MatrixLogarithm.h
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00001 // This file is part of Eigen, a lightweight C++ template library
00002 // for linear algebra.
00003 //
00004 // Copyright (C) 2011 Jitse Niesen <jitse@maths.leeds.ac.uk>
00005 // Copyright (C) 2011 Chen-Pang He <jdh8@ms63.hinet.net>
00006 //
00007 // This Source Code Form is subject to the terms of the Mozilla
00008 // Public License v. 2.0. If a copy of the MPL was not distributed
00009 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
00010 
00011 #ifndef EIGEN_MATRIX_LOGARITHM
00012 #define EIGEN_MATRIX_LOGARITHM
00013 
00014 #ifndef M_PI
00015 #define M_PI 3.141592653589793238462643383279503L
00016 #endif
00017 
00018 namespace Eigen { 
00019 
00030 template <typename MatrixType>
00031 class MatrixLogarithmAtomic
00032 {
00033 public:
00034 
00035   typedef typename MatrixType::Scalar Scalar;
00036   // typedef typename MatrixType::Index Index;
00037   typedef typename NumTraits<Scalar>::Real RealScalar;
00038   // typedef typename internal::stem_function<Scalar>::type StemFunction;
00039   // typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, 1> VectorType;
00040 
00042   MatrixLogarithmAtomic() { }
00043 
00048   MatrixType compute(const MatrixType& A);
00049 
00050 private:
00051 
00052   void compute2x2(const MatrixType& A, MatrixType& result);
00053   void computeBig(const MatrixType& A, MatrixType& result);
00054   int getPadeDegree(float normTminusI);
00055   int getPadeDegree(double normTminusI);
00056   int getPadeDegree(long double normTminusI);
00057   void computePade(MatrixType& result, const MatrixType& T, int degree);
00058   void computePade3(MatrixType& result, const MatrixType& T);
00059   void computePade4(MatrixType& result, const MatrixType& T);
00060   void computePade5(MatrixType& result, const MatrixType& T);
00061   void computePade6(MatrixType& result, const MatrixType& T);
00062   void computePade7(MatrixType& result, const MatrixType& T);
00063   void computePade8(MatrixType& result, const MatrixType& T);
00064   void computePade9(MatrixType& result, const MatrixType& T);
00065   void computePade10(MatrixType& result, const MatrixType& T);
00066   void computePade11(MatrixType& result, const MatrixType& T);
00067 
00068   static const int minPadeDegree = 3;
00069   static const int maxPadeDegree = std::numeric_limits<RealScalar>::digits<= 24?  5:  // single precision
00070                                    std::numeric_limits<RealScalar>::digits<= 53?  7:  // double precision
00071                                    std::numeric_limits<RealScalar>::digits<= 64?  8:  // extended precision
00072                                    std::numeric_limits<RealScalar>::digits<=106? 10:  // double-double
00073                                                                                  11;  // quadruple precision
00074 
00075   // Prevent copying
00076   MatrixLogarithmAtomic(const MatrixLogarithmAtomic&);
00077   MatrixLogarithmAtomic& operator=(const MatrixLogarithmAtomic&);
00078 };
00079 
00081 template <typename MatrixType>
00082 MatrixType MatrixLogarithmAtomic<MatrixType>::compute(const MatrixType& A)
00083 {
00084   using std::log;
00085   MatrixType result(A.rows(), A.rows());
00086   if (A.rows() == 1)
00087     result(0,0) = log(A(0,0));
00088   else if (A.rows() == 2)
00089     compute2x2(A, result);
00090   else
00091     computeBig(A, result);
00092   return result;
00093 }
00094 
00096 template <typename MatrixType>
00097 void MatrixLogarithmAtomic<MatrixType>::compute2x2(const MatrixType& A, MatrixType& result)
00098 {
00099   using std::abs;
00100   using std::ceil;
00101   using std::imag;
00102   using std::log;
00103 
00104   Scalar logA00 = log(A(0,0));
00105   Scalar logA11 = log(A(1,1));
00106 
00107   result(0,0) = logA00;
00108   result(1,0) = Scalar(0);
00109   result(1,1) = logA11;
00110 
00111   if (A(0,0) == A(1,1)) {
00112     result(0,1) = A(0,1) / A(0,0);
00113   } else if ((abs(A(0,0)) < 0.5*abs(A(1,1))) || (abs(A(0,0)) > 2*abs(A(1,1)))) {
00114     result(0,1) = A(0,1) * (logA11 - logA00) / (A(1,1) - A(0,0));
00115   } else {
00116     // computation in previous branch is inaccurate if A(1,1) \approx A(0,0)
00117     int unwindingNumber = static_cast<int>(ceil((imag(logA11 - logA00) - M_PI) / (2*M_PI)));
00118     Scalar y = A(1,1) - A(0,0), x = A(1,1) + A(0,0);
00119     result(0,1) = A(0,1) * (Scalar(2) * numext::atanh2(y,x) + Scalar(0,2*M_PI*unwindingNumber)) / y;
00120   }
00121 }
00122 
00125 template <typename MatrixType>
00126 void MatrixLogarithmAtomic<MatrixType>::computeBig(const MatrixType& A, MatrixType& result)
00127 {
00128   using std::pow;
00129   int numberOfSquareRoots = 0;
00130   int numberOfExtraSquareRoots = 0;
00131   int degree;
00132   MatrixType T = A, sqrtT;
00133   const RealScalar maxNormForPade = maxPadeDegree<= 5? 5.3149729967117310e-1:                     // single precision
00134                                     maxPadeDegree<= 7? 2.6429608311114350e-1:                     // double precision
00135                                     maxPadeDegree<= 8? 2.32777776523703892094e-1L:                // extended precision
00136                                     maxPadeDegree<=10? 1.05026503471351080481093652651105e-1L:    // double-double
00137                                                        1.1880960220216759245467951592883642e-1L;  // quadruple precision
00138 
00139   while (true) {
00140     RealScalar normTminusI = (T - MatrixType::Identity(T.rows(), T.rows())).cwiseAbs().colwise().sum().maxCoeff();
00141     if (normTminusI < maxNormForPade) {
00142       degree = getPadeDegree(normTminusI);
00143       int degree2 = getPadeDegree(normTminusI / RealScalar(2));
00144       if ((degree - degree2 <= 1) || (numberOfExtraSquareRoots == 1)) 
00145         break;
00146       ++numberOfExtraSquareRoots;
00147     }
00148     MatrixSquareRootTriangular<MatrixType>(T).compute(sqrtT);
00149     T = sqrtT.template triangularView<Upper>();
00150     ++numberOfSquareRoots;
00151   }
00152 
00153   computePade(result, T, degree);
00154   result *= pow(RealScalar(2), numberOfSquareRoots);
00155 }
00156 
00157 /* \brief Get suitable degree for Pade approximation. (specialized for RealScalar = float) */
00158 template <typename MatrixType>
00159 int MatrixLogarithmAtomic<MatrixType>::getPadeDegree(float normTminusI)
00160 {
00161   const float maxNormForPade[] = { 2.5111573934555054e-1 /* degree = 3 */ , 4.0535837411880493e-1,
00162             5.3149729967117310e-1 };
00163   int degree = 3;
00164   for (; degree <= maxPadeDegree; ++degree) 
00165     if (normTminusI <= maxNormForPade[degree - minPadeDegree])
00166       break;
00167   return degree;
00168 }
00169 
00170 /* \brief Get suitable degree for Pade approximation. (specialized for RealScalar = double) */
00171 template <typename MatrixType>
00172 int MatrixLogarithmAtomic<MatrixType>::getPadeDegree(double normTminusI)
00173 {
00174   const double maxNormForPade[] = { 1.6206284795015624e-2 /* degree = 3 */ , 5.3873532631381171e-2,
00175             1.1352802267628681e-1, 1.8662860613541288e-1, 2.642960831111435e-1 };
00176   int degree = 3;
00177   for (; degree <= maxPadeDegree; ++degree)
00178     if (normTminusI <= maxNormForPade[degree - minPadeDegree])
00179       break;
00180   return degree;
00181 }
00182 
00183 /* \brief Get suitable degree for Pade approximation. (specialized for RealScalar = long double) */
00184 template <typename MatrixType>
00185 int MatrixLogarithmAtomic<MatrixType>::getPadeDegree(long double normTminusI)
00186 {
00187 #if   LDBL_MANT_DIG == 53         // double precision
00188   const long double maxNormForPade[] = { 1.6206284795015624e-2L /* degree = 3 */ , 5.3873532631381171e-2L,
00189             1.1352802267628681e-1L, 1.8662860613541288e-1L, 2.642960831111435e-1L };
00190 #elif LDBL_MANT_DIG <= 64         // extended precision
00191   const long double maxNormForPade[] = { 5.48256690357782863103e-3L /* degree = 3 */, 2.34559162387971167321e-2L,
00192             5.84603923897347449857e-2L, 1.08486423756725170223e-1L, 1.68385767881294446649e-1L,
00193             2.32777776523703892094e-1L };
00194 #elif LDBL_MANT_DIG <= 106        // double-double
00195   const long double maxNormForPade[] = { 8.58970550342939562202529664318890e-5L /* degree = 3 */,
00196             9.34074328446359654039446552677759e-4L, 4.26117194647672175773064114582860e-3L,
00197             1.21546224740281848743149666560464e-2L, 2.61100544998339436713088248557444e-2L,
00198             4.66170074627052749243018566390567e-2L, 7.32585144444135027565872014932387e-2L,
00199             1.05026503471351080481093652651105e-1L };
00200 #else                             // quadruple precision
00201   const long double maxNormForPade[] = { 4.7419931187193005048501568167858103e-5L /* degree = 3 */,
00202             5.8853168473544560470387769480192666e-4L, 2.9216120366601315391789493628113520e-3L,
00203             8.8415758124319434347116734705174308e-3L, 1.9850836029449446668518049562565291e-2L,
00204             3.6688019729653446926585242192447447e-2L, 5.9290962294020186998954055264528393e-2L,
00205             8.6998436081634343903250580992127677e-2L, 1.1880960220216759245467951592883642e-1L };
00206 #endif
00207   int degree = 3;
00208   for (; degree <= maxPadeDegree; ++degree)
00209     if (normTminusI <= maxNormForPade[degree - minPadeDegree])
00210       break;
00211   return degree;
00212 }
00213 
00214 /* \brief Compute Pade approximation to matrix logarithm */
00215 template <typename MatrixType>
00216 void MatrixLogarithmAtomic<MatrixType>::computePade(MatrixType& result, const MatrixType& T, int degree)
00217 {
00218   switch (degree) {
00219     case 3:  computePade3(result, T);  break;
00220     case 4:  computePade4(result, T);  break;
00221     case 5:  computePade5(result, T);  break;
00222     case 6:  computePade6(result, T);  break;
00223     case 7:  computePade7(result, T);  break;
00224     case 8:  computePade8(result, T);  break;
00225     case 9:  computePade9(result, T);  break;
00226     case 10: computePade10(result, T); break;
00227     case 11: computePade11(result, T); break;
00228     default: assert(false); // should never happen
00229   }
00230 } 
00231 
00232 template <typename MatrixType>
00233 void MatrixLogarithmAtomic<MatrixType>::computePade3(MatrixType& result, const MatrixType& T)
00234 {
00235   const int degree = 3;
00236   const RealScalar nodes[]   = { 0.1127016653792583114820734600217600L, 0.5000000000000000000000000000000000L,
00237             0.8872983346207416885179265399782400L };
00238   const RealScalar weights[] = { 0.2777777777777777777777777777777778L, 0.4444444444444444444444444444444444L,
00239             0.2777777777777777777777777777777778L };
00240   eigen_assert(degree <= maxPadeDegree);
00241   MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
00242   result.setZero(T.rows(), T.rows());
00243   for (int k = 0; k < degree; ++k)
00244     result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
00245                            .template triangularView<Upper>().solve(TminusI);
00246 }
00247 
00248 template <typename MatrixType>
00249 void MatrixLogarithmAtomic<MatrixType>::computePade4(MatrixType& result, const MatrixType& T)
00250 {
00251   const int degree = 4;
00252   const RealScalar nodes[]   = { 0.0694318442029737123880267555535953L, 0.3300094782075718675986671204483777L,
00253             0.6699905217924281324013328795516223L, 0.9305681557970262876119732444464048L };
00254   const RealScalar weights[] = { 0.1739274225687269286865319746109997L, 0.3260725774312730713134680253890003L,
00255             0.3260725774312730713134680253890003L, 0.1739274225687269286865319746109997L };
00256   eigen_assert(degree <= maxPadeDegree);
00257   MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
00258   result.setZero(T.rows(), T.rows());
00259   for (int k = 0; k < degree; ++k)
00260     result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
00261                            .template triangularView<Upper>().solve(TminusI);
00262 }
00263 
00264 template <typename MatrixType>
00265 void MatrixLogarithmAtomic<MatrixType>::computePade5(MatrixType& result, const MatrixType& T)
00266 {
00267   const int degree = 5;
00268   const RealScalar nodes[]   = { 0.0469100770306680036011865608503035L, 0.2307653449471584544818427896498956L,
00269             0.5000000000000000000000000000000000L, 0.7692346550528415455181572103501044L,
00270             0.9530899229693319963988134391496965L };
00271   const RealScalar weights[] = { 0.1184634425280945437571320203599587L, 0.2393143352496832340206457574178191L,
00272             0.2844444444444444444444444444444444L, 0.2393143352496832340206457574178191L,
00273             0.1184634425280945437571320203599587L };
00274   eigen_assert(degree <= maxPadeDegree);
00275   MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
00276   result.setZero(T.rows(), T.rows());
00277   for (int k = 0; k < degree; ++k)
00278     result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
00279                            .template triangularView<Upper>().solve(TminusI);
00280 }
00281 
00282 template <typename MatrixType>
00283 void MatrixLogarithmAtomic<MatrixType>::computePade6(MatrixType& result, const MatrixType& T)
00284 {
00285   const int degree = 6;
00286   const RealScalar nodes[]   = { 0.0337652428984239860938492227530027L, 0.1693953067668677431693002024900473L,
00287             0.3806904069584015456847491391596440L, 0.6193095930415984543152508608403560L,
00288             0.8306046932331322568306997975099527L, 0.9662347571015760139061507772469973L };
00289   const RealScalar weights[] = { 0.0856622461895851725201480710863665L, 0.1803807865240693037849167569188581L,
00290             0.2339569672863455236949351719947755L, 0.2339569672863455236949351719947755L,
00291             0.1803807865240693037849167569188581L, 0.0856622461895851725201480710863665L };
00292   eigen_assert(degree <= maxPadeDegree);
00293   MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
00294   result.setZero(T.rows(), T.rows());
00295   for (int k = 0; k < degree; ++k)
00296     result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
00297                            .template triangularView<Upper>().solve(TminusI);
00298 }
00299 
00300 template <typename MatrixType>
00301 void MatrixLogarithmAtomic<MatrixType>::computePade7(MatrixType& result, const MatrixType& T)
00302 {
00303   const int degree = 7;
00304   const RealScalar nodes[]   = { 0.0254460438286207377369051579760744L, 0.1292344072003027800680676133596058L,
00305             0.2970774243113014165466967939615193L, 0.5000000000000000000000000000000000L,
00306             0.7029225756886985834533032060384807L, 0.8707655927996972199319323866403942L,
00307             0.9745539561713792622630948420239256L };
00308   const RealScalar weights[] = { 0.0647424830844348466353057163395410L, 0.1398526957446383339507338857118898L,
00309             0.1909150252525594724751848877444876L, 0.2089795918367346938775510204081633L,
00310             0.1909150252525594724751848877444876L, 0.1398526957446383339507338857118898L,
00311             0.0647424830844348466353057163395410L };
00312   eigen_assert(degree <= maxPadeDegree);
00313   MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
00314   result.setZero(T.rows(), T.rows());
00315   for (int k = 0; k < degree; ++k)
00316     result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
00317                            .template triangularView<Upper>().solve(TminusI);
00318 }
00319 
00320 template <typename MatrixType>
00321 void MatrixLogarithmAtomic<MatrixType>::computePade8(MatrixType& result, const MatrixType& T)
00322 {
00323   const int degree = 8;
00324   const RealScalar nodes[]   = { 0.0198550717512318841582195657152635L, 0.1016667612931866302042230317620848L,
00325             0.2372337950418355070911304754053768L, 0.4082826787521750975302619288199080L,
00326             0.5917173212478249024697380711800920L, 0.7627662049581644929088695245946232L,
00327             0.8983332387068133697957769682379152L, 0.9801449282487681158417804342847365L };
00328   const RealScalar weights[] = { 0.0506142681451881295762656771549811L, 0.1111905172266872352721779972131204L,
00329             0.1568533229389436436689811009933007L, 0.1813418916891809914825752246385978L,
00330             0.1813418916891809914825752246385978L, 0.1568533229389436436689811009933007L,
00331             0.1111905172266872352721779972131204L, 0.0506142681451881295762656771549811L };
00332   eigen_assert(degree <= maxPadeDegree);
00333   MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
00334   result.setZero(T.rows(), T.rows());
00335   for (int k = 0; k < degree; ++k)
00336     result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
00337                            .template triangularView<Upper>().solve(TminusI);
00338 }
00339 
00340 template <typename MatrixType>
00341 void MatrixLogarithmAtomic<MatrixType>::computePade9(MatrixType& result, const MatrixType& T)
00342 {
00343   const int degree = 9;
00344   const RealScalar nodes[]   = { 0.0159198802461869550822118985481636L, 0.0819844463366821028502851059651326L,
00345             0.1933142836497048013456489803292629L, 0.3378732882980955354807309926783317L,
00346             0.5000000000000000000000000000000000L, 0.6621267117019044645192690073216683L,
00347             0.8066857163502951986543510196707371L, 0.9180155536633178971497148940348674L,
00348             0.9840801197538130449177881014518364L };
00349   const RealScalar weights[] = { 0.0406371941807872059859460790552618L, 0.0903240803474287020292360156214564L,
00350             0.1303053482014677311593714347093164L, 0.1561735385200014200343152032922218L,
00351             0.1651196775006298815822625346434870L, 0.1561735385200014200343152032922218L,
00352             0.1303053482014677311593714347093164L, 0.0903240803474287020292360156214564L,
00353             0.0406371941807872059859460790552618L };
00354   eigen_assert(degree <= maxPadeDegree);
00355   MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
00356   result.setZero(T.rows(), T.rows());
00357   for (int k = 0; k < degree; ++k)
00358     result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
00359                            .template triangularView<Upper>().solve(TminusI);
00360 }
00361 
00362 template <typename MatrixType>
00363 void MatrixLogarithmAtomic<MatrixType>::computePade10(MatrixType& result, const MatrixType& T)
00364 {
00365   const int degree = 10;
00366   const RealScalar nodes[]   = { 0.0130467357414141399610179939577740L, 0.0674683166555077446339516557882535L,
00367             0.1602952158504877968828363174425632L, 0.2833023029353764046003670284171079L,
00368             0.4255628305091843945575869994351400L, 0.5744371694908156054424130005648600L,
00369             0.7166976970646235953996329715828921L, 0.8397047841495122031171636825574368L,
00370             0.9325316833444922553660483442117465L, 0.9869532642585858600389820060422260L };
00371   const RealScalar weights[] = { 0.0333356721543440687967844049466659L, 0.0747256745752902965728881698288487L,
00372             0.1095431812579910219977674671140816L, 0.1346333596549981775456134607847347L,
00373             0.1477621123573764350869464973256692L, 0.1477621123573764350869464973256692L,
00374             0.1346333596549981775456134607847347L, 0.1095431812579910219977674671140816L,
00375             0.0747256745752902965728881698288487L, 0.0333356721543440687967844049466659L };
00376   eigen_assert(degree <= maxPadeDegree);
00377   MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
00378   result.setZero(T.rows(), T.rows());
00379   for (int k = 0; k < degree; ++k)
00380     result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
00381                            .template triangularView<Upper>().solve(TminusI);
00382 }
00383 
00384 template <typename MatrixType>
00385 void MatrixLogarithmAtomic<MatrixType>::computePade11(MatrixType& result, const MatrixType& T)
00386 {
00387   const int degree = 11;
00388   const RealScalar nodes[]   = { 0.0108856709269715035980309994385713L, 0.0564687001159523504624211153480364L,
00389             0.1349239972129753379532918739844233L, 0.2404519353965940920371371652706952L,
00390             0.3652284220238275138342340072995692L, 0.5000000000000000000000000000000000L,
00391             0.6347715779761724861657659927004308L, 0.7595480646034059079628628347293048L,
00392             0.8650760027870246620467081260155767L, 0.9435312998840476495375788846519636L,
00393             0.9891143290730284964019690005614287L };
00394   const RealScalar weights[] = { 0.0278342835580868332413768602212743L, 0.0627901847324523123173471496119701L,
00395             0.0931451054638671257130488207158280L, 0.1165968822959952399592618524215876L,
00396             0.1314022722551233310903444349452546L, 0.1364625433889503153572417641681711L,
00397             0.1314022722551233310903444349452546L, 0.1165968822959952399592618524215876L,
00398             0.0931451054638671257130488207158280L, 0.0627901847324523123173471496119701L,
00399             0.0278342835580868332413768602212743L };
00400   eigen_assert(degree <= maxPadeDegree);
00401   MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
00402   result.setZero(T.rows(), T.rows());
00403   for (int k = 0; k < degree; ++k)
00404     result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
00405                            .template triangularView<Upper>().solve(TminusI);
00406 }
00407 
00420 template<typename Derived> class MatrixLogarithmReturnValue
00421 : public ReturnByValue<MatrixLogarithmReturnValue<Derived> >
00422 {
00423 public:
00424 
00425   typedef typename Derived::Scalar Scalar;
00426   typedef typename Derived::Index Index;
00427 
00432   MatrixLogarithmReturnValue(const Derived& A) : m_A(A) { }
00433   
00438   template <typename ResultType>
00439   inline void evalTo(ResultType& result) const
00440   {
00441     typedef typename Derived::PlainObject PlainObject;
00442     typedef internal::traits<PlainObject> Traits;
00443     static const int RowsAtCompileTime = Traits::RowsAtCompileTime;
00444     static const int ColsAtCompileTime = Traits::ColsAtCompileTime;
00445     static const int Options = PlainObject::Options;
00446     typedef std::complex<typename NumTraits<Scalar>::Real> ComplexScalar;
00447     typedef Matrix<ComplexScalar, Dynamic, Dynamic, Options, RowsAtCompileTime, ColsAtCompileTime> DynMatrixType;
00448     typedef MatrixLogarithmAtomic<DynMatrixType> AtomicType;
00449     AtomicType atomic;
00450     
00451     const PlainObject Aevaluated = m_A.eval();
00452     MatrixFunction<PlainObject, AtomicType> mf(Aevaluated, atomic);
00453     mf.compute(result);
00454   }
00455 
00456   Index rows() const { return m_A.rows(); }
00457   Index cols() const { return m_A.cols(); }
00458   
00459 private:
00460   typename internal::nested<Derived>::type m_A;
00461   
00462   MatrixLogarithmReturnValue& operator=(const MatrixLogarithmReturnValue&);
00463 };
00464 
00465 namespace internal {
00466   template<typename Derived>
00467   struct traits<MatrixLogarithmReturnValue<Derived> >
00468   {
00469     typedef typename Derived::PlainObject ReturnType;
00470   };
00471 }
00472 
00473 
00474 /********** MatrixBase method **********/
00475 
00476 
00477 template <typename Derived>
00478 const MatrixLogarithmReturnValue<Derived> MatrixBase<Derived>::log() const
00479 {
00480   eigen_assert(rows() == cols());
00481   return MatrixLogarithmReturnValue<Derived>(derived());
00482 }
00483 
00484 } // end namespace Eigen
00485 
00486 #endif // EIGEN_MATRIX_LOGARITHM


acado
Author(s): Milan Vukov, Rien Quirynen
autogenerated on Sat Jun 8 2019 19:38:07