17faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez// This file is part of Eigen, a lightweight C++ template library 27faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez// for linear algebra. 37faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez// 47faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez// Copyright (C) 2012, 2013 Chen-Pang He <jdh8@ms63.hinet.net> 57faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez// 67faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez// This Source Code Form is subject to the terms of the Mozilla 77faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez// Public License v. 2.0. If a copy of the MPL was not distributed 87faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez// with this file, You can obtain one at http://mozilla.org/MPL/2.0/. 97faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 107faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez#ifndef EIGEN_MATRIX_POWER 117faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez#define EIGEN_MATRIX_POWER 127faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 137faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeznamespace Eigen { 147faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 157faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename MatrixType> class MatrixPower; 167faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 177faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename MatrixType> 187faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandezclass MatrixPowerRetval : public ReturnByValue< MatrixPowerRetval<MatrixType> > 197faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez{ 207faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez public: 217faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez typedef typename MatrixType::RealScalar RealScalar; 227faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez typedef typename MatrixType::Index Index; 237faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 247faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez MatrixPowerRetval(MatrixPower<MatrixType>& pow, RealScalar p) : m_pow(pow), m_p(p) 257faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez { } 267faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 277faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez template<typename ResultType> 287faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez inline void evalTo(ResultType& res) const 297faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez { m_pow.compute(res, m_p); } 307faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 317faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez Index rows() const { return m_pow.rows(); } 327faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez Index cols() const { return m_pow.cols(); } 337faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 347faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez private: 357faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez MatrixPower<MatrixType>& m_pow; 367faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const RealScalar m_p; 377faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez MatrixPowerRetval& operator=(const MatrixPowerRetval&); 387faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez}; 397faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 407faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename MatrixType> 417faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandezclass MatrixPowerAtomic 427faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez{ 437faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez private: 447faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez enum { 457faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez RowsAtCompileTime = MatrixType::RowsAtCompileTime, 467faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime 477faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez }; 487faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez typedef typename MatrixType::Scalar Scalar; 497faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez typedef typename MatrixType::RealScalar RealScalar; 507faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez typedef std::complex<RealScalar> ComplexScalar; 517faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez typedef typename MatrixType::Index Index; 527faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez typedef Array<Scalar, RowsAtCompileTime, 1, ColMajor, MaxRowsAtCompileTime> ArrayType; 537faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 547faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const MatrixType& m_A; 557faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez RealScalar m_p; 567faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 577faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez void computePade(int degree, const MatrixType& IminusT, MatrixType& res) const; 587faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez void compute2x2(MatrixType& res, RealScalar p) const; 597faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez void computeBig(MatrixType& res) const; 607faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez static int getPadeDegree(float normIminusT); 617faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez static int getPadeDegree(double normIminusT); 627faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez static int getPadeDegree(long double normIminusT); 637faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez static ComplexScalar computeSuperDiag(const ComplexScalar&, const ComplexScalar&, RealScalar p); 647faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez static RealScalar computeSuperDiag(RealScalar, RealScalar, RealScalar p); 657faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 667faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez public: 677faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez MatrixPowerAtomic(const MatrixType& T, RealScalar p); 687faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez void compute(MatrixType& res) const; 697faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez}; 707faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 717faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename MatrixType> 727faaa9f3f0df9d23790277834d426c3d992ac3baCarlos HernandezMatrixPowerAtomic<MatrixType>::MatrixPowerAtomic(const MatrixType& T, RealScalar p) : 737faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez m_A(T), m_p(p) 747faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez{ eigen_assert(T.rows() == T.cols()); } 757faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 767faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename MatrixType> 777faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandezvoid MatrixPowerAtomic<MatrixType>::compute(MatrixType& res) const 787faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez{ 797faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez res.resizeLike(m_A); 807faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez switch (m_A.rows()) { 817faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez case 0: 827faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez break; 837faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez case 1: 847faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez res(0,0) = std::pow(m_A(0,0), m_p); 857faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez break; 867faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez case 2: 877faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez compute2x2(res, m_p); 887faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez break; 897faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez default: 907faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez computeBig(res); 917faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez } 927faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez} 937faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 947faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename MatrixType> 957faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandezvoid MatrixPowerAtomic<MatrixType>::computePade(int degree, const MatrixType& IminusT, MatrixType& res) const 967faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez{ 977faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez int i = degree<<1; 987faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez res = (m_p-degree) / ((i-1)<<1) * IminusT; 997faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez for (--i; i; --i) { 1007faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez res = (MatrixType::Identity(IminusT.rows(), IminusT.cols()) + res).template triangularView<Upper>() 1017faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez .solve((i==1 ? -m_p : i&1 ? (-m_p-(i>>1))/(i<<1) : (m_p-(i>>1))/((i-1)<<1)) * IminusT).eval(); 1027faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez } 1037faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez res += MatrixType::Identity(IminusT.rows(), IminusT.cols()); 1047faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez} 1057faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 1067faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez// This function assumes that res has the correct size (see bug 614) 1077faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename MatrixType> 1087faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandezvoid MatrixPowerAtomic<MatrixType>::compute2x2(MatrixType& res, RealScalar p) const 1097faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez{ 1107faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez using std::abs; 1117faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez using std::pow; 1127faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 1137faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez ArrayType logTdiag = m_A.diagonal().array().log(); 1147faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez res.coeffRef(0,0) = pow(m_A.coeff(0,0), p); 1157faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 1167faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez for (Index i=1; i < m_A.cols(); ++i) { 1177faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez res.coeffRef(i,i) = pow(m_A.coeff(i,i), p); 1187faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez if (m_A.coeff(i-1,i-1) == m_A.coeff(i,i)) 1197faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez res.coeffRef(i-1,i) = p * pow(m_A.coeff(i,i), p-1); 1207faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez else if (2*abs(m_A.coeff(i-1,i-1)) < abs(m_A.coeff(i,i)) || 2*abs(m_A.coeff(i,i)) < abs(m_A.coeff(i-1,i-1))) 1217faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez res.coeffRef(i-1,i) = (res.coeff(i,i)-res.coeff(i-1,i-1)) / (m_A.coeff(i,i)-m_A.coeff(i-1,i-1)); 1227faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez else 1237faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez res.coeffRef(i-1,i) = computeSuperDiag(m_A.coeff(i,i), m_A.coeff(i-1,i-1), p); 1247faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez res.coeffRef(i-1,i) *= m_A.coeff(i-1,i); 1257faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez } 1267faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez} 1277faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 1287faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename MatrixType> 1297faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandezvoid MatrixPowerAtomic<MatrixType>::computeBig(MatrixType& res) const 1307faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez{ 1317faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const int digits = std::numeric_limits<RealScalar>::digits; 1327faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const RealScalar maxNormForPade = digits <= 24? 4.3386528e-1f: // sigle precision 1337faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez digits <= 53? 2.789358995219730e-1: // double precision 1347faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez digits <= 64? 2.4471944416607995472e-1L: // extended precision 1357faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez digits <= 106? 1.1016843812851143391275867258512e-1L: // double-double 1367faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 9.134603732914548552537150753385375e-2L; // quadruple precision 1377faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez MatrixType IminusT, sqrtT, T = m_A.template triangularView<Upper>(); 1387faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez RealScalar normIminusT; 1397faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez int degree, degree2, numberOfSquareRoots = 0; 1407faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez bool hasExtraSquareRoot = false; 1417faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 1427faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez /* FIXME 1437faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * For singular T, norm(I - T) >= 1 but maxNormForPade < 1, leads to infinite 1447faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * loop. We should move 0 eigenvalues to bottom right corner. We need not 1457faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * worry about tiny values (e.g. 1e-300) because they will reach 1 if 1467faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * repetitively sqrt'ed. 1477faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * 1487faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * If the 0 eigenvalues are semisimple, they can form a 0 matrix at the 1497faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * bottom right corner. 1507faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * 1517faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * [ T A ]^p [ T^p (T^-1 T^p A) ] 1527faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * [ ] = [ ] 1537faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * [ 0 0 ] [ 0 0 ] 1547faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez */ 1557faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez for (Index i=0; i < m_A.cols(); ++i) 1567faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez eigen_assert(m_A(i,i) != RealScalar(0)); 1577faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 1587faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez while (true) { 1597faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez IminusT = MatrixType::Identity(m_A.rows(), m_A.cols()) - T; 1607faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez normIminusT = IminusT.cwiseAbs().colwise().sum().maxCoeff(); 1617faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez if (normIminusT < maxNormForPade) { 1627faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez degree = getPadeDegree(normIminusT); 1637faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez degree2 = getPadeDegree(normIminusT/2); 1647faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez if (degree - degree2 <= 1 || hasExtraSquareRoot) 1657faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez break; 1667faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez hasExtraSquareRoot = true; 1677faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez } 1687faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez MatrixSquareRootTriangular<MatrixType>(T).compute(sqrtT); 1697faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez T = sqrtT.template triangularView<Upper>(); 1707faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez ++numberOfSquareRoots; 1717faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez } 1727faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez computePade(degree, IminusT, res); 1737faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 1747faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez for (; numberOfSquareRoots; --numberOfSquareRoots) { 1757faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez compute2x2(res, std::ldexp(m_p, -numberOfSquareRoots)); 1767faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez res = res.template triangularView<Upper>() * res; 1777faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez } 1787faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez compute2x2(res, m_p); 1797faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez} 1807faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 1817faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename MatrixType> 1827faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandezinline int MatrixPowerAtomic<MatrixType>::getPadeDegree(float normIminusT) 1837faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez{ 1847faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const float maxNormForPade[] = { 2.8064004e-1f /* degree = 3 */ , 4.3386528e-1f }; 1857faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez int degree = 3; 1867faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez for (; degree <= 4; ++degree) 1877faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez if (normIminusT <= maxNormForPade[degree - 3]) 1887faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez break; 1897faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez return degree; 1907faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez} 1917faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 1927faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename MatrixType> 1937faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandezinline int MatrixPowerAtomic<MatrixType>::getPadeDegree(double normIminusT) 1947faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez{ 1957faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const double maxNormForPade[] = { 1.884160592658218e-2 /* degree = 3 */ , 6.038881904059573e-2, 1.239917516308172e-1, 1967faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 1.999045567181744e-1, 2.789358995219730e-1 }; 1977faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez int degree = 3; 1987faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez for (; degree <= 7; ++degree) 1997faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez if (normIminusT <= maxNormForPade[degree - 3]) 2007faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez break; 2017faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez return degree; 2027faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez} 2037faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 2047faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename MatrixType> 2057faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandezinline int MatrixPowerAtomic<MatrixType>::getPadeDegree(long double normIminusT) 2067faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez{ 2077faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez#if LDBL_MANT_DIG == 53 2087faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const int maxPadeDegree = 7; 2097faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const double maxNormForPade[] = { 1.884160592658218e-2L /* degree = 3 */ , 6.038881904059573e-2L, 1.239917516308172e-1L, 2107faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 1.999045567181744e-1L, 2.789358995219730e-1L }; 2117faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez#elif LDBL_MANT_DIG <= 64 2127faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const int maxPadeDegree = 8; 2137faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const double maxNormForPade[] = { 6.3854693117491799460e-3L /* degree = 3 */ , 2.6394893435456973676e-2L, 2147faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 6.4216043030404063729e-2L, 1.1701165502926694307e-1L, 1.7904284231268670284e-1L, 2.4471944416607995472e-1L }; 2157faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez#elif LDBL_MANT_DIG <= 106 2167faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const int maxPadeDegree = 10; 2177faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const double maxNormForPade[] = { 1.0007161601787493236741409687186e-4L /* degree = 3 */ , 2187faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 1.0007161601787493236741409687186e-3L, 4.7069769360887572939882574746264e-3L, 1.3220386624169159689406653101695e-2L, 2197faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 2.8063482381631737920612944054906e-2L, 4.9625993951953473052385361085058e-2L, 7.7367040706027886224557538328171e-2L, 2207faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 1.1016843812851143391275867258512e-1L }; 2217faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez#else 2227faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const int maxPadeDegree = 10; 2237faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const double maxNormForPade[] = { 5.524506147036624377378713555116378e-5L /* degree = 3 */ , 2247faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 6.640600568157479679823602193345995e-4L, 3.227716520106894279249709728084626e-3L, 2257faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 9.619593944683432960546978734646284e-3L, 2.134595382433742403911124458161147e-2L, 2267faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 3.908166513900489428442993794761185e-2L, 6.266780814639442865832535460550138e-2L, 2277faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 9.134603732914548552537150753385375e-2L }; 2287faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez#endif 2297faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez int degree = 3; 2307faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez for (; degree <= maxPadeDegree; ++degree) 2317faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez if (normIminusT <= maxNormForPade[degree - 3]) 2327faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez break; 2337faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez return degree; 2347faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez} 2357faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 2367faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename MatrixType> 2377faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandezinline typename MatrixPowerAtomic<MatrixType>::ComplexScalar 2387faaa9f3f0df9d23790277834d426c3d992ac3baCarlos HernandezMatrixPowerAtomic<MatrixType>::computeSuperDiag(const ComplexScalar& curr, const ComplexScalar& prev, RealScalar p) 2397faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez{ 2407faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez ComplexScalar logCurr = std::log(curr); 2417faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez ComplexScalar logPrev = std::log(prev); 2427faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez int unwindingNumber = std::ceil((numext::imag(logCurr - logPrev) - M_PI) / (2*M_PI)); 2437faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez ComplexScalar w = numext::atanh2(curr - prev, curr + prev) + ComplexScalar(0, M_PI*unwindingNumber); 2447faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez return RealScalar(2) * std::exp(RealScalar(0.5) * p * (logCurr + logPrev)) * std::sinh(p * w) / (curr - prev); 2457faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez} 2467faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 2477faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename MatrixType> 2487faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandezinline typename MatrixPowerAtomic<MatrixType>::RealScalar 2497faaa9f3f0df9d23790277834d426c3d992ac3baCarlos HernandezMatrixPowerAtomic<MatrixType>::computeSuperDiag(RealScalar curr, RealScalar prev, RealScalar p) 2507faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez{ 2517faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez RealScalar w = numext::atanh2(curr - prev, curr + prev); 2527faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez return 2 * std::exp(p * (std::log(curr) + std::log(prev)) / 2) * std::sinh(p * w) / (curr - prev); 2537faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez} 2547faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 2557faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez/** 2567faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * \ingroup MatrixFunctions_Module 2577faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * 2587faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * \brief Class for computing matrix powers. 2597faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * 2607faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * \tparam MatrixType type of the base, expected to be an instantiation 2617faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * of the Matrix class template. 2627faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * 2637faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * This class is capable of computing real/complex matrices raised to 2647faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * an arbitrary real power. Meanwhile, it saves the result of Schur 2657faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * decomposition if an non-integral power has even been calculated. 2667faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * Therefore, if you want to compute multiple (>= 2) matrix powers 2677faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * for the same matrix, using the class directly is more efficient than 2687faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * calling MatrixBase::pow(). 2697faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * 2707faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * Example: 2717faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * \include MatrixPower_optimal.cpp 2727faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * Output: \verbinclude MatrixPower_optimal.out 2737faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez */ 2747faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename MatrixType> 2757faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandezclass MatrixPower 2767faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez{ 2777faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez private: 2787faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez enum { 2797faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez RowsAtCompileTime = MatrixType::RowsAtCompileTime, 2807faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez ColsAtCompileTime = MatrixType::ColsAtCompileTime, 2817faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime, 2827faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime 2837faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez }; 2847faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez typedef typename MatrixType::Scalar Scalar; 2857faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez typedef typename MatrixType::RealScalar RealScalar; 2867faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez typedef typename MatrixType::Index Index; 2877faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 2887faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez public: 2897faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez /** 2907faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * \brief Constructor. 2917faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * 2927faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * \param[in] A the base of the matrix power. 2937faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * 2947faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * The class stores a reference to A, so it should not be changed 2957faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * (or destroyed) before evaluation. 2967faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez */ 2977faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez explicit MatrixPower(const MatrixType& A) : m_A(A), m_conditionNumber(0) 2987faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez { eigen_assert(A.rows() == A.cols()); } 2997faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 3007faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez /** 3017faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * \brief Returns the matrix power. 3027faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * 3037faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * \param[in] p exponent, a real scalar. 3047faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * \return The expression \f$ A^p \f$, where A is specified in the 3057faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * constructor. 3067faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez */ 3077faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const MatrixPowerRetval<MatrixType> operator()(RealScalar p) 3087faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez { return MatrixPowerRetval<MatrixType>(*this, p); } 3097faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 3107faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez /** 3117faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * \brief Compute the matrix power. 3127faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * 3137faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * \param[in] p exponent, a real scalar. 3147faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * \param[out] res \f$ A^p \f$ where A is specified in the 3157faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * constructor. 3167faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez */ 3177faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez template<typename ResultType> 3187faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez void compute(ResultType& res, RealScalar p); 3197faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 3207faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez Index rows() const { return m_A.rows(); } 3217faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez Index cols() const { return m_A.cols(); } 3227faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 3237faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez private: 3247faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez typedef std::complex<RealScalar> ComplexScalar; 3257faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, MatrixType::Options, 3267faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez MaxRowsAtCompileTime, MaxColsAtCompileTime> ComplexMatrix; 3277faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 3287faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez typename MatrixType::Nested m_A; 3297faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez MatrixType m_tmp; 3307faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez ComplexMatrix m_T, m_U, m_fT; 3317faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez RealScalar m_conditionNumber; 3327faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 3337faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez RealScalar modfAndInit(RealScalar, RealScalar*); 3347faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 3357faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez template<typename ResultType> 3367faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez void computeIntPower(ResultType&, RealScalar); 3377faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 3387faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez template<typename ResultType> 3397faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez void computeFracPower(ResultType&, RealScalar); 3407faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 3417faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez template<int Rows, int Cols, int Options, int MaxRows, int MaxCols> 3427faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez static void revertSchur( 3437faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez Matrix<ComplexScalar, Rows, Cols, Options, MaxRows, MaxCols>& res, 3447faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const ComplexMatrix& T, 3457faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const ComplexMatrix& U); 3467faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 3477faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez template<int Rows, int Cols, int Options, int MaxRows, int MaxCols> 3487faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez static void revertSchur( 3497faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez Matrix<RealScalar, Rows, Cols, Options, MaxRows, MaxCols>& res, 3507faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const ComplexMatrix& T, 3517faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const ComplexMatrix& U); 3527faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez}; 3537faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 3547faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename MatrixType> 3557faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename ResultType> 3567faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandezvoid MatrixPower<MatrixType>::compute(ResultType& res, RealScalar p) 3577faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez{ 3587faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez switch (cols()) { 3597faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez case 0: 3607faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez break; 3617faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez case 1: 3627faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez res(0,0) = std::pow(m_A.coeff(0,0), p); 3637faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez break; 3647faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez default: 3657faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez RealScalar intpart, x = modfAndInit(p, &intpart); 3667faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez computeIntPower(res, intpart); 3677faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez computeFracPower(res, x); 3687faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez } 3697faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez} 3707faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 3717faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename MatrixType> 3727faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztypename MatrixPower<MatrixType>::RealScalar 3737faaa9f3f0df9d23790277834d426c3d992ac3baCarlos HernandezMatrixPower<MatrixType>::modfAndInit(RealScalar x, RealScalar* intpart) 3747faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez{ 3757faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez typedef Array<RealScalar, RowsAtCompileTime, 1, ColMajor, MaxRowsAtCompileTime> RealArray; 3767faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 3777faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez *intpart = std::floor(x); 3787faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez RealScalar res = x - *intpart; 3797faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 3807faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez if (!m_conditionNumber && res) { 3817faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const ComplexSchur<MatrixType> schurOfA(m_A); 3827faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez m_T = schurOfA.matrixT(); 3837faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez m_U = schurOfA.matrixU(); 3847faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 3857faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const RealArray absTdiag = m_T.diagonal().array().abs(); 3867faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez m_conditionNumber = absTdiag.maxCoeff() / absTdiag.minCoeff(); 3877faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez } 3887faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 3897faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez if (res>RealScalar(0.5) && res>(1-res)*std::pow(m_conditionNumber, res)) { 3907faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez --res; 3917faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez ++*intpart; 3927faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez } 3937faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez return res; 3947faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez} 3957faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 3967faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename MatrixType> 3977faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename ResultType> 3987faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandezvoid MatrixPower<MatrixType>::computeIntPower(ResultType& res, RealScalar p) 3997faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez{ 4007faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez RealScalar pp = std::abs(p); 4017faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 4027faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez if (p<0) m_tmp = m_A.inverse(); 4037faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez else m_tmp = m_A; 4047faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 4057faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez res = MatrixType::Identity(rows(), cols()); 4067faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez while (pp >= 1) { 4077faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez if (std::fmod(pp, 2) >= 1) 4087faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez res = m_tmp * res; 4097faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez m_tmp *= m_tmp; 4107faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez pp /= 2; 4117faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez } 4127faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez} 4137faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 4147faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename MatrixType> 4157faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename ResultType> 4167faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandezvoid MatrixPower<MatrixType>::computeFracPower(ResultType& res, RealScalar p) 4177faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez{ 4187faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez if (p) { 4197faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez eigen_assert(m_conditionNumber); 4207faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez MatrixPowerAtomic<ComplexMatrix>(m_T, p).compute(m_fT); 4217faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez revertSchur(m_tmp, m_fT, m_U); 4227faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez res = m_tmp * res; 4237faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez } 4247faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez} 4257faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 4267faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename MatrixType> 4277faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<int Rows, int Cols, int Options, int MaxRows, int MaxCols> 4287faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandezinline void MatrixPower<MatrixType>::revertSchur( 4297faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez Matrix<ComplexScalar, Rows, Cols, Options, MaxRows, MaxCols>& res, 4307faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const ComplexMatrix& T, 4317faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const ComplexMatrix& U) 4327faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez{ res.noalias() = U * (T.template triangularView<Upper>() * U.adjoint()); } 4337faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 4347faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename MatrixType> 4357faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<int Rows, int Cols, int Options, int MaxRows, int MaxCols> 4367faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandezinline void MatrixPower<MatrixType>::revertSchur( 4377faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez Matrix<RealScalar, Rows, Cols, Options, MaxRows, MaxCols>& res, 4387faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const ComplexMatrix& T, 4397faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const ComplexMatrix& U) 4407faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez{ res.noalias() = (U * (T.template triangularView<Upper>() * U.adjoint())).real(); } 4417faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 4427faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez/** 4437faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * \ingroup MatrixFunctions_Module 4447faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * 4457faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * \brief Proxy for the matrix power of some matrix (expression). 4467faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * 4477faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * \tparam Derived type of the base, a matrix (expression). 4487faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * 4497faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * This class holds the arguments to the matrix power until it is 4507faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * assigned or evaluated for some other reason (so the argument 4517faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * should not be changed in the meantime). It is the return type of 4527faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * MatrixBase::pow() and related functions and most of the 4537faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * time this is the only way it is used. 4547faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez */ 4557faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename Derived> 4567faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandezclass MatrixPowerReturnValue : public ReturnByValue< MatrixPowerReturnValue<Derived> > 4577faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez{ 4587faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez public: 4597faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez typedef typename Derived::PlainObject PlainObject; 4607faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez typedef typename Derived::RealScalar RealScalar; 4617faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez typedef typename Derived::Index Index; 4627faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 4637faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez /** 4647faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * \brief Constructor. 4657faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * 4667faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * \param[in] A %Matrix (expression), the base of the matrix power. 4677faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * \param[in] p scalar, the exponent of the matrix power. 4687faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez */ 4697faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez MatrixPowerReturnValue(const Derived& A, RealScalar p) : m_A(A), m_p(p) 4707faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez { } 4717faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 4727faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez /** 4737faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * \brief Compute the matrix power. 4747faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * 4757faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * \param[out] result \f$ A^p \f$ where \p A and \p p are as in the 4767faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez * constructor. 4777faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez */ 4787faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez template<typename ResultType> 4797faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez inline void evalTo(ResultType& res) const 4807faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez { MatrixPower<PlainObject>(m_A.eval()).compute(res, m_p); } 4817faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 4827faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez Index rows() const { return m_A.rows(); } 4837faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez Index cols() const { return m_A.cols(); } 4847faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 4857faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez private: 4867faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const Derived& m_A; 4877faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez const RealScalar m_p; 4887faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez MatrixPowerReturnValue& operator=(const MatrixPowerReturnValue&); 4897faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez}; 4907faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 4917faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeznamespace internal { 4927faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 4937faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename MatrixPowerType> 4947faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandezstruct traits< MatrixPowerRetval<MatrixPowerType> > 4957faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez{ typedef typename MatrixPowerType::PlainObject ReturnType; }; 4967faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 4977faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename Derived> 4987faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandezstruct traits< MatrixPowerReturnValue<Derived> > 4997faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez{ typedef typename Derived::PlainObject ReturnType; }; 5007faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 5017faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez} 5027faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 5037faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandeztemplate<typename Derived> 5047faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandezconst MatrixPowerReturnValue<Derived> MatrixBase<Derived>::pow(const RealScalar& p) const 5057faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez{ return MatrixPowerReturnValue<Derived>(derived(), p); } 5067faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 5077faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez} // namespace Eigen 5087faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez 5097faaa9f3f0df9d23790277834d426c3d992ac3baCarlos Hernandez#endif // EIGEN_MATRIX_POWER 510