1// This file is part of Eigen, a lightweight C++ template library
2// for linear algebra.
3//
4// Copyright (C) 2011 Jitse Niesen <jitse@maths.leeds.ac.uk>
5// Copyright (C) 2011 Chen-Pang He <jdh8@ms63.hinet.net>
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
18namespace Eigen {
19
20/** \ingroup MatrixFunctions_Module
21  * \class MatrixLogarithmAtomic
22  * \brief Helper class for computing matrix logarithm of atomic matrices.
23  *
24  * \internal
25  * Here, an atomic matrix is a triangular matrix whose diagonal
26  * entries are close to each other.
27  *
28  * \sa class MatrixFunctionAtomic, MatrixBase::log()
29  */
30template <typename MatrixType>
31class MatrixLogarithmAtomic
32{
33public:
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
41  /** \brief Constructor. */
42  MatrixLogarithmAtomic() { }
43
44  /** \brief Compute matrix logarithm of atomic matrix
45    * \param[in]  A  argument of matrix logarithm, should be upper triangular and atomic
46    * \returns  The logarithm of \p A.
47    */
48  MatrixType compute(const MatrixType& A);
49
50private:
51
52  void compute2x2(const MatrixType& A, MatrixType& result);
53  void computeBig(const MatrixType& A, MatrixType& result);
54  static Scalar atanh(Scalar x);
55  int getPadeDegree(float normTminusI);
56  int getPadeDegree(double normTminusI);
57  int getPadeDegree(long double normTminusI);
58  void computePade(MatrixType& result, const MatrixType& T, int degree);
59  void computePade3(MatrixType& result, const MatrixType& T);
60  void computePade4(MatrixType& result, const MatrixType& T);
61  void computePade5(MatrixType& result, const MatrixType& T);
62  void computePade6(MatrixType& result, const MatrixType& T);
63  void computePade7(MatrixType& result, const MatrixType& T);
64  void computePade8(MatrixType& result, const MatrixType& T);
65  void computePade9(MatrixType& result, const MatrixType& T);
66  void computePade10(MatrixType& result, const MatrixType& T);
67  void computePade11(MatrixType& result, const MatrixType& T);
68
69  static const int minPadeDegree = 3;
70  static const int maxPadeDegree = std::numeric_limits<RealScalar>::digits<= 24?  5:      // single precision
71                                   std::numeric_limits<RealScalar>::digits<= 53?  7:      // double precision
72                                   std::numeric_limits<RealScalar>::digits<= 64?  8:      // extended precision
73                                   std::numeric_limits<RealScalar>::digits<=106? 10: 11;  // double-double or quadruple precision
74
75  // Prevent copying
76  MatrixLogarithmAtomic(const MatrixLogarithmAtomic&);
77  MatrixLogarithmAtomic& operator=(const MatrixLogarithmAtomic&);
78};
79
80/** \brief Compute logarithm of triangular matrix with clustered eigenvalues. */
81template <typename MatrixType>
82MatrixType 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
95/** \brief Compute atanh (inverse hyperbolic tangent). */
96template <typename MatrixType>
97typename MatrixType::Scalar MatrixLogarithmAtomic<MatrixType>::atanh(typename MatrixType::Scalar x)
98{
99  using std::abs;
100  using std::sqrt;
101  if (abs(x) > sqrt(NumTraits<Scalar>::epsilon()))
102    return Scalar(0.5) * log((Scalar(1) + x) / (Scalar(1) - x));
103  else
104    return x + x*x*x / Scalar(3);
105}
106
107/** \brief Compute logarithm of 2x2 triangular matrix. */
108template <typename MatrixType>
109void MatrixLogarithmAtomic<MatrixType>::compute2x2(const MatrixType& A, MatrixType& result)
110{
111  using std::abs;
112  using std::ceil;
113  using std::imag;
114  using std::log;
115
116  Scalar logA00 = log(A(0,0));
117  Scalar logA11 = log(A(1,1));
118
119  result(0,0) = logA00;
120  result(1,0) = Scalar(0);
121  result(1,1) = logA11;
122
123  if (A(0,0) == A(1,1)) {
124    result(0,1) = A(0,1) / A(0,0);
125  } else if ((abs(A(0,0)) < 0.5*abs(A(1,1))) || (abs(A(0,0)) > 2*abs(A(1,1)))) {
126    result(0,1) = A(0,1) * (logA11 - logA00) / (A(1,1) - A(0,0));
127  } else {
128    // computation in previous branch is inaccurate if A(1,1) \approx A(0,0)
129    int unwindingNumber = static_cast<int>(ceil((imag(logA11 - logA00) - M_PI) / (2*M_PI)));
130    Scalar z = (A(1,1) - A(0,0)) / (A(1,1) + A(0,0));
131    result(0,1) = A(0,1) * (Scalar(2) * atanh(z) + Scalar(0,2*M_PI*unwindingNumber)) / (A(1,1) - A(0,0));
132  }
133}
134
135/** \brief Compute logarithm of triangular matrices with size > 2.
136  * \details This uses a inverse scale-and-square algorithm. */
137template <typename MatrixType>
138void MatrixLogarithmAtomic<MatrixType>::computeBig(const MatrixType& A, MatrixType& result)
139{
140  int numberOfSquareRoots = 0;
141  int numberOfExtraSquareRoots = 0;
142  int degree;
143  MatrixType T = A;
144  const RealScalar maxNormForPade = maxPadeDegree<= 5? 5.3149729967117310e-1:                     // single precision
145                                    maxPadeDegree<= 7? 2.6429608311114350e-1:                     // double precision
146                                    maxPadeDegree<= 8? 2.32777776523703892094e-1L:                // extended precision
147                                    maxPadeDegree<=10? 1.05026503471351080481093652651105e-1L:    // double-double
148                                                       1.1880960220216759245467951592883642e-1L;  // quadruple precision
149
150  while (true) {
151    RealScalar normTminusI = (T - MatrixType::Identity(T.rows(), T.rows())).cwiseAbs().colwise().sum().maxCoeff();
152    if (normTminusI < maxNormForPade) {
153      degree = getPadeDegree(normTminusI);
154      int degree2 = getPadeDegree(normTminusI / RealScalar(2));
155      if ((degree - degree2 <= 1) || (numberOfExtraSquareRoots == 1))
156	break;
157      ++numberOfExtraSquareRoots;
158    }
159    MatrixType sqrtT;
160    MatrixSquareRootTriangular<MatrixType>(T).compute(sqrtT);
161    T = sqrtT;
162    ++numberOfSquareRoots;
163  }
164
165  computePade(result, T, degree);
166  result *= pow(RealScalar(2), numberOfSquareRoots);
167}
168
169/* \brief Get suitable degree for Pade approximation. (specialized for RealScalar = float) */
170template <typename MatrixType>
171int MatrixLogarithmAtomic<MatrixType>::getPadeDegree(float normTminusI)
172{
173  const float maxNormForPade[] = { 2.5111573934555054e-1 /* degree = 3 */ , 4.0535837411880493e-1,
174            5.3149729967117310e-1 };
175  for (int degree = 3; degree <= maxPadeDegree; ++degree)
176    if (normTminusI <= maxNormForPade[degree - minPadeDegree])
177      return degree;
178  assert(false); // this line should never be reached
179}
180
181/* \brief Get suitable degree for Pade approximation. (specialized for RealScalar = double) */
182template <typename MatrixType>
183int MatrixLogarithmAtomic<MatrixType>::getPadeDegree(double normTminusI)
184{
185  const double maxNormForPade[] = { 1.6206284795015624e-2 /* degree = 3 */ , 5.3873532631381171e-2,
186            1.1352802267628681e-1, 1.8662860613541288e-1, 2.642960831111435e-1 };
187  for (int degree = 3; degree <= maxPadeDegree; ++degree)
188    if (normTminusI <= maxNormForPade[degree - minPadeDegree])
189      return degree;
190  assert(false); // this line should never be reached
191}
192
193/* \brief Get suitable degree for Pade approximation. (specialized for RealScalar = long double) */
194template <typename MatrixType>
195int MatrixLogarithmAtomic<MatrixType>::getPadeDegree(long double normTminusI)
196{
197#if   LDBL_MANT_DIG == 53         // double precision
198  const double maxNormForPade[] = { 1.6206284795015624e-2 /* degree = 3 */ , 5.3873532631381171e-2,
199            1.1352802267628681e-1, 1.8662860613541288e-1, 2.642960831111435e-1 };
200#elif LDBL_MANT_DIG <= 64         // extended precision
201  const double maxNormForPade[] = { 5.48256690357782863103e-3 /* degree = 3 */, 2.34559162387971167321e-2,
202            5.84603923897347449857e-2, 1.08486423756725170223e-1, 1.68385767881294446649e-1,
203            2.32777776523703892094e-1 };
204#elif LDBL_MANT_DIG <= 106        // double-double
205  const double maxNormForPade[] = { 8.58970550342939562202529664318890e-5 /* degree = 3 */,
206            9.34074328446359654039446552677759e-4, 4.26117194647672175773064114582860e-3,
207            1.21546224740281848743149666560464e-2, 2.61100544998339436713088248557444e-2,
208            4.66170074627052749243018566390567e-2, 7.32585144444135027565872014932387e-2,
209            1.05026503471351080481093652651105e-1 };
210#else                             // quadruple precision
211  const double maxNormForPade[] = { 4.7419931187193005048501568167858103e-5 /* degree = 3 */,
212            5.8853168473544560470387769480192666e-4, 2.9216120366601315391789493628113520e-3,
213            8.8415758124319434347116734705174308e-3, 1.9850836029449446668518049562565291e-2,
214            3.6688019729653446926585242192447447e-2, 5.9290962294020186998954055264528393e-2,
215            8.6998436081634343903250580992127677e-2, 1.1880960220216759245467951592883642e-1 };
216#endif
217  for (int degree = 3; degree <= maxPadeDegree; ++degree)
218    if (normTminusI <= maxNormForPade[degree - minPadeDegree])
219      return degree;
220  assert(false); // this line should never be reached
221}
222
223/* \brief Compute Pade approximation to matrix logarithm */
224template <typename MatrixType>
225void MatrixLogarithmAtomic<MatrixType>::computePade(MatrixType& result, const MatrixType& T, int degree)
226{
227  switch (degree) {
228    case 3:  computePade3(result, T);  break;
229    case 4:  computePade4(result, T);  break;
230    case 5:  computePade5(result, T);  break;
231    case 6:  computePade6(result, T);  break;
232    case 7:  computePade7(result, T);  break;
233    case 8:  computePade8(result, T);  break;
234    case 9:  computePade9(result, T);  break;
235    case 10: computePade10(result, T); break;
236    case 11: computePade11(result, T); break;
237    default: assert(false); // should never happen
238  }
239}
240
241template <typename MatrixType>
242void MatrixLogarithmAtomic<MatrixType>::computePade3(MatrixType& result, const MatrixType& T)
243{
244  const int degree = 3;
245  const RealScalar nodes[]   = { 0.1127016653792583114820734600217600L, 0.5000000000000000000000000000000000L,
246            0.8872983346207416885179265399782400L };
247  const RealScalar weights[] = { 0.2777777777777777777777777777777778L, 0.4444444444444444444444444444444444L,
248            0.2777777777777777777777777777777778L };
249  assert(degree <= maxPadeDegree);
250  MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
251  result.setZero(T.rows(), T.rows());
252  for (int k = 0; k < degree; ++k)
253    result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
254                           .template triangularView<Upper>().solve(TminusI);
255}
256
257template <typename MatrixType>
258void MatrixLogarithmAtomic<MatrixType>::computePade4(MatrixType& result, const MatrixType& T)
259{
260  const int degree = 4;
261  const RealScalar nodes[]   = { 0.0694318442029737123880267555535953L, 0.3300094782075718675986671204483777L,
262            0.6699905217924281324013328795516223L, 0.9305681557970262876119732444464048L };
263  const RealScalar weights[] = { 0.1739274225687269286865319746109997L, 0.3260725774312730713134680253890003L,
264            0.3260725774312730713134680253890003L, 0.1739274225687269286865319746109997L };
265  assert(degree <= maxPadeDegree);
266  MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
267  result.setZero(T.rows(), T.rows());
268  for (int k = 0; k < degree; ++k)
269    result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
270                           .template triangularView<Upper>().solve(TminusI);
271}
272
273template <typename MatrixType>
274void MatrixLogarithmAtomic<MatrixType>::computePade5(MatrixType& result, const MatrixType& T)
275{
276  const int degree = 5;
277  const RealScalar nodes[]   = { 0.0469100770306680036011865608503035L, 0.2307653449471584544818427896498956L,
278            0.5000000000000000000000000000000000L, 0.7692346550528415455181572103501044L,
279            0.9530899229693319963988134391496965L };
280  const RealScalar weights[] = { 0.1184634425280945437571320203599587L, 0.2393143352496832340206457574178191L,
281            0.2844444444444444444444444444444444L, 0.2393143352496832340206457574178191L,
282            0.1184634425280945437571320203599587L };
283  assert(degree <= maxPadeDegree);
284  MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
285  result.setZero(T.rows(), T.rows());
286  for (int k = 0; k < degree; ++k)
287    result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
288                           .template triangularView<Upper>().solve(TminusI);
289}
290
291template <typename MatrixType>
292void MatrixLogarithmAtomic<MatrixType>::computePade6(MatrixType& result, const MatrixType& T)
293{
294  const int degree = 6;
295  const RealScalar nodes[]   = { 0.0337652428984239860938492227530027L, 0.1693953067668677431693002024900473L,
296            0.3806904069584015456847491391596440L, 0.6193095930415984543152508608403560L,
297		        0.8306046932331322568306997975099527L, 0.9662347571015760139061507772469973L };
298  const RealScalar weights[] = { 0.0856622461895851725201480710863665L, 0.1803807865240693037849167569188581L,
299            0.2339569672863455236949351719947755L, 0.2339569672863455236949351719947755L,
300 		        0.1803807865240693037849167569188581L, 0.0856622461895851725201480710863665L };
301  assert(degree <= maxPadeDegree);
302  MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
303  result.setZero(T.rows(), T.rows());
304  for (int k = 0; k < degree; ++k)
305    result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
306                           .template triangularView<Upper>().solve(TminusI);
307}
308
309template <typename MatrixType>
310void MatrixLogarithmAtomic<MatrixType>::computePade7(MatrixType& result, const MatrixType& T)
311{
312  const int degree = 7;
313  const RealScalar nodes[]   = { 0.0254460438286207377369051579760744L, 0.1292344072003027800680676133596058L,
314            0.2970774243113014165466967939615193L, 0.5000000000000000000000000000000000L,
315            0.7029225756886985834533032060384807L, 0.8707655927996972199319323866403942L,
316            0.9745539561713792622630948420239256L };
317  const RealScalar weights[] = { 0.0647424830844348466353057163395410L, 0.1398526957446383339507338857118898L,
318            0.1909150252525594724751848877444876L, 0.2089795918367346938775510204081633L,
319            0.1909150252525594724751848877444876L, 0.1398526957446383339507338857118898L,
320            0.0647424830844348466353057163395410L };
321  assert(degree <= maxPadeDegree);
322  MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
323  result.setZero(T.rows(), T.rows());
324  for (int k = 0; k < degree; ++k)
325    result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
326                           .template triangularView<Upper>().solve(TminusI);
327}
328
329template <typename MatrixType>
330void MatrixLogarithmAtomic<MatrixType>::computePade8(MatrixType& result, const MatrixType& T)
331{
332  const int degree = 8;
333  const RealScalar nodes[]   = { 0.0198550717512318841582195657152635L, 0.1016667612931866302042230317620848L,
334            0.2372337950418355070911304754053768L, 0.4082826787521750975302619288199080L,
335            0.5917173212478249024697380711800920L, 0.7627662049581644929088695245946232L,
336            0.8983332387068133697957769682379152L, 0.9801449282487681158417804342847365L };
337  const RealScalar weights[] = { 0.0506142681451881295762656771549811L, 0.1111905172266872352721779972131204L,
338            0.1568533229389436436689811009933007L, 0.1813418916891809914825752246385978L,
339            0.1813418916891809914825752246385978L, 0.1568533229389436436689811009933007L,
340            0.1111905172266872352721779972131204L, 0.0506142681451881295762656771549811L };
341  assert(degree <= maxPadeDegree);
342  MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
343  result.setZero(T.rows(), T.rows());
344  for (int k = 0; k < degree; ++k)
345    result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
346                           .template triangularView<Upper>().solve(TminusI);
347}
348
349template <typename MatrixType>
350void MatrixLogarithmAtomic<MatrixType>::computePade9(MatrixType& result, const MatrixType& T)
351{
352  const int degree = 9;
353  const RealScalar nodes[]   = { 0.0159198802461869550822118985481636L, 0.0819844463366821028502851059651326L,
354            0.1933142836497048013456489803292629L, 0.3378732882980955354807309926783317L,
355            0.5000000000000000000000000000000000L, 0.6621267117019044645192690073216683L,
356            0.8066857163502951986543510196707371L, 0.9180155536633178971497148940348674L,
357            0.9840801197538130449177881014518364L };
358  const RealScalar weights[] = { 0.0406371941807872059859460790552618L, 0.0903240803474287020292360156214564L,
359            0.1303053482014677311593714347093164L, 0.1561735385200014200343152032922218L,
360            0.1651196775006298815822625346434870L, 0.1561735385200014200343152032922218L,
361            0.1303053482014677311593714347093164L, 0.0903240803474287020292360156214564L,
362            0.0406371941807872059859460790552618L };
363  assert(degree <= maxPadeDegree);
364  MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
365  result.setZero(T.rows(), T.rows());
366  for (int k = 0; k < degree; ++k)
367    result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
368                           .template triangularView<Upper>().solve(TminusI);
369}
370
371template <typename MatrixType>
372void MatrixLogarithmAtomic<MatrixType>::computePade10(MatrixType& result, const MatrixType& T)
373{
374  const int degree = 10;
375  const RealScalar nodes[]   = { 0.0130467357414141399610179939577740L, 0.0674683166555077446339516557882535L,
376            0.1602952158504877968828363174425632L, 0.2833023029353764046003670284171079L,
377            0.4255628305091843945575869994351400L, 0.5744371694908156054424130005648600L,
378            0.7166976970646235953996329715828921L, 0.8397047841495122031171636825574368L,
379            0.9325316833444922553660483442117465L, 0.9869532642585858600389820060422260L };
380  const RealScalar weights[] = { 0.0333356721543440687967844049466659L, 0.0747256745752902965728881698288487L,
381            0.1095431812579910219977674671140816L, 0.1346333596549981775456134607847347L,
382            0.1477621123573764350869464973256692L, 0.1477621123573764350869464973256692L,
383            0.1346333596549981775456134607847347L, 0.1095431812579910219977674671140816L,
384            0.0747256745752902965728881698288487L, 0.0333356721543440687967844049466659L };
385  assert(degree <= maxPadeDegree);
386  MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
387  result.setZero(T.rows(), T.rows());
388  for (int k = 0; k < degree; ++k)
389    result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
390                           .template triangularView<Upper>().solve(TminusI);
391}
392
393template <typename MatrixType>
394void MatrixLogarithmAtomic<MatrixType>::computePade11(MatrixType& result, const MatrixType& T)
395{
396  const int degree = 11;
397  const RealScalar nodes[]   = { 0.0108856709269715035980309994385713L, 0.0564687001159523504624211153480364L,
398            0.1349239972129753379532918739844233L, 0.2404519353965940920371371652706952L,
399            0.3652284220238275138342340072995692L, 0.5000000000000000000000000000000000L,
400            0.6347715779761724861657659927004308L, 0.7595480646034059079628628347293048L,
401            0.8650760027870246620467081260155767L, 0.9435312998840476495375788846519636L,
402            0.9891143290730284964019690005614287L };
403  const RealScalar weights[] = { 0.0278342835580868332413768602212743L, 0.0627901847324523123173471496119701L,
404            0.0931451054638671257130488207158280L, 0.1165968822959952399592618524215876L,
405            0.1314022722551233310903444349452546L, 0.1364625433889503153572417641681711L,
406            0.1314022722551233310903444349452546L, 0.1165968822959952399592618524215876L,
407            0.0931451054638671257130488207158280L, 0.0627901847324523123173471496119701L,
408            0.0278342835580868332413768602212743L };
409  assert(degree <= maxPadeDegree);
410  MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
411  result.setZero(T.rows(), T.rows());
412  for (int k = 0; k < degree; ++k)
413    result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
414                           .template triangularView<Upper>().solve(TminusI);
415}
416
417/** \ingroup MatrixFunctions_Module
418  *
419  * \brief Proxy for the matrix logarithm of some matrix (expression).
420  *
421  * \tparam Derived  Type of the argument to the matrix function.
422  *
423  * This class holds the argument to the matrix function until it is
424  * assigned or evaluated for some other reason (so the argument
425  * should not be changed in the meantime). It is the return type of
426  * matrixBase::matrixLogarithm() and most of the time this is the
427  * only way it is used.
428  */
429template<typename Derived> class MatrixLogarithmReturnValue
430: public ReturnByValue<MatrixLogarithmReturnValue<Derived> >
431{
432public:
433
434  typedef typename Derived::Scalar Scalar;
435  typedef typename Derived::Index Index;
436
437  /** \brief Constructor.
438    *
439    * \param[in]  A  %Matrix (expression) forming the argument of the matrix logarithm.
440    */
441  MatrixLogarithmReturnValue(const Derived& A) : m_A(A) { }
442
443  /** \brief Compute the matrix logarithm.
444    *
445    * \param[out]  result  Logarithm of \p A, where \A is as specified in the constructor.
446    */
447  template <typename ResultType>
448  inline void evalTo(ResultType& result) const
449  {
450    typedef typename Derived::PlainObject PlainObject;
451    typedef internal::traits<PlainObject> Traits;
452    static const int RowsAtCompileTime = Traits::RowsAtCompileTime;
453    static const int ColsAtCompileTime = Traits::ColsAtCompileTime;
454    static const int Options = PlainObject::Options;
455    typedef std::complex<typename NumTraits<Scalar>::Real> ComplexScalar;
456    typedef Matrix<ComplexScalar, Dynamic, Dynamic, Options, RowsAtCompileTime, ColsAtCompileTime> DynMatrixType;
457    typedef MatrixLogarithmAtomic<DynMatrixType> AtomicType;
458    AtomicType atomic;
459
460    const PlainObject Aevaluated = m_A.eval();
461    MatrixFunction<PlainObject, AtomicType> mf(Aevaluated, atomic);
462    mf.compute(result);
463  }
464
465  Index rows() const { return m_A.rows(); }
466  Index cols() const { return m_A.cols(); }
467
468private:
469  typename internal::nested<Derived>::type m_A;
470
471  MatrixLogarithmReturnValue& operator=(const MatrixLogarithmReturnValue&);
472};
473
474namespace internal {
475  template<typename Derived>
476  struct traits<MatrixLogarithmReturnValue<Derived> >
477  {
478    typedef typename Derived::PlainObject ReturnType;
479  };
480}
481
482
483/********** MatrixBase method **********/
484
485
486template <typename Derived>
487const MatrixLogarithmReturnValue<Derived> MatrixBase<Derived>::log() const
488{
489  eigen_assert(rows() == cols());
490  return MatrixLogarithmReturnValue<Derived>(derived());
491}
492
493} // end namespace Eigen
494
495#endif // EIGEN_MATRIX_LOGARITHM
496