[136] | 1 | // This file is part of Eigen, a lightweight C++ template library
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| 2 | // for linear algebra.
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| 3 | //
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| 4 | // Copyright (C) 2012, 2013 Chen-Pang He <jdh8@ms63.hinet.net>
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| 5 | //
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| 6 | // This Source Code Form is subject to the terms of the Mozilla
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| 7 | // Public License v. 2.0. If a copy of the MPL was not distributed
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| 8 | // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
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| 9 |
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| 10 | #ifndef EIGEN_MATRIX_POWER
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| 11 | #define EIGEN_MATRIX_POWER
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| 12 |
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| 13 | namespace Eigen {
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| 14 |
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| 15 | template<typename MatrixType> class MatrixPower;
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| 16 |
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| 17 | template<typename MatrixType>
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| 18 | class MatrixPowerRetval : public ReturnByValue< MatrixPowerRetval<MatrixType> >
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| 19 | {
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| 20 | public:
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| 21 | typedef typename MatrixType::RealScalar RealScalar;
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| 22 | typedef typename MatrixType::Index Index;
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| 23 |
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| 24 | MatrixPowerRetval(MatrixPower<MatrixType>& pow, RealScalar p) : m_pow(pow), m_p(p)
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| 25 | { }
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| 26 |
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| 27 | template<typename ResultType>
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| 28 | inline void evalTo(ResultType& res) const
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| 29 | { m_pow.compute(res, m_p); }
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| 30 |
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| 31 | Index rows() const { return m_pow.rows(); }
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| 32 | Index cols() const { return m_pow.cols(); }
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| 33 |
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| 34 | private:
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| 35 | MatrixPower<MatrixType>& m_pow;
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| 36 | const RealScalar m_p;
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| 37 | MatrixPowerRetval& operator=(const MatrixPowerRetval&);
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| 38 | };
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| 39 |
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| 40 | template<typename MatrixType>
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| 41 | class MatrixPowerAtomic
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| 42 | {
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| 43 | private:
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| 44 | enum {
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| 45 | RowsAtCompileTime = MatrixType::RowsAtCompileTime,
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| 46 | MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime
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| 47 | };
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| 48 | typedef typename MatrixType::Scalar Scalar;
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| 49 | typedef typename MatrixType::RealScalar RealScalar;
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| 50 | typedef std::complex<RealScalar> ComplexScalar;
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| 51 | typedef typename MatrixType::Index Index;
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| 52 | typedef Array<Scalar, RowsAtCompileTime, 1, ColMajor, MaxRowsAtCompileTime> ArrayType;
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| 53 |
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| 54 | const MatrixType& m_A;
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| 55 | RealScalar m_p;
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| 56 |
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| 57 | void computePade(int degree, const MatrixType& IminusT, MatrixType& res) const;
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| 58 | void compute2x2(MatrixType& res, RealScalar p) const;
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| 59 | void computeBig(MatrixType& res) const;
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| 60 | static int getPadeDegree(float normIminusT);
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| 61 | static int getPadeDegree(double normIminusT);
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| 62 | static int getPadeDegree(long double normIminusT);
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| 63 | static ComplexScalar computeSuperDiag(const ComplexScalar&, const ComplexScalar&, RealScalar p);
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| 64 | static RealScalar computeSuperDiag(RealScalar, RealScalar, RealScalar p);
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| 65 |
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| 66 | public:
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| 67 | MatrixPowerAtomic(const MatrixType& T, RealScalar p);
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| 68 | void compute(MatrixType& res) const;
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| 69 | };
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| 70 |
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| 71 | template<typename MatrixType>
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| 72 | MatrixPowerAtomic<MatrixType>::MatrixPowerAtomic(const MatrixType& T, RealScalar p) :
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| 73 | m_A(T), m_p(p)
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| 74 | { eigen_assert(T.rows() == T.cols()); }
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| 75 |
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| 76 | template<typename MatrixType>
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| 77 | void MatrixPowerAtomic<MatrixType>::compute(MatrixType& res) const
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| 78 | {
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| 79 | res.resizeLike(m_A);
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| 80 | switch (m_A.rows()) {
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| 81 | case 0:
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| 82 | break;
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| 83 | case 1:
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| 84 | res(0,0) = std::pow(m_A(0,0), m_p);
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| 85 | break;
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| 86 | case 2:
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| 87 | compute2x2(res, m_p);
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| 88 | break;
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| 89 | default:
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| 90 | computeBig(res);
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| 91 | }
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| 92 | }
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| 93 |
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| 94 | template<typename MatrixType>
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| 95 | void MatrixPowerAtomic<MatrixType>::computePade(int degree, const MatrixType& IminusT, MatrixType& res) const
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| 96 | {
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| 97 | int i = degree<<1;
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| 98 | res = (m_p-degree) / ((i-1)<<1) * IminusT;
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| 99 | for (--i; i; --i) {
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| 100 | res = (MatrixType::Identity(IminusT.rows(), IminusT.cols()) + res).template triangularView<Upper>()
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| 101 | .solve((i==1 ? -m_p : i&1 ? (-m_p-(i>>1))/(i<<1) : (m_p-(i>>1))/((i-1)<<1)) * IminusT).eval();
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| 102 | }
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| 103 | res += MatrixType::Identity(IminusT.rows(), IminusT.cols());
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| 104 | }
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| 105 |
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| 106 | // This function assumes that res has the correct size (see bug 614)
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| 107 | template<typename MatrixType>
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| 108 | void MatrixPowerAtomic<MatrixType>::compute2x2(MatrixType& res, RealScalar p) const
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| 109 | {
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| 110 | using std::abs;
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| 111 | using std::pow;
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| 112 |
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| 113 | res.coeffRef(0,0) = pow(m_A.coeff(0,0), p);
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| 114 |
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| 115 | for (Index i=1; i < m_A.cols(); ++i) {
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| 116 | res.coeffRef(i,i) = pow(m_A.coeff(i,i), p);
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| 117 | if (m_A.coeff(i-1,i-1) == m_A.coeff(i,i))
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| 118 | res.coeffRef(i-1,i) = p * pow(m_A.coeff(i,i), p-1);
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| 119 | 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)))
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| 120 | 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));
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| 121 | else
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| 122 | res.coeffRef(i-1,i) = computeSuperDiag(m_A.coeff(i,i), m_A.coeff(i-1,i-1), p);
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| 123 | res.coeffRef(i-1,i) *= m_A.coeff(i-1,i);
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| 124 | }
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| 125 | }
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| 126 |
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| 127 | template<typename MatrixType>
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| 128 | void MatrixPowerAtomic<MatrixType>::computeBig(MatrixType& res) const
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| 129 | {
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| 130 | const int digits = std::numeric_limits<RealScalar>::digits;
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| 131 | const RealScalar maxNormForPade = digits <= 24? 4.3386528e-1f: // sigle precision
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| 132 | digits <= 53? 2.789358995219730e-1: // double precision
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| 133 | digits <= 64? 2.4471944416607995472e-1L: // extended precision
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| 134 | digits <= 106? 1.1016843812851143391275867258512e-1L: // double-double
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| 135 | 9.134603732914548552537150753385375e-2L; // quadruple precision
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| 136 | MatrixType IminusT, sqrtT, T = m_A.template triangularView<Upper>();
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| 137 | RealScalar normIminusT;
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| 138 | int degree, degree2, numberOfSquareRoots = 0;
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| 139 | bool hasExtraSquareRoot = false;
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| 140 |
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| 141 | /* FIXME
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| 142 | * For singular T, norm(I - T) >= 1 but maxNormForPade < 1, leads to infinite
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| 143 | * loop. We should move 0 eigenvalues to bottom right corner. We need not
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| 144 | * worry about tiny values (e.g. 1e-300) because they will reach 1 if
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| 145 | * repetitively sqrt'ed.
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| 146 | *
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| 147 | * If the 0 eigenvalues are semisimple, they can form a 0 matrix at the
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| 148 | * bottom right corner.
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| 149 | *
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| 150 | * [ T A ]^p [ T^p (T^-1 T^p A) ]
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| 151 | * [ ] = [ ]
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| 152 | * [ 0 0 ] [ 0 0 ]
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| 153 | */
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| 154 | for (Index i=0; i < m_A.cols(); ++i)
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| 155 | eigen_assert(m_A(i,i) != RealScalar(0));
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| 156 |
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| 157 | while (true) {
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| 158 | IminusT = MatrixType::Identity(m_A.rows(), m_A.cols()) - T;
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| 159 | normIminusT = IminusT.cwiseAbs().colwise().sum().maxCoeff();
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| 160 | if (normIminusT < maxNormForPade) {
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| 161 | degree = getPadeDegree(normIminusT);
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| 162 | degree2 = getPadeDegree(normIminusT/2);
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| 163 | if (degree - degree2 <= 1 || hasExtraSquareRoot)
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| 164 | break;
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| 165 | hasExtraSquareRoot = true;
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| 166 | }
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| 167 | MatrixSquareRootTriangular<MatrixType>(T).compute(sqrtT);
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| 168 | T = sqrtT.template triangularView<Upper>();
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| 169 | ++numberOfSquareRoots;
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| 170 | }
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| 171 | computePade(degree, IminusT, res);
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| 172 |
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| 173 | for (; numberOfSquareRoots; --numberOfSquareRoots) {
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| 174 | compute2x2(res, std::ldexp(m_p, -numberOfSquareRoots));
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| 175 | res = res.template triangularView<Upper>() * res;
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| 176 | }
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| 177 | compute2x2(res, m_p);
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| 178 | }
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| 179 |
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| 180 | template<typename MatrixType>
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| 181 | inline int MatrixPowerAtomic<MatrixType>::getPadeDegree(float normIminusT)
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| 182 | {
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| 183 | const float maxNormForPade[] = { 2.8064004e-1f /* degree = 3 */ , 4.3386528e-1f };
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| 184 | int degree = 3;
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| 185 | for (; degree <= 4; ++degree)
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| 186 | if (normIminusT <= maxNormForPade[degree - 3])
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| 187 | break;
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| 188 | return degree;
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| 189 | }
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| 190 |
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| 191 | template<typename MatrixType>
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| 192 | inline int MatrixPowerAtomic<MatrixType>::getPadeDegree(double normIminusT)
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| 193 | {
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| 194 | const double maxNormForPade[] = { 1.884160592658218e-2 /* degree = 3 */ , 6.038881904059573e-2, 1.239917516308172e-1,
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| 195 | 1.999045567181744e-1, 2.789358995219730e-1 };
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| 196 | int degree = 3;
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| 197 | for (; degree <= 7; ++degree)
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| 198 | if (normIminusT <= maxNormForPade[degree - 3])
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| 199 | break;
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| 200 | return degree;
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| 201 | }
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| 202 |
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| 203 | template<typename MatrixType>
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| 204 | inline int MatrixPowerAtomic<MatrixType>::getPadeDegree(long double normIminusT)
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| 205 | {
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| 206 | #if LDBL_MANT_DIG == 53
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| 207 | const int maxPadeDegree = 7;
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| 208 | const double maxNormForPade[] = { 1.884160592658218e-2L /* degree = 3 */ , 6.038881904059573e-2L, 1.239917516308172e-1L,
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| 209 | 1.999045567181744e-1L, 2.789358995219730e-1L };
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| 210 | #elif LDBL_MANT_DIG <= 64
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| 211 | const int maxPadeDegree = 8;
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| 212 | const double maxNormForPade[] = { 6.3854693117491799460e-3L /* degree = 3 */ , 2.6394893435456973676e-2L,
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| 213 | 6.4216043030404063729e-2L, 1.1701165502926694307e-1L, 1.7904284231268670284e-1L, 2.4471944416607995472e-1L };
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| 214 | #elif LDBL_MANT_DIG <= 106
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| 215 | const int maxPadeDegree = 10;
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| 216 | const double maxNormForPade[] = { 1.0007161601787493236741409687186e-4L /* degree = 3 */ ,
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| 217 | 1.0007161601787493236741409687186e-3L, 4.7069769360887572939882574746264e-3L, 1.3220386624169159689406653101695e-2L,
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| 218 | 2.8063482381631737920612944054906e-2L, 4.9625993951953473052385361085058e-2L, 7.7367040706027886224557538328171e-2L,
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| 219 | 1.1016843812851143391275867258512e-1L };
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| 220 | #else
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| 221 | const int maxPadeDegree = 10;
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| 222 | const double maxNormForPade[] = { 5.524506147036624377378713555116378e-5L /* degree = 3 */ ,
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| 223 | 6.640600568157479679823602193345995e-4L, 3.227716520106894279249709728084626e-3L,
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| 224 | 9.619593944683432960546978734646284e-3L, 2.134595382433742403911124458161147e-2L,
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| 225 | 3.908166513900489428442993794761185e-2L, 6.266780814639442865832535460550138e-2L,
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| 226 | 9.134603732914548552537150753385375e-2L };
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| 227 | #endif
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| 228 | int degree = 3;
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| 229 | for (; degree <= maxPadeDegree; ++degree)
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| 230 | if (normIminusT <= maxNormForPade[degree - 3])
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| 231 | break;
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| 232 | return degree;
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| 233 | }
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| 234 |
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| 235 | template<typename MatrixType>
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| 236 | inline typename MatrixPowerAtomic<MatrixType>::ComplexScalar
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| 237 | MatrixPowerAtomic<MatrixType>::computeSuperDiag(const ComplexScalar& curr, const ComplexScalar& prev, RealScalar p)
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| 238 | {
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| 239 | ComplexScalar logCurr = std::log(curr);
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| 240 | ComplexScalar logPrev = std::log(prev);
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| 241 | int unwindingNumber = std::ceil((numext::imag(logCurr - logPrev) - M_PI) / (2*M_PI));
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| 242 | ComplexScalar w = numext::atanh2(curr - prev, curr + prev) + ComplexScalar(0, M_PI*unwindingNumber);
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| 243 | return RealScalar(2) * std::exp(RealScalar(0.5) * p * (logCurr + logPrev)) * std::sinh(p * w) / (curr - prev);
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| 244 | }
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| 245 |
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| 246 | template<typename MatrixType>
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| 247 | inline typename MatrixPowerAtomic<MatrixType>::RealScalar
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| 248 | MatrixPowerAtomic<MatrixType>::computeSuperDiag(RealScalar curr, RealScalar prev, RealScalar p)
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| 249 | {
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| 250 | RealScalar w = numext::atanh2(curr - prev, curr + prev);
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| 251 | return 2 * std::exp(p * (std::log(curr) + std::log(prev)) / 2) * std::sinh(p * w) / (curr - prev);
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| 252 | }
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| 253 |
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| 254 | /**
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| 255 | * \ingroup MatrixFunctions_Module
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| 256 | *
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| 257 | * \brief Class for computing matrix powers.
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| 258 | *
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| 259 | * \tparam MatrixType type of the base, expected to be an instantiation
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| 260 | * of the Matrix class template.
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| 261 | *
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| 262 | * This class is capable of computing real/complex matrices raised to
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| 263 | * an arbitrary real power. Meanwhile, it saves the result of Schur
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| 264 | * decomposition if an non-integral power has even been calculated.
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| 265 | * Therefore, if you want to compute multiple (>= 2) matrix powers
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| 266 | * for the same matrix, using the class directly is more efficient than
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| 267 | * calling MatrixBase::pow().
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| 268 | *
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| 269 | * Example:
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| 270 | * \include MatrixPower_optimal.cpp
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| 271 | * Output: \verbinclude MatrixPower_optimal.out
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| 272 | */
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| 273 | template<typename MatrixType>
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| 274 | class MatrixPower
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| 275 | {
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| 276 | private:
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| 277 | enum {
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| 278 | RowsAtCompileTime = MatrixType::RowsAtCompileTime,
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| 279 | ColsAtCompileTime = MatrixType::ColsAtCompileTime,
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| 280 | MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
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| 281 | MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
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| 282 | };
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| 283 | typedef typename MatrixType::Scalar Scalar;
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| 284 | typedef typename MatrixType::RealScalar RealScalar;
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| 285 | typedef typename MatrixType::Index Index;
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| 286 |
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| 287 | public:
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| 288 | /**
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| 289 | * \brief Constructor.
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| 290 | *
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| 291 | * \param[in] A the base of the matrix power.
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| 292 | *
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| 293 | * The class stores a reference to A, so it should not be changed
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| 294 | * (or destroyed) before evaluation.
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| 295 | */
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| 296 | explicit MatrixPower(const MatrixType& A) : m_A(A), m_conditionNumber(0)
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| 297 | { eigen_assert(A.rows() == A.cols()); }
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| 298 |
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| 299 | /**
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| 300 | * \brief Returns the matrix power.
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| 301 | *
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| 302 | * \param[in] p exponent, a real scalar.
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| 303 | * \return The expression \f$ A^p \f$, where A is specified in the
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| 304 | * constructor.
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| 305 | */
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| 306 | const MatrixPowerRetval<MatrixType> operator()(RealScalar p)
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| 307 | { return MatrixPowerRetval<MatrixType>(*this, p); }
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| 308 |
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| 309 | /**
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| 310 | * \brief Compute the matrix power.
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| 311 | *
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| 312 | * \param[in] p exponent, a real scalar.
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| 313 | * \param[out] res \f$ A^p \f$ where A is specified in the
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| 314 | * constructor.
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| 315 | */
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| 316 | template<typename ResultType>
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| 317 | void compute(ResultType& res, RealScalar p);
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| 318 |
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| 319 | Index rows() const { return m_A.rows(); }
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| 320 | Index cols() const { return m_A.cols(); }
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| 321 |
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| 322 | private:
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| 323 | typedef std::complex<RealScalar> ComplexScalar;
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| 324 | typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, MatrixType::Options,
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| 325 | MaxRowsAtCompileTime, MaxColsAtCompileTime> ComplexMatrix;
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| 326 |
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| 327 | typename MatrixType::Nested m_A;
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| 328 | MatrixType m_tmp;
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| 329 | ComplexMatrix m_T, m_U, m_fT;
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| 330 | RealScalar m_conditionNumber;
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| 331 |
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| 332 | RealScalar modfAndInit(RealScalar, RealScalar*);
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| 333 |
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| 334 | template<typename ResultType>
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| 335 | void computeIntPower(ResultType&, RealScalar);
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| 336 |
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| 337 | template<typename ResultType>
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| 338 | void computeFracPower(ResultType&, RealScalar);
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| 339 |
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| 340 | template<int Rows, int Cols, int Options, int MaxRows, int MaxCols>
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| 341 | static void revertSchur(
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| 342 | Matrix<ComplexScalar, Rows, Cols, Options, MaxRows, MaxCols>& res,
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| 343 | const ComplexMatrix& T,
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| 344 | const ComplexMatrix& U);
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| 345 |
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| 346 | template<int Rows, int Cols, int Options, int MaxRows, int MaxCols>
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| 347 | static void revertSchur(
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| 348 | Matrix<RealScalar, Rows, Cols, Options, MaxRows, MaxCols>& res,
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| 349 | const ComplexMatrix& T,
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| 350 | const ComplexMatrix& U);
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| 351 | };
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| 352 |
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| 353 | template<typename MatrixType>
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| 354 | template<typename ResultType>
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| 355 | void MatrixPower<MatrixType>::compute(ResultType& res, RealScalar p)
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| 356 | {
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| 357 | switch (cols()) {
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| 358 | case 0:
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| 359 | break;
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| 360 | case 1:
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| 361 | res(0,0) = std::pow(m_A.coeff(0,0), p);
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| 362 | break;
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| 363 | default:
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| 364 | RealScalar intpart, x = modfAndInit(p, &intpart);
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| 365 | computeIntPower(res, intpart);
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| 366 | computeFracPower(res, x);
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| 367 | }
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| 368 | }
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| 369 |
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| 370 | template<typename MatrixType>
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| 371 | typename MatrixPower<MatrixType>::RealScalar
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| 372 | MatrixPower<MatrixType>::modfAndInit(RealScalar x, RealScalar* intpart)
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| 373 | {
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| 374 | typedef Array<RealScalar, RowsAtCompileTime, 1, ColMajor, MaxRowsAtCompileTime> RealArray;
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| 375 |
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| 376 | *intpart = std::floor(x);
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| 377 | RealScalar res = x - *intpart;
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| 378 |
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| 379 | if (!m_conditionNumber && res) {
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| 380 | const ComplexSchur<MatrixType> schurOfA(m_A);
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| 381 | m_T = schurOfA.matrixT();
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| 382 | m_U = schurOfA.matrixU();
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| 383 |
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| 384 | const RealArray absTdiag = m_T.diagonal().array().abs();
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| 385 | m_conditionNumber = absTdiag.maxCoeff() / absTdiag.minCoeff();
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| 386 | }
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| 387 |
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| 388 | if (res>RealScalar(0.5) && res>(1-res)*std::pow(m_conditionNumber, res)) {
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| 389 | --res;
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| 390 | ++*intpart;
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| 391 | }
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| 392 | return res;
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| 393 | }
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| 394 |
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| 395 | template<typename MatrixType>
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| 396 | template<typename ResultType>
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| 397 | void MatrixPower<MatrixType>::computeIntPower(ResultType& res, RealScalar p)
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| 398 | {
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| 399 | RealScalar pp = std::abs(p);
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| 400 |
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| 401 | if (p<0) m_tmp = m_A.inverse();
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| 402 | else m_tmp = m_A;
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| 403 |
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| 404 | res = MatrixType::Identity(rows(), cols());
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| 405 | while (pp >= 1) {
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| 406 | if (std::fmod(pp, 2) >= 1)
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| 407 | res = m_tmp * res;
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| 408 | m_tmp *= m_tmp;
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| 409 | pp /= 2;
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| 410 | }
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| 411 | }
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| 412 |
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| 413 | template<typename MatrixType>
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| 414 | template<typename ResultType>
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| 415 | void MatrixPower<MatrixType>::computeFracPower(ResultType& res, RealScalar p)
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| 416 | {
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| 417 | if (p) {
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| 418 | eigen_assert(m_conditionNumber);
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| 419 | MatrixPowerAtomic<ComplexMatrix>(m_T, p).compute(m_fT);
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| 420 | revertSchur(m_tmp, m_fT, m_U);
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| 421 | res = m_tmp * res;
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| 422 | }
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| 423 | }
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| 424 |
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| 425 | template<typename MatrixType>
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| 426 | template<int Rows, int Cols, int Options, int MaxRows, int MaxCols>
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| 427 | inline void MatrixPower<MatrixType>::revertSchur(
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| 428 | Matrix<ComplexScalar, Rows, Cols, Options, MaxRows, MaxCols>& res,
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| 429 | const ComplexMatrix& T,
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| 430 | const ComplexMatrix& U)
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| 431 | { res.noalias() = U * (T.template triangularView<Upper>() * U.adjoint()); }
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| 432 |
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| 433 | template<typename MatrixType>
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| 434 | template<int Rows, int Cols, int Options, int MaxRows, int MaxCols>
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| 435 | inline void MatrixPower<MatrixType>::revertSchur(
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| 436 | Matrix<RealScalar, Rows, Cols, Options, MaxRows, MaxCols>& res,
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| 437 | const ComplexMatrix& T,
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| 438 | const ComplexMatrix& U)
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| 439 | { res.noalias() = (U * (T.template triangularView<Upper>() * U.adjoint())).real(); }
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| 440 |
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| 441 | /**
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| 442 | * \ingroup MatrixFunctions_Module
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| 443 | *
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| 444 | * \brief Proxy for the matrix power of some matrix (expression).
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| 445 | *
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| 446 | * \tparam Derived type of the base, a matrix (expression).
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| 447 | *
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| 448 | * This class holds the arguments to the matrix power until it is
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| 449 | * assigned or evaluated for some other reason (so the argument
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| 450 | * should not be changed in the meantime). It is the return type of
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| 451 | * MatrixBase::pow() and related functions and most of the
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| 452 | * time this is the only way it is used.
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| 453 | */
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| 454 | template<typename Derived>
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| 455 | class MatrixPowerReturnValue : public ReturnByValue< MatrixPowerReturnValue<Derived> >
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| 456 | {
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| 457 | public:
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| 458 | typedef typename Derived::PlainObject PlainObject;
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| 459 | typedef typename Derived::RealScalar RealScalar;
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| 460 | typedef typename Derived::Index Index;
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| 461 |
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| 462 | /**
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| 463 | * \brief Constructor.
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| 464 | *
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| 465 | * \param[in] A %Matrix (expression), the base of the matrix power.
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| 466 | * \param[in] p scalar, the exponent of the matrix power.
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| 467 | */
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| 468 | MatrixPowerReturnValue(const Derived& A, RealScalar p) : m_A(A), m_p(p)
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| 469 | { }
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| 470 |
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| 471 | /**
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| 472 | * \brief Compute the matrix power.
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| 473 | *
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| 474 | * \param[out] result \f$ A^p \f$ where \p A and \p p are as in the
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| 475 | * constructor.
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| 476 | */
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| 477 | template<typename ResultType>
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| 478 | inline void evalTo(ResultType& res) const
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| 479 | { MatrixPower<PlainObject>(m_A.eval()).compute(res, m_p); }
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| 480 |
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| 481 | Index rows() const { return m_A.rows(); }
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| 482 | Index cols() const { return m_A.cols(); }
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| 483 |
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| 484 | private:
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| 485 | const Derived& m_A;
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| 486 | const RealScalar m_p;
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| 487 | MatrixPowerReturnValue& operator=(const MatrixPowerReturnValue&);
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| 488 | };
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| 489 |
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| 490 | namespace internal {
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| 491 |
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| 492 | template<typename MatrixPowerType>
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| 493 | struct traits< MatrixPowerRetval<MatrixPowerType> >
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| 494 | { typedef typename MatrixPowerType::PlainObject ReturnType; };
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| 495 |
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| 496 | template<typename Derived>
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| 497 | struct traits< MatrixPowerReturnValue<Derived> >
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| 498 | { typedef typename Derived::PlainObject ReturnType; };
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| 499 |
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| 500 | }
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| 501 |
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| 502 | template<typename Derived>
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| 503 | const MatrixPowerReturnValue<Derived> MatrixBase<Derived>::pow(const RealScalar& p) const
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| 504 | { return MatrixPowerReturnValue<Derived>(derived(), p); }
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| 505 |
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| 506 | } // namespace Eigen
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| 507 |
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| 508 | #endif // EIGEN_MATRIX_POWER
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