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378 lines
17 KiB
C++
378 lines
17 KiB
C++
// This file is part of Eigen, a lightweight C++ template library
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// for linear algebra.
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//
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// Copyright (C) 2011, 2013 Jitse Niesen <jitse@maths.leeds.ac.uk>
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// Copyright (C) 2011 Chen-Pang He <jdh8@ms63.hinet.net>
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//
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// This Source Code Form is subject to the terms of the Mozilla
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// Public License v. 2.0. If a copy of the MPL was not distributed
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// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
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#ifndef EIGEN_MATRIX_LOGARITHM
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#define EIGEN_MATRIX_LOGARITHM
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#ifndef M_PI
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#define M_PI 3.141592653589793238462643383279503L
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#endif
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namespace Eigen {
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namespace internal {
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template <typename Scalar>
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struct matrix_log_min_pade_degree
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{
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static const int value = 3;
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};
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template <typename Scalar>
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struct matrix_log_max_pade_degree
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{
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typedef typename NumTraits<Scalar>::Real RealScalar;
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static const int value = std::numeric_limits<RealScalar>::digits<= 24? 5: // single precision
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std::numeric_limits<RealScalar>::digits<= 53? 7: // double precision
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std::numeric_limits<RealScalar>::digits<= 64? 8: // extended precision
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std::numeric_limits<RealScalar>::digits<=106? 10: // double-double
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11; // quadruple precision
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};
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/** \brief Compute logarithm of 2x2 triangular matrix. */
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template <typename MatrixType>
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void matrix_log_compute_2x2(const MatrixType& A, MatrixType& result)
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{
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typedef typename MatrixType::Scalar Scalar;
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using std::abs;
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using std::ceil;
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using std::imag;
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using std::log;
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Scalar logA00 = log(A(0,0));
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Scalar logA11 = log(A(1,1));
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result(0,0) = logA00;
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result(1,0) = Scalar(0);
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result(1,1) = logA11;
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Scalar y = A(1,1) - A(0,0);
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if (y==Scalar(0))
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{
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result(0,1) = A(0,1) / A(0,0);
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}
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else if ((abs(A(0,0)) < 0.5*abs(A(1,1))) || (abs(A(0,0)) > 2*abs(A(1,1))))
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{
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result(0,1) = A(0,1) * (logA11 - logA00) / y;
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}
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else
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{
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// computation in previous branch is inaccurate if A(1,1) \approx A(0,0)
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int unwindingNumber = static_cast<int>(ceil((imag(logA11 - logA00) - M_PI) / (2*M_PI)));
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result(0,1) = A(0,1) * (numext::log1p(y/A(0,0)) + Scalar(0,2*M_PI*unwindingNumber)) / y;
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}
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}
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/* \brief Get suitable degree for Pade approximation. (specialized for RealScalar = float) */
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inline int matrix_log_get_pade_degree(float normTminusI)
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{
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const float maxNormForPade[] = { 2.5111573934555054e-1 /* degree = 3 */ , 4.0535837411880493e-1,
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5.3149729967117310e-1 };
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const int minPadeDegree = matrix_log_min_pade_degree<float>::value;
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const int maxPadeDegree = matrix_log_max_pade_degree<float>::value;
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int degree = minPadeDegree;
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for (; degree <= maxPadeDegree; ++degree)
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if (normTminusI <= maxNormForPade[degree - minPadeDegree])
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break;
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return degree;
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}
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/* \brief Get suitable degree for Pade approximation. (specialized for RealScalar = double) */
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inline int matrix_log_get_pade_degree(double normTminusI)
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{
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const double maxNormForPade[] = { 1.6206284795015624e-2 /* degree = 3 */ , 5.3873532631381171e-2,
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1.1352802267628681e-1, 1.8662860613541288e-1, 2.642960831111435e-1 };
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const int minPadeDegree = matrix_log_min_pade_degree<double>::value;
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const int maxPadeDegree = matrix_log_max_pade_degree<double>::value;
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int degree = minPadeDegree;
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for (; degree <= maxPadeDegree; ++degree)
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if (normTminusI <= maxNormForPade[degree - minPadeDegree])
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break;
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return degree;
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}
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/* \brief Get suitable degree for Pade approximation. (specialized for RealScalar = long double) */
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inline int matrix_log_get_pade_degree(long double normTminusI)
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{
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#if LDBL_MANT_DIG == 53 // double precision
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const long double maxNormForPade[] = { 1.6206284795015624e-2L /* degree = 3 */ , 5.3873532631381171e-2L,
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1.1352802267628681e-1L, 1.8662860613541288e-1L, 2.642960831111435e-1L };
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#elif LDBL_MANT_DIG <= 64 // extended precision
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const long double maxNormForPade[] = { 5.48256690357782863103e-3L /* degree = 3 */, 2.34559162387971167321e-2L,
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5.84603923897347449857e-2L, 1.08486423756725170223e-1L, 1.68385767881294446649e-1L,
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2.32777776523703892094e-1L };
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#elif LDBL_MANT_DIG <= 106 // double-double
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const long double maxNormForPade[] = { 8.58970550342939562202529664318890e-5L /* degree = 3 */,
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9.34074328446359654039446552677759e-4L, 4.26117194647672175773064114582860e-3L,
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1.21546224740281848743149666560464e-2L, 2.61100544998339436713088248557444e-2L,
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4.66170074627052749243018566390567e-2L, 7.32585144444135027565872014932387e-2L,
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1.05026503471351080481093652651105e-1L };
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#else // quadruple precision
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const long double maxNormForPade[] = { 4.7419931187193005048501568167858103e-5L /* degree = 3 */,
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5.8853168473544560470387769480192666e-4L, 2.9216120366601315391789493628113520e-3L,
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8.8415758124319434347116734705174308e-3L, 1.9850836029449446668518049562565291e-2L,
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3.6688019729653446926585242192447447e-2L, 5.9290962294020186998954055264528393e-2L,
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8.6998436081634343903250580992127677e-2L, 1.1880960220216759245467951592883642e-1L };
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#endif
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const int minPadeDegree = matrix_log_min_pade_degree<long double>::value;
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const int maxPadeDegree = matrix_log_max_pade_degree<long double>::value;
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int degree = minPadeDegree;
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for (; degree <= maxPadeDegree; ++degree)
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if (normTminusI <= maxNormForPade[degree - minPadeDegree])
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break;
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return degree;
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}
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/* \brief Compute Pade approximation to matrix logarithm */
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template <typename MatrixType>
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void matrix_log_compute_pade(MatrixType& result, const MatrixType& T, int degree)
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{
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typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
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const int minPadeDegree = 3;
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const int maxPadeDegree = 11;
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assert(degree >= minPadeDegree && degree <= maxPadeDegree);
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const RealScalar nodes[][maxPadeDegree] = {
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{ 0.1127016653792583114820734600217600L, 0.5000000000000000000000000000000000L, // degree 3
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0.8872983346207416885179265399782400L },
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{ 0.0694318442029737123880267555535953L, 0.3300094782075718675986671204483777L, // degree 4
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0.6699905217924281324013328795516223L, 0.9305681557970262876119732444464048L },
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{ 0.0469100770306680036011865608503035L, 0.2307653449471584544818427896498956L, // degree 5
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0.5000000000000000000000000000000000L, 0.7692346550528415455181572103501044L,
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0.9530899229693319963988134391496965L },
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{ 0.0337652428984239860938492227530027L, 0.1693953067668677431693002024900473L, // degree 6
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0.3806904069584015456847491391596440L, 0.6193095930415984543152508608403560L,
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0.8306046932331322568306997975099527L, 0.9662347571015760139061507772469973L },
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{ 0.0254460438286207377369051579760744L, 0.1292344072003027800680676133596058L, // degree 7
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0.2970774243113014165466967939615193L, 0.5000000000000000000000000000000000L,
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0.7029225756886985834533032060384807L, 0.8707655927996972199319323866403942L,
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0.9745539561713792622630948420239256L },
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{ 0.0198550717512318841582195657152635L, 0.1016667612931866302042230317620848L, // degree 8
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0.2372337950418355070911304754053768L, 0.4082826787521750975302619288199080L,
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0.5917173212478249024697380711800920L, 0.7627662049581644929088695245946232L,
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0.8983332387068133697957769682379152L, 0.9801449282487681158417804342847365L },
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{ 0.0159198802461869550822118985481636L, 0.0819844463366821028502851059651326L, // degree 9
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0.1933142836497048013456489803292629L, 0.3378732882980955354807309926783317L,
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0.5000000000000000000000000000000000L, 0.6621267117019044645192690073216683L,
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0.8066857163502951986543510196707371L, 0.9180155536633178971497148940348674L,
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0.9840801197538130449177881014518364L },
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{ 0.0130467357414141399610179939577740L, 0.0674683166555077446339516557882535L, // degree 10
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0.1602952158504877968828363174425632L, 0.2833023029353764046003670284171079L,
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0.4255628305091843945575869994351400L, 0.5744371694908156054424130005648600L,
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0.7166976970646235953996329715828921L, 0.8397047841495122031171636825574368L,
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0.9325316833444922553660483442117465L, 0.9869532642585858600389820060422260L },
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{ 0.0108856709269715035980309994385713L, 0.0564687001159523504624211153480364L, // degree 11
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0.1349239972129753379532918739844233L, 0.2404519353965940920371371652706952L,
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0.3652284220238275138342340072995692L, 0.5000000000000000000000000000000000L,
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0.6347715779761724861657659927004308L, 0.7595480646034059079628628347293048L,
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0.8650760027870246620467081260155767L, 0.9435312998840476495375788846519636L,
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0.9891143290730284964019690005614287L } };
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const RealScalar weights[][maxPadeDegree] = {
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{ 0.2777777777777777777777777777777778L, 0.4444444444444444444444444444444444L, // degree 3
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0.2777777777777777777777777777777778L },
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{ 0.1739274225687269286865319746109997L, 0.3260725774312730713134680253890003L, // degree 4
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0.3260725774312730713134680253890003L, 0.1739274225687269286865319746109997L },
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{ 0.1184634425280945437571320203599587L, 0.2393143352496832340206457574178191L, // degree 5
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0.2844444444444444444444444444444444L, 0.2393143352496832340206457574178191L,
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0.1184634425280945437571320203599587L },
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{ 0.0856622461895851725201480710863665L, 0.1803807865240693037849167569188581L, // degree 6
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0.2339569672863455236949351719947755L, 0.2339569672863455236949351719947755L,
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0.1803807865240693037849167569188581L, 0.0856622461895851725201480710863665L },
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{ 0.0647424830844348466353057163395410L, 0.1398526957446383339507338857118898L, // degree 7
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0.1909150252525594724751848877444876L, 0.2089795918367346938775510204081633L,
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0.1909150252525594724751848877444876L, 0.1398526957446383339507338857118898L,
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0.0647424830844348466353057163395410L },
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{ 0.0506142681451881295762656771549811L, 0.1111905172266872352721779972131204L, // degree 8
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0.1568533229389436436689811009933007L, 0.1813418916891809914825752246385978L,
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0.1813418916891809914825752246385978L, 0.1568533229389436436689811009933007L,
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0.1111905172266872352721779972131204L, 0.0506142681451881295762656771549811L },
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{ 0.0406371941807872059859460790552618L, 0.0903240803474287020292360156214564L, // degree 9
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0.1303053482014677311593714347093164L, 0.1561735385200014200343152032922218L,
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0.1651196775006298815822625346434870L, 0.1561735385200014200343152032922218L,
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0.1303053482014677311593714347093164L, 0.0903240803474287020292360156214564L,
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0.0406371941807872059859460790552618L },
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{ 0.0333356721543440687967844049466659L, 0.0747256745752902965728881698288487L, // degree 10
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0.1095431812579910219977674671140816L, 0.1346333596549981775456134607847347L,
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0.1477621123573764350869464973256692L, 0.1477621123573764350869464973256692L,
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0.1346333596549981775456134607847347L, 0.1095431812579910219977674671140816L,
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0.0747256745752902965728881698288487L, 0.0333356721543440687967844049466659L },
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{ 0.0278342835580868332413768602212743L, 0.0627901847324523123173471496119701L, // degree 11
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0.0931451054638671257130488207158280L, 0.1165968822959952399592618524215876L,
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0.1314022722551233310903444349452546L, 0.1364625433889503153572417641681711L,
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0.1314022722551233310903444349452546L, 0.1165968822959952399592618524215876L,
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0.0931451054638671257130488207158280L, 0.0627901847324523123173471496119701L,
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0.0278342835580868332413768602212743L } };
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MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
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result.setZero(T.rows(), T.rows());
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for (int k = 0; k < degree; ++k) {
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RealScalar weight = weights[degree-minPadeDegree][k];
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RealScalar node = nodes[degree-minPadeDegree][k];
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result += weight * (MatrixType::Identity(T.rows(), T.rows()) + node * TminusI)
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.template triangularView<Upper>().solve(TminusI);
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}
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}
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/** \brief Compute logarithm of triangular matrices with size > 2.
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* \details This uses a inverse scale-and-square algorithm. */
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template <typename MatrixType>
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void matrix_log_compute_big(const MatrixType& A, MatrixType& result)
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{
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typedef typename MatrixType::Scalar Scalar;
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typedef typename NumTraits<Scalar>::Real RealScalar;
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using std::pow;
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int numberOfSquareRoots = 0;
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int numberOfExtraSquareRoots = 0;
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int degree;
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MatrixType T = A, sqrtT;
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int maxPadeDegree = matrix_log_max_pade_degree<Scalar>::value;
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const RealScalar maxNormForPade = maxPadeDegree<= 5? 5.3149729967117310e-1: // single precision
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maxPadeDegree<= 7? 2.6429608311114350e-1: // double precision
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maxPadeDegree<= 8? 2.32777776523703892094e-1L: // extended precision
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maxPadeDegree<=10? 1.05026503471351080481093652651105e-1L: // double-double
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1.1880960220216759245467951592883642e-1L; // quadruple precision
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while (true) {
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RealScalar normTminusI = (T - MatrixType::Identity(T.rows(), T.rows())).cwiseAbs().colwise().sum().maxCoeff();
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if (normTminusI < maxNormForPade) {
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degree = matrix_log_get_pade_degree(normTminusI);
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int degree2 = matrix_log_get_pade_degree(normTminusI / RealScalar(2));
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if ((degree - degree2 <= 1) || (numberOfExtraSquareRoots == 1))
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break;
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++numberOfExtraSquareRoots;
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}
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matrix_sqrt_triangular(T, sqrtT);
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T = sqrtT.template triangularView<Upper>();
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++numberOfSquareRoots;
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}
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matrix_log_compute_pade(result, T, degree);
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result *= pow(RealScalar(2), numberOfSquareRoots);
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}
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/** \ingroup MatrixFunctions_Module
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* \class MatrixLogarithmAtomic
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* \brief Helper class for computing matrix logarithm of atomic matrices.
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*
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* Here, an atomic matrix is a triangular matrix whose diagonal entries are close to each other.
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*
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* \sa class MatrixFunctionAtomic, MatrixBase::log()
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*/
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template <typename MatrixType>
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class MatrixLogarithmAtomic
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{
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public:
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/** \brief Compute matrix logarithm of atomic matrix
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* \param[in] A argument of matrix logarithm, should be upper triangular and atomic
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* \returns The logarithm of \p A.
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*/
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MatrixType compute(const MatrixType& A);
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};
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template <typename MatrixType>
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MatrixType MatrixLogarithmAtomic<MatrixType>::compute(const MatrixType& A)
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{
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using std::log;
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MatrixType result(A.rows(), A.rows());
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if (A.rows() == 1)
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result(0,0) = log(A(0,0));
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else if (A.rows() == 2)
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matrix_log_compute_2x2(A, result);
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else
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matrix_log_compute_big(A, result);
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return result;
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}
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} // end of namespace internal
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/** \ingroup MatrixFunctions_Module
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*
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* \brief Proxy for the matrix logarithm of some matrix (expression).
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*
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* \tparam Derived Type of the argument to the matrix function.
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*
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* This class holds the argument to the matrix function until it is
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* assigned or evaluated for some other reason (so the argument
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* should not be changed in the meantime). It is the return type of
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* MatrixBase::log() and most of the time this is the only way it
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* is used.
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*/
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template<typename Derived> class MatrixLogarithmReturnValue
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: public ReturnByValue<MatrixLogarithmReturnValue<Derived> >
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{
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public:
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typedef typename Derived::Scalar Scalar;
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typedef typename Derived::Index Index;
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protected:
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typedef typename internal::nested<Derived>::type DerivedNested;
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public:
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/** \brief Constructor.
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*
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* \param[in] A %Matrix (expression) forming the argument of the matrix logarithm.
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*/
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explicit MatrixLogarithmReturnValue(const Derived& A) : m_A(A) { }
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/** \brief Compute the matrix logarithm.
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*
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* \param[out] result Logarithm of \p A, where \A is as specified in the constructor.
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*/
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template <typename ResultType>
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inline void evalTo(ResultType& result) const
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{
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typedef typename internal::nested_eval<Derived, 10>::type DerivedEvalType;
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typedef typename internal::remove_all<DerivedEvalType>::type DerivedEvalTypeClean;
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typedef internal::traits<DerivedEvalTypeClean> Traits;
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static const int RowsAtCompileTime = Traits::RowsAtCompileTime;
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static const int ColsAtCompileTime = Traits::ColsAtCompileTime;
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static const int Options = DerivedEvalTypeClean::Options;
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typedef std::complex<typename NumTraits<Scalar>::Real> ComplexScalar;
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typedef Matrix<ComplexScalar, Dynamic, Dynamic, Options, RowsAtCompileTime, ColsAtCompileTime> DynMatrixType;
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typedef internal::MatrixLogarithmAtomic<DynMatrixType> AtomicType;
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AtomicType atomic;
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internal::matrix_function_compute<DerivedEvalTypeClean>::run(m_A, atomic, result);
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}
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Index rows() const { return m_A.rows(); }
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Index cols() const { return m_A.cols(); }
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private:
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const DerivedNested m_A;
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};
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namespace internal {
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template<typename Derived>
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struct traits<MatrixLogarithmReturnValue<Derived> >
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{
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typedef typename Derived::PlainObject ReturnType;
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};
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}
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/********** MatrixBase method **********/
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template <typename Derived>
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const MatrixLogarithmReturnValue<Derived> MatrixBase<Derived>::log() const
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{
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eigen_assert(rows() == cols());
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return MatrixLogarithmReturnValue<Derived>(derived());
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}
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} // end namespace Eigen
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#endif // EIGEN_MATRIX_LOGARITHM
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