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 <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 
18 namespace Eigen {
19 
30 template <typename MatrixType>
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 
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  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
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 typename 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 
108 template <typename MatrixType>
109 void 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 
137 template <typename MatrixType>
138 void 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) */
170 template <typename MatrixType>
171 int 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) */
182 template <typename MatrixType>
183 int 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) */
194 template <typename MatrixType>
195 int MatrixLogarithmAtomic<MatrixType>::getPadeDegree(long double normTminusI)
196 {
197 #if LDBL_MANT_DIG == 53 // double precision
198  const long double maxNormForPade[] = { 1.6206284795015624e-2L /* degree = 3 */ , 5.3873532631381171e-2L,
199  1.1352802267628681e-1L, 1.8662860613541288e-1L, 2.642960831111435e-1L };
200 #elif LDBL_MANT_DIG <= 64 // extended precision
201  const long double maxNormForPade[] = { 5.48256690357782863103e-3L /* degree = 3 */, 2.34559162387971167321e-2L,
202  5.84603923897347449857e-2L, 1.08486423756725170223e-1L, 1.68385767881294446649e-1L,
203  2.32777776523703892094e-1L };
204 #elif LDBL_MANT_DIG <= 106 // double-double
205  const long double maxNormForPade[] = { 8.58970550342939562202529664318890e-5L /* degree = 3 */,
206  9.34074328446359654039446552677759e-4L, 4.26117194647672175773064114582860e-3L,
207  1.21546224740281848743149666560464e-2L, 2.61100544998339436713088248557444e-2L,
208  4.66170074627052749243018566390567e-2L, 7.32585144444135027565872014932387e-2L,
209  1.05026503471351080481093652651105e-1L };
210 #else // quadruple precision
211  const long double maxNormForPade[] = { 4.7419931187193005048501568167858103e-5L /* degree = 3 */,
212  5.8853168473544560470387769480192666e-4L, 2.9216120366601315391789493628113520e-3L,
213  8.8415758124319434347116734705174308e-3L, 1.9850836029449446668518049562565291e-2L,
214  3.6688019729653446926585242192447447e-2L, 5.9290962294020186998954055264528393e-2L,
215  8.6998436081634343903250580992127677e-2L, 1.1880960220216759245467951592883642e-1L };
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 */
224 template <typename MatrixType>
225 void 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 
241 template <typename MatrixType>
242 void 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 
257 template <typename MatrixType>
258 void 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 
273 template <typename MatrixType>
274 void 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 
291 template <typename MatrixType>
292 void 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 
309 template <typename MatrixType>
310 void 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 
329 template <typename MatrixType>
330 void 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 
349 template <typename MatrixType>
350 void 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 
371 template <typename MatrixType>
372 void 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 
393 template <typename MatrixType>
394 void 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 
429 template<typename Derived> class MatrixLogarithmReturnValue
430 : public ReturnByValue<MatrixLogarithmReturnValue<Derived> >
431 {
432 public:
433 
434  typedef typename Derived::Scalar Scalar;
435  typedef typename Derived::Index Index;
436 
441  MatrixLogarithmReturnValue(const Derived& A) : m_A(A) { }
442 
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;
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 
468 private:
469  typename internal::nested<Derived>::type m_A;
470 
472 };
473 
474 namespace 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 
486 template <typename Derived>
487 const 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