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MatrixLogarithm.h
1 // This file is part of Eigen, a lightweight C++ template library
2 // for linear algebra.
3 //
4 // Copyright (C) 2011 Jitse Niesen <[email protected]>
5 // Copyright (C) 2011 Chen-Pang He <[email protected]>
6 //
7 // This Source Code Form is subject to the terms of the Mozilla
8 // Public License v. 2.0. If a copy of the MPL was not distributed
9 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
10 
11 #ifndef EIGEN_MATRIX_LOGARITHM
12 #define EIGEN_MATRIX_LOGARITHM
13 
14 #ifndef M_PI
15 #define M_PI 3.141592653589793238462643383279503L
16 #endif
17 
18 namespace Eigen {
19 
30 template <typename MatrixType>
31 class MatrixLogarithmAtomic
32 {
33 public:
34 
35  typedef typename MatrixType::Scalar Scalar;
36  // typedef typename MatrixType::Index Index;
37  typedef typename NumTraits<Scalar>::Real RealScalar;
38  // typedef typename internal::stem_function<Scalar>::type StemFunction;
39  // typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, 1> VectorType;
40 
42  MatrixLogarithmAtomic() { }
43 
48  MatrixType compute(const MatrixType& A);
49 
50 private:
51 
52  void compute2x2(const MatrixType& A, MatrixType& result);
53  void computeBig(const MatrixType& A, MatrixType& result);
54  int getPadeDegree(float normTminusI);
55  int getPadeDegree(double normTminusI);
56  int getPadeDegree(long double normTminusI);
57  void computePade(MatrixType& result, const MatrixType& T, int degree);
58  void computePade3(MatrixType& result, const MatrixType& T);
59  void computePade4(MatrixType& result, const MatrixType& T);
60  void computePade5(MatrixType& result, const MatrixType& T);
61  void computePade6(MatrixType& result, const MatrixType& T);
62  void computePade7(MatrixType& result, const MatrixType& T);
63  void computePade8(MatrixType& result, const MatrixType& T);
64  void computePade9(MatrixType& result, const MatrixType& T);
65  void computePade10(MatrixType& result, const MatrixType& T);
66  void computePade11(MatrixType& result, const MatrixType& T);
67 
68  static const int minPadeDegree = 3;
69  static const int maxPadeDegree = std::numeric_limits<RealScalar>::digits<= 24? 5: // single precision
70  std::numeric_limits<RealScalar>::digits<= 53? 7: // double precision
71  std::numeric_limits<RealScalar>::digits<= 64? 8: // extended precision
72  std::numeric_limits<RealScalar>::digits<=106? 10: // double-double
73  11; // quadruple precision
74 
75  // Prevent copying
76  MatrixLogarithmAtomic(const MatrixLogarithmAtomic&);
77  MatrixLogarithmAtomic& operator=(const MatrixLogarithmAtomic&);
78 };
79 
81 template <typename MatrixType>
82 MatrixType MatrixLogarithmAtomic<MatrixType>::compute(const MatrixType& A)
83 {
84  using std::log;
85  MatrixType result(A.rows(), A.rows());
86  if (A.rows() == 1)
87  result(0,0) = log(A(0,0));
88  else if (A.rows() == 2)
89  compute2x2(A, result);
90  else
91  computeBig(A, result);
92  return result;
93 }
94 
96 template <typename MatrixType>
97 void MatrixLogarithmAtomic<MatrixType>::compute2x2(const MatrixType& A, MatrixType& result)
98 {
99  using std::abs;
100  using std::ceil;
101  using std::imag;
102  using std::log;
103 
104  Scalar logA00 = log(A(0,0));
105  Scalar logA11 = log(A(1,1));
106 
107  result(0,0) = logA00;
108  result(1,0) = Scalar(0);
109  result(1,1) = logA11;
110 
111  if (A(0,0) == A(1,1)) {
112  result(0,1) = A(0,1) / A(0,0);
113  } else if ((abs(A(0,0)) < 0.5*abs(A(1,1))) || (abs(A(0,0)) > 2*abs(A(1,1)))) {
114  result(0,1) = A(0,1) * (logA11 - logA00) / (A(1,1) - A(0,0));
115  } else {
116  // computation in previous branch is inaccurate if A(1,1) \approx A(0,0)
117  int unwindingNumber = static_cast<int>(ceil((imag(logA11 - logA00) - M_PI) / (2*M_PI)));
118  Scalar y = A(1,1) - A(0,0), x = A(1,1) + A(0,0);
119  result(0,1) = A(0,1) * (Scalar(2) * numext::atanh2(y,x) + Scalar(0,2*M_PI*unwindingNumber)) / y;
120  }
121 }
122 
125 template <typename MatrixType>
126 void MatrixLogarithmAtomic<MatrixType>::computeBig(const MatrixType& A, MatrixType& result)
127 {
128  using std::pow;
129  int numberOfSquareRoots = 0;
130  int numberOfExtraSquareRoots = 0;
131  int degree;
132  MatrixType T = A, sqrtT;
133  const RealScalar maxNormForPade = maxPadeDegree<= 5? 5.3149729967117310e-1: // single precision
134  maxPadeDegree<= 7? 2.6429608311114350e-1: // double precision
135  maxPadeDegree<= 8? 2.32777776523703892094e-1L: // extended precision
136  maxPadeDegree<=10? 1.05026503471351080481093652651105e-1L: // double-double
137  1.1880960220216759245467951592883642e-1L; // quadruple precision
138 
139  while (true) {
140  RealScalar normTminusI = (T - MatrixType::Identity(T.rows(), T.rows())).cwiseAbs().colwise().sum().maxCoeff();
141  if (normTminusI < maxNormForPade) {
142  degree = getPadeDegree(normTminusI);
143  int degree2 = getPadeDegree(normTminusI / RealScalar(2));
144  if ((degree - degree2 <= 1) || (numberOfExtraSquareRoots == 1))
145  break;
146  ++numberOfExtraSquareRoots;
147  }
148  MatrixSquareRootTriangular<MatrixType>(T).compute(sqrtT);
149  T = sqrtT.template triangularView<Upper>();
150  ++numberOfSquareRoots;
151  }
152 
153  computePade(result, T, degree);
154  result *= pow(RealScalar(2), numberOfSquareRoots);
155 }
156 
157 /* \brief Get suitable degree for Pade approximation. (specialized for RealScalar = float) */
158 template <typename MatrixType>
159 int MatrixLogarithmAtomic<MatrixType>::getPadeDegree(float normTminusI)
160 {
161  const float maxNormForPade[] = { 2.5111573934555054e-1 /* degree = 3 */ , 4.0535837411880493e-1,
162  5.3149729967117310e-1 };
163  int degree = 3;
164  for (; degree <= maxPadeDegree; ++degree)
165  if (normTminusI <= maxNormForPade[degree - minPadeDegree])
166  break;
167  return degree;
168 }
169 
170 /* \brief Get suitable degree for Pade approximation. (specialized for RealScalar = double) */
171 template <typename MatrixType>
172 int MatrixLogarithmAtomic<MatrixType>::getPadeDegree(double normTminusI)
173 {
174  const double maxNormForPade[] = { 1.6206284795015624e-2 /* degree = 3 */ , 5.3873532631381171e-2,
175  1.1352802267628681e-1, 1.8662860613541288e-1, 2.642960831111435e-1 };
176  int degree = 3;
177  for (; degree <= maxPadeDegree; ++degree)
178  if (normTminusI <= maxNormForPade[degree - minPadeDegree])
179  break;
180  return degree;
181 }
182 
183 /* \brief Get suitable degree for Pade approximation. (specialized for RealScalar = long double) */
184 template <typename MatrixType>
185 int MatrixLogarithmAtomic<MatrixType>::getPadeDegree(long double normTminusI)
186 {
187 #if LDBL_MANT_DIG == 53 // double precision
188  const long double maxNormForPade[] = { 1.6206284795015624e-2L /* degree = 3 */ , 5.3873532631381171e-2L,
189  1.1352802267628681e-1L, 1.8662860613541288e-1L, 2.642960831111435e-1L };
190 #elif LDBL_MANT_DIG <= 64 // extended precision
191  const long double maxNormForPade[] = { 5.48256690357782863103e-3L /* degree = 3 */, 2.34559162387971167321e-2L,
192  5.84603923897347449857e-2L, 1.08486423756725170223e-1L, 1.68385767881294446649e-1L,
193  2.32777776523703892094e-1L };
194 #elif LDBL_MANT_DIG <= 106 // double-double
195  const long double maxNormForPade[] = { 8.58970550342939562202529664318890e-5L /* degree = 3 */,
196  9.34074328446359654039446552677759e-4L, 4.26117194647672175773064114582860e-3L,
197  1.21546224740281848743149666560464e-2L, 2.61100544998339436713088248557444e-2L,
198  4.66170074627052749243018566390567e-2L, 7.32585144444135027565872014932387e-2L,
199  1.05026503471351080481093652651105e-1L };
200 #else // quadruple precision
201  const long double maxNormForPade[] = { 4.7419931187193005048501568167858103e-5L /* degree = 3 */,
202  5.8853168473544560470387769480192666e-4L, 2.9216120366601315391789493628113520e-3L,
203  8.8415758124319434347116734705174308e-3L, 1.9850836029449446668518049562565291e-2L,
204  3.6688019729653446926585242192447447e-2L, 5.9290962294020186998954055264528393e-2L,
205  8.6998436081634343903250580992127677e-2L, 1.1880960220216759245467951592883642e-1L };
206 #endif
207  int degree = 3;
208  for (; degree <= maxPadeDegree; ++degree)
209  if (normTminusI <= maxNormForPade[degree - minPadeDegree])
210  break;
211  return degree;
212 }
213 
214 /* \brief Compute Pade approximation to matrix logarithm */
215 template <typename MatrixType>
216 void MatrixLogarithmAtomic<MatrixType>::computePade(MatrixType& result, const MatrixType& T, int degree)
217 {
218  switch (degree) {
219  case 3: computePade3(result, T); break;
220  case 4: computePade4(result, T); break;
221  case 5: computePade5(result, T); break;
222  case 6: computePade6(result, T); break;
223  case 7: computePade7(result, T); break;
224  case 8: computePade8(result, T); break;
225  case 9: computePade9(result, T); break;
226  case 10: computePade10(result, T); break;
227  case 11: computePade11(result, T); break;
228  default: assert(false); // should never happen
229  }
230 }
231 
232 template <typename MatrixType>
233 void MatrixLogarithmAtomic<MatrixType>::computePade3(MatrixType& result, const MatrixType& T)
234 {
235  const int degree = 3;
236  const RealScalar nodes[] = { 0.1127016653792583114820734600217600L, 0.5000000000000000000000000000000000L,
237  0.8872983346207416885179265399782400L };
238  const RealScalar weights[] = { 0.2777777777777777777777777777777778L, 0.4444444444444444444444444444444444L,
239  0.2777777777777777777777777777777778L };
240  eigen_assert(degree <= maxPadeDegree);
241  MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
242  result.setZero(T.rows(), T.rows());
243  for (int k = 0; k < degree; ++k)
244  result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
245  .template triangularView<Upper>().solve(TminusI);
246 }
247 
248 template <typename MatrixType>
249 void MatrixLogarithmAtomic<MatrixType>::computePade4(MatrixType& result, const MatrixType& T)
250 {
251  const int degree = 4;
252  const RealScalar nodes[] = { 0.0694318442029737123880267555535953L, 0.3300094782075718675986671204483777L,
253  0.6699905217924281324013328795516223L, 0.9305681557970262876119732444464048L };
254  const RealScalar weights[] = { 0.1739274225687269286865319746109997L, 0.3260725774312730713134680253890003L,
255  0.3260725774312730713134680253890003L, 0.1739274225687269286865319746109997L };
256  eigen_assert(degree <= maxPadeDegree);
257  MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
258  result.setZero(T.rows(), T.rows());
259  for (int k = 0; k < degree; ++k)
260  result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
261  .template triangularView<Upper>().solve(TminusI);
262 }
263 
264 template <typename MatrixType>
265 void MatrixLogarithmAtomic<MatrixType>::computePade5(MatrixType& result, const MatrixType& T)
266 {
267  const int degree = 5;
268  const RealScalar nodes[] = { 0.0469100770306680036011865608503035L, 0.2307653449471584544818427896498956L,
269  0.5000000000000000000000000000000000L, 0.7692346550528415455181572103501044L,
270  0.9530899229693319963988134391496965L };
271  const RealScalar weights[] = { 0.1184634425280945437571320203599587L, 0.2393143352496832340206457574178191L,
272  0.2844444444444444444444444444444444L, 0.2393143352496832340206457574178191L,
273  0.1184634425280945437571320203599587L };
274  eigen_assert(degree <= maxPadeDegree);
275  MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
276  result.setZero(T.rows(), T.rows());
277  for (int k = 0; k < degree; ++k)
278  result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
279  .template triangularView<Upper>().solve(TminusI);
280 }
281 
282 template <typename MatrixType>
283 void MatrixLogarithmAtomic<MatrixType>::computePade6(MatrixType& result, const MatrixType& T)
284 {
285  const int degree = 6;
286  const RealScalar nodes[] = { 0.0337652428984239860938492227530027L, 0.1693953067668677431693002024900473L,
287  0.3806904069584015456847491391596440L, 0.6193095930415984543152508608403560L,
288  0.8306046932331322568306997975099527L, 0.9662347571015760139061507772469973L };
289  const RealScalar weights[] = { 0.0856622461895851725201480710863665L, 0.1803807865240693037849167569188581L,
290  0.2339569672863455236949351719947755L, 0.2339569672863455236949351719947755L,
291  0.1803807865240693037849167569188581L, 0.0856622461895851725201480710863665L };
292  eigen_assert(degree <= maxPadeDegree);
293  MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
294  result.setZero(T.rows(), T.rows());
295  for (int k = 0; k < degree; ++k)
296  result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
297  .template triangularView<Upper>().solve(TminusI);
298 }
299 
300 template <typename MatrixType>
301 void MatrixLogarithmAtomic<MatrixType>::computePade7(MatrixType& result, const MatrixType& T)
302 {
303  const int degree = 7;
304  const RealScalar nodes[] = { 0.0254460438286207377369051579760744L, 0.1292344072003027800680676133596058L,
305  0.2970774243113014165466967939615193L, 0.5000000000000000000000000000000000L,
306  0.7029225756886985834533032060384807L, 0.8707655927996972199319323866403942L,
307  0.9745539561713792622630948420239256L };
308  const RealScalar weights[] = { 0.0647424830844348466353057163395410L, 0.1398526957446383339507338857118898L,
309  0.1909150252525594724751848877444876L, 0.2089795918367346938775510204081633L,
310  0.1909150252525594724751848877444876L, 0.1398526957446383339507338857118898L,
311  0.0647424830844348466353057163395410L };
312  eigen_assert(degree <= maxPadeDegree);
313  MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
314  result.setZero(T.rows(), T.rows());
315  for (int k = 0; k < degree; ++k)
316  result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
317  .template triangularView<Upper>().solve(TminusI);
318 }
319 
320 template <typename MatrixType>
321 void MatrixLogarithmAtomic<MatrixType>::computePade8(MatrixType& result, const MatrixType& T)
322 {
323  const int degree = 8;
324  const RealScalar nodes[] = { 0.0198550717512318841582195657152635L, 0.1016667612931866302042230317620848L,
325  0.2372337950418355070911304754053768L, 0.4082826787521750975302619288199080L,
326  0.5917173212478249024697380711800920L, 0.7627662049581644929088695245946232L,
327  0.8983332387068133697957769682379152L, 0.9801449282487681158417804342847365L };
328  const RealScalar weights[] = { 0.0506142681451881295762656771549811L, 0.1111905172266872352721779972131204L,
329  0.1568533229389436436689811009933007L, 0.1813418916891809914825752246385978L,
330  0.1813418916891809914825752246385978L, 0.1568533229389436436689811009933007L,
331  0.1111905172266872352721779972131204L, 0.0506142681451881295762656771549811L };
332  eigen_assert(degree <= maxPadeDegree);
333  MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
334  result.setZero(T.rows(), T.rows());
335  for (int k = 0; k < degree; ++k)
336  result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
337  .template triangularView<Upper>().solve(TminusI);
338 }
339 
340 template <typename MatrixType>
341 void MatrixLogarithmAtomic<MatrixType>::computePade9(MatrixType& result, const MatrixType& T)
342 {
343  const int degree = 9;
344  const RealScalar nodes[] = { 0.0159198802461869550822118985481636L, 0.0819844463366821028502851059651326L,
345  0.1933142836497048013456489803292629L, 0.3378732882980955354807309926783317L,
346  0.5000000000000000000000000000000000L, 0.6621267117019044645192690073216683L,
347  0.8066857163502951986543510196707371L, 0.9180155536633178971497148940348674L,
348  0.9840801197538130449177881014518364L };
349  const RealScalar weights[] = { 0.0406371941807872059859460790552618L, 0.0903240803474287020292360156214564L,
350  0.1303053482014677311593714347093164L, 0.1561735385200014200343152032922218L,
351  0.1651196775006298815822625346434870L, 0.1561735385200014200343152032922218L,
352  0.1303053482014677311593714347093164L, 0.0903240803474287020292360156214564L,
353  0.0406371941807872059859460790552618L };
354  eigen_assert(degree <= maxPadeDegree);
355  MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
356  result.setZero(T.rows(), T.rows());
357  for (int k = 0; k < degree; ++k)
358  result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
359  .template triangularView<Upper>().solve(TminusI);
360 }
361 
362 template <typename MatrixType>
363 void MatrixLogarithmAtomic<MatrixType>::computePade10(MatrixType& result, const MatrixType& T)
364 {
365  const int degree = 10;
366  const RealScalar nodes[] = { 0.0130467357414141399610179939577740L, 0.0674683166555077446339516557882535L,
367  0.1602952158504877968828363174425632L, 0.2833023029353764046003670284171079L,
368  0.4255628305091843945575869994351400L, 0.5744371694908156054424130005648600L,
369  0.7166976970646235953996329715828921L, 0.8397047841495122031171636825574368L,
370  0.9325316833444922553660483442117465L, 0.9869532642585858600389820060422260L };
371  const RealScalar weights[] = { 0.0333356721543440687967844049466659L, 0.0747256745752902965728881698288487L,
372  0.1095431812579910219977674671140816L, 0.1346333596549981775456134607847347L,
373  0.1477621123573764350869464973256692L, 0.1477621123573764350869464973256692L,
374  0.1346333596549981775456134607847347L, 0.1095431812579910219977674671140816L,
375  0.0747256745752902965728881698288487L, 0.0333356721543440687967844049466659L };
376  eigen_assert(degree <= maxPadeDegree);
377  MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
378  result.setZero(T.rows(), T.rows());
379  for (int k = 0; k < degree; ++k)
380  result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
381  .template triangularView<Upper>().solve(TminusI);
382 }
383 
384 template <typename MatrixType>
385 void MatrixLogarithmAtomic<MatrixType>::computePade11(MatrixType& result, const MatrixType& T)
386 {
387  const int degree = 11;
388  const RealScalar nodes[] = { 0.0108856709269715035980309994385713L, 0.0564687001159523504624211153480364L,
389  0.1349239972129753379532918739844233L, 0.2404519353965940920371371652706952L,
390  0.3652284220238275138342340072995692L, 0.5000000000000000000000000000000000L,
391  0.6347715779761724861657659927004308L, 0.7595480646034059079628628347293048L,
392  0.8650760027870246620467081260155767L, 0.9435312998840476495375788846519636L,
393  0.9891143290730284964019690005614287L };
394  const RealScalar weights[] = { 0.0278342835580868332413768602212743L, 0.0627901847324523123173471496119701L,
395  0.0931451054638671257130488207158280L, 0.1165968822959952399592618524215876L,
396  0.1314022722551233310903444349452546L, 0.1364625433889503153572417641681711L,
397  0.1314022722551233310903444349452546L, 0.1165968822959952399592618524215876L,
398  0.0931451054638671257130488207158280L, 0.0627901847324523123173471496119701L,
399  0.0278342835580868332413768602212743L };
400  eigen_assert(degree <= maxPadeDegree);
401  MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
402  result.setZero(T.rows(), T.rows());
403  for (int k = 0; k < degree; ++k)
404  result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
405  .template triangularView<Upper>().solve(TminusI);
406 }
407 
420 template<typename Derived> class MatrixLogarithmReturnValue
421 : public ReturnByValue<MatrixLogarithmReturnValue<Derived> >
422 {
423 public:
424 
425  typedef typename Derived::Scalar Scalar;
426  typedef typename Derived::Index Index;
427 
432  MatrixLogarithmReturnValue(const Derived& A) : m_A(A) { }
433 
438  template <typename ResultType>
439  inline void evalTo(ResultType& result) const
440  {
441  typedef typename Derived::PlainObject PlainObject;
442  typedef internal::traits<PlainObject> Traits;
443  static const int RowsAtCompileTime = Traits::RowsAtCompileTime;
444  static const int ColsAtCompileTime = Traits::ColsAtCompileTime;
445  static const int Options = PlainObject::Options;
446  typedef std::complex<typename NumTraits<Scalar>::Real> ComplexScalar;
447  typedef Matrix<ComplexScalar, Dynamic, Dynamic, Options, RowsAtCompileTime, ColsAtCompileTime> DynMatrixType;
448  typedef MatrixLogarithmAtomic<DynMatrixType> AtomicType;
449  AtomicType atomic;
450 
451  const PlainObject Aevaluated = m_A.eval();
452  MatrixFunction<PlainObject, AtomicType> mf(Aevaluated, atomic);
453  mf.compute(result);
454  }
455 
456  Index rows() const { return m_A.rows(); }
457  Index cols() const { return m_A.cols(); }
458 
459 private:
460  typename internal::nested<Derived>::type m_A;
461 
462  MatrixLogarithmReturnValue& operator=(const MatrixLogarithmReturnValue&);
463 };
464 
465 namespace internal {
466  template<typename Derived>
467  struct traits<MatrixLogarithmReturnValue<Derived> >
468  {
469  typedef typename Derived::PlainObject ReturnType;
470  };
471 }
472 
473 
474 /********** MatrixBase method **********/
475 
476 
477 template <typename Derived>
478 const MatrixLogarithmReturnValue<Derived> MatrixBase<Derived>::log() const
479 {
480  eigen_assert(rows() == cols());
481  return MatrixLogarithmReturnValue<Derived>(derived());
482 }
483 
484 } // end namespace Eigen
485 
486 #endif // EIGEN_MATRIX_LOGARITHM
Eigen::MatrixLogarithmReturnValue::evalTo
void evalTo(ResultType &result) const
Compute the matrix logarithm.
Definition: MatrixLogarithm.h:439
Eigen::MatrixFunction
Class for computing matrix functions.
Definition: MatrixFunction.h:37
Eigen::MatrixLogarithmAtomic::compute
MatrixType compute(const MatrixType &A)
Compute matrix logarithm of atomic matrix.
Definition: MatrixLogarithm.h:82
Eigen::MatrixFunction::compute
void compute(ResultType &result)
Compute the matrix function.
Eigen::MatrixLogarithmAtomic
Helper class for computing matrix logarithm of atomic matrices.
Definition: MatrixLogarithm.h:31
Eigen::MatrixLogarithmReturnValue::MatrixLogarithmReturnValue
MatrixLogarithmReturnValue(const Derived &A)
Constructor.
Definition: MatrixLogarithm.h:432
Eigen::MatrixLogarithmReturnValue
Proxy for the matrix logarithm of some matrix (expression).
Definition: MatrixLogarithm.h:420
Eigen::MatrixLogarithmAtomic::MatrixLogarithmAtomic
MatrixLogarithmAtomic()
Constructor.
Definition: MatrixLogarithm.h:42