);
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}
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template<typename Derived>
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class MatrixPowerProductBase : public MatrixBase<Derived>
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{
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public:
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typedef MatrixBase<Derived> Base;
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typedef typename Base::PlainObject PlainObject;
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EIGEN_DENSE_PUBLIC_INTERFACE(MatrixPowerProductBase)
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inline Index rows() const { return derived().rows(); }
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inline Index cols() const { return derived().cols(); }
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template<typename ResultType>
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inline void evalTo(ResultType& res) const { derived().evalTo(res); }
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const PlainObject& eval() const
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{
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m_result.resize(rows(), cols());
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derived().evalTo(m_result);
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return m_result;
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}
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operator const PlainObject&() const
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{ return eval(); }
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protected:
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mutable PlainObject m_result;
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};
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} // namespace Eigen
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#endif // EIGEN_MATRIX_POWER
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