MatrixLogarithm.h
1 // This file is part of Eigen, a lightweight C++ template library
2 // for linear algebra.
3 //
4 // Copyright (C) 2011, 2013 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 
20 namespace internal {
21 
22 template <typename Scalar>
23 struct matrix_log_min_pade_degree
24 {
25  static const int value = 3;
26 };
27 
28 template <typename Scalar>
29 struct matrix_log_max_pade_degree
30 {
31  typedef typename NumTraits<Scalar>::Real RealScalar;
32  static const int value = std::numeric_limits<RealScalar>::digits<= 24? 5: // single precision
33  std::numeric_limits<RealScalar>::digits<= 53? 7: // double precision
34  std::numeric_limits<RealScalar>::digits<= 64? 8: // extended precision
35  std::numeric_limits<RealScalar>::digits<=106? 10: // double-double
36  11; // quadruple precision
37 };
38 
40 template <typename MatrixType>
41 void matrix_log_compute_2x2(const MatrixType& A, MatrixType& result)
42 {
43  typedef typename MatrixType::Scalar Scalar;
44  using std::abs;
45  using std::ceil;
46  using std::imag;
47  using std::log;
48 
49  Scalar logA00 = log(A(0,0));
50  Scalar logA11 = log(A(1,1));
51 
52  result(0,0) = logA00;
53  result(1,0) = Scalar(0);
54  result(1,1) = logA11;
55 
56  Scalar y = A(1,1) - A(0,0);
57  if (y==Scalar(0))
58  {
59  result(0,1) = A(0,1) / A(0,0);
60  }
61  else if ((abs(A(0,0)) < 0.5*abs(A(1,1))) || (abs(A(0,0)) > 2*abs(A(1,1))))
62  {
63  result(0,1) = A(0,1) * (logA11 - logA00) / y;
64  }
65  else
66  {
67  // computation in previous branch is inaccurate if A(1,1) \approx A(0,0)
68  int unwindingNumber = static_cast<int>(ceil((imag(logA11 - logA00) - M_PI) / (2*M_PI)));
69  result(0,1) = A(0,1) * (numext::log1p(y/A(0,0)) + Scalar(0,2*M_PI*unwindingNumber)) / y;
70  }
71 }
72 
73 /* \brief Get suitable degree for Pade approximation. (specialized for RealScalar = float) */
74 inline int matrix_log_get_pade_degree(float normTminusI)
75 {
76  const float maxNormForPade[] = { 2.5111573934555054e-1 /* degree = 3 */ , 4.0535837411880493e-1,
77  5.3149729967117310e-1 };
78  const int minPadeDegree = matrix_log_min_pade_degree<float>::value;
79  const int maxPadeDegree = matrix_log_max_pade_degree<float>::value;
80  int degree = minPadeDegree;
81  for (; degree <= maxPadeDegree; ++degree)
82  if (normTminusI <= maxNormForPade[degree - minPadeDegree])
83  break;
84  return degree;
85 }
86 
87 /* \brief Get suitable degree for Pade approximation. (specialized for RealScalar = double) */
88 inline int matrix_log_get_pade_degree(double normTminusI)
89 {
90  const double maxNormForPade[] = { 1.6206284795015624e-2 /* degree = 3 */ , 5.3873532631381171e-2,
91  1.1352802267628681e-1, 1.8662860613541288e-1, 2.642960831111435e-1 };
92  const int minPadeDegree = matrix_log_min_pade_degree<double>::value;
93  const int maxPadeDegree = matrix_log_max_pade_degree<double>::value;
94  int degree = minPadeDegree;
95  for (; degree <= maxPadeDegree; ++degree)
96  if (normTminusI <= maxNormForPade[degree - minPadeDegree])
97  break;
98  return degree;
99 }
100 
101 /* \brief Get suitable degree for Pade approximation. (specialized for RealScalar = long double) */
102 inline int matrix_log_get_pade_degree(long double normTminusI)
103 {
104 #if LDBL_MANT_DIG == 53 // double precision
105  const long double maxNormForPade[] = { 1.6206284795015624e-2L /* degree = 3 */ , 5.3873532631381171e-2L,
106  1.1352802267628681e-1L, 1.8662860613541288e-1L, 2.642960831111435e-1L };
107 #elif LDBL_MANT_DIG <= 64 // extended precision
108  const long double maxNormForPade[] = { 5.48256690357782863103e-3L /* degree = 3 */, 2.34559162387971167321e-2L,
109  5.84603923897347449857e-2L, 1.08486423756725170223e-1L, 1.68385767881294446649e-1L,
110  2.32777776523703892094e-1L };
111 #elif LDBL_MANT_DIG <= 106 // double-double
112  const long double maxNormForPade[] = { 8.58970550342939562202529664318890e-5L /* degree = 3 */,
113  9.34074328446359654039446552677759e-4L, 4.26117194647672175773064114582860e-3L,
114  1.21546224740281848743149666560464e-2L, 2.61100544998339436713088248557444e-2L,
115  4.66170074627052749243018566390567e-2L, 7.32585144444135027565872014932387e-2L,
116  1.05026503471351080481093652651105e-1L };
117 #else // quadruple precision
118  const long double maxNormForPade[] = { 4.7419931187193005048501568167858103e-5L /* degree = 3 */,
119  5.8853168473544560470387769480192666e-4L, 2.9216120366601315391789493628113520e-3L,
120  8.8415758124319434347116734705174308e-3L, 1.9850836029449446668518049562565291e-2L,
121  3.6688019729653446926585242192447447e-2L, 5.9290962294020186998954055264528393e-2L,
122  8.6998436081634343903250580992127677e-2L, 1.1880960220216759245467951592883642e-1L };
123 #endif
124  const int minPadeDegree = matrix_log_min_pade_degree<long double>::value;
125  const int maxPadeDegree = matrix_log_max_pade_degree<long double>::value;
126  int degree = minPadeDegree;
127  for (; degree <= maxPadeDegree; ++degree)
128  if (normTminusI <= maxNormForPade[degree - minPadeDegree])
129  break;
130  return degree;
131 }
132 
133 /* \brief Compute Pade approximation to matrix logarithm */
134 template <typename MatrixType>
135 void matrix_log_compute_pade(MatrixType& result, const MatrixType& T, int degree)
136 {
137  typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
138  const int minPadeDegree = 3;
139  const int maxPadeDegree = 11;
140  assert(degree >= minPadeDegree && degree <= maxPadeDegree);
141 
142  const RealScalar nodes[][maxPadeDegree] = {
143  { 0.1127016653792583114820734600217600L, 0.5000000000000000000000000000000000L, // degree 3
144  0.8872983346207416885179265399782400L },
145  { 0.0694318442029737123880267555535953L, 0.3300094782075718675986671204483777L, // degree 4
146  0.6699905217924281324013328795516223L, 0.9305681557970262876119732444464048L },
147  { 0.0469100770306680036011865608503035L, 0.2307653449471584544818427896498956L, // degree 5
148  0.5000000000000000000000000000000000L, 0.7692346550528415455181572103501044L,
149  0.9530899229693319963988134391496965L },
150  { 0.0337652428984239860938492227530027L, 0.1693953067668677431693002024900473L, // degree 6
151  0.3806904069584015456847491391596440L, 0.6193095930415984543152508608403560L,
152  0.8306046932331322568306997975099527L, 0.9662347571015760139061507772469973L },
153  { 0.0254460438286207377369051579760744L, 0.1292344072003027800680676133596058L, // degree 7
154  0.2970774243113014165466967939615193L, 0.5000000000000000000000000000000000L,
155  0.7029225756886985834533032060384807L, 0.8707655927996972199319323866403942L,
156  0.9745539561713792622630948420239256L },
157  { 0.0198550717512318841582195657152635L, 0.1016667612931866302042230317620848L, // degree 8
158  0.2372337950418355070911304754053768L, 0.4082826787521750975302619288199080L,
159  0.5917173212478249024697380711800920L, 0.7627662049581644929088695245946232L,
160  0.8983332387068133697957769682379152L, 0.9801449282487681158417804342847365L },
161  { 0.0159198802461869550822118985481636L, 0.0819844463366821028502851059651326L, // degree 9
162  0.1933142836497048013456489803292629L, 0.3378732882980955354807309926783317L,
163  0.5000000000000000000000000000000000L, 0.6621267117019044645192690073216683L,
164  0.8066857163502951986543510196707371L, 0.9180155536633178971497148940348674L,
165  0.9840801197538130449177881014518364L },
166  { 0.0130467357414141399610179939577740L, 0.0674683166555077446339516557882535L, // degree 10
167  0.1602952158504877968828363174425632L, 0.2833023029353764046003670284171079L,
168  0.4255628305091843945575869994351400L, 0.5744371694908156054424130005648600L,
169  0.7166976970646235953996329715828921L, 0.8397047841495122031171636825574368L,
170  0.9325316833444922553660483442117465L, 0.9869532642585858600389820060422260L },
171  { 0.0108856709269715035980309994385713L, 0.0564687001159523504624211153480364L, // degree 11
172  0.1349239972129753379532918739844233L, 0.2404519353965940920371371652706952L,
173  0.3652284220238275138342340072995692L, 0.5000000000000000000000000000000000L,
174  0.6347715779761724861657659927004308L, 0.7595480646034059079628628347293048L,
175  0.8650760027870246620467081260155767L, 0.9435312998840476495375788846519636L,
176  0.9891143290730284964019690005614287L } };
177 
178  const RealScalar weights[][maxPadeDegree] = {
179  { 0.2777777777777777777777777777777778L, 0.4444444444444444444444444444444444L, // degree 3
180  0.2777777777777777777777777777777778L },
181  { 0.1739274225687269286865319746109997L, 0.3260725774312730713134680253890003L, // degree 4
182  0.3260725774312730713134680253890003L, 0.1739274225687269286865319746109997L },
183  { 0.1184634425280945437571320203599587L, 0.2393143352496832340206457574178191L, // degree 5
184  0.2844444444444444444444444444444444L, 0.2393143352496832340206457574178191L,
185  0.1184634425280945437571320203599587L },
186  { 0.0856622461895851725201480710863665L, 0.1803807865240693037849167569188581L, // degree 6
187  0.2339569672863455236949351719947755L, 0.2339569672863455236949351719947755L,
188  0.1803807865240693037849167569188581L, 0.0856622461895851725201480710863665L },
189  { 0.0647424830844348466353057163395410L, 0.1398526957446383339507338857118898L, // degree 7
190  0.1909150252525594724751848877444876L, 0.2089795918367346938775510204081633L,
191  0.1909150252525594724751848877444876L, 0.1398526957446383339507338857118898L,
192  0.0647424830844348466353057163395410L },
193  { 0.0506142681451881295762656771549811L, 0.1111905172266872352721779972131204L, // degree 8
194  0.1568533229389436436689811009933007L, 0.1813418916891809914825752246385978L,
195  0.1813418916891809914825752246385978L, 0.1568533229389436436689811009933007L,
196  0.1111905172266872352721779972131204L, 0.0506142681451881295762656771549811L },
197  { 0.0406371941807872059859460790552618L, 0.0903240803474287020292360156214564L, // degree 9
198  0.1303053482014677311593714347093164L, 0.1561735385200014200343152032922218L,
199  0.1651196775006298815822625346434870L, 0.1561735385200014200343152032922218L,
200  0.1303053482014677311593714347093164L, 0.0903240803474287020292360156214564L,
201  0.0406371941807872059859460790552618L },
202  { 0.0333356721543440687967844049466659L, 0.0747256745752902965728881698288487L, // degree 10
203  0.1095431812579910219977674671140816L, 0.1346333596549981775456134607847347L,
204  0.1477621123573764350869464973256692L, 0.1477621123573764350869464973256692L,
205  0.1346333596549981775456134607847347L, 0.1095431812579910219977674671140816L,
206  0.0747256745752902965728881698288487L, 0.0333356721543440687967844049466659L },
207  { 0.0278342835580868332413768602212743L, 0.0627901847324523123173471496119701L, // degree 11
208  0.0931451054638671257130488207158280L, 0.1165968822959952399592618524215876L,
209  0.1314022722551233310903444349452546L, 0.1364625433889503153572417641681711L,
210  0.1314022722551233310903444349452546L, 0.1165968822959952399592618524215876L,
211  0.0931451054638671257130488207158280L, 0.0627901847324523123173471496119701L,
212  0.0278342835580868332413768602212743L } };
213 
214  MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
215  result.setZero(T.rows(), T.rows());
216  for (int k = 0; k < degree; ++k) {
217  RealScalar weight = weights[degree-minPadeDegree][k];
218  RealScalar node = nodes[degree-minPadeDegree][k];
219  result += weight * (MatrixType::Identity(T.rows(), T.rows()) + node * TminusI)
220  .template triangularView<Upper>().solve(TminusI);
221  }
222 }
223 
226 template <typename MatrixType>
227 void matrix_log_compute_big(const MatrixType& A, MatrixType& result)
228 {
229  typedef typename MatrixType::Scalar Scalar;
230  typedef typename NumTraits<Scalar>::Real RealScalar;
231  using std::pow;
232 
233  int numberOfSquareRoots = 0;
234  int numberOfExtraSquareRoots = 0;
235  int degree;
236  MatrixType T = A, sqrtT;
237 
238  int maxPadeDegree = matrix_log_max_pade_degree<Scalar>::value;
239  const RealScalar maxNormForPade = maxPadeDegree<= 5? 5.3149729967117310e-1: // single precision
240  maxPadeDegree<= 7? 2.6429608311114350e-1: // double precision
241  maxPadeDegree<= 8? 2.32777776523703892094e-1L: // extended precision
242  maxPadeDegree<=10? 1.05026503471351080481093652651105e-1L: // double-double
243  1.1880960220216759245467951592883642e-1L; // quadruple precision
244 
245  while (true) {
246  RealScalar normTminusI = (T - MatrixType::Identity(T.rows(), T.rows())).cwiseAbs().colwise().sum().maxCoeff();
247  if (normTminusI < maxNormForPade) {
248  degree = matrix_log_get_pade_degree(normTminusI);
249  int degree2 = matrix_log_get_pade_degree(normTminusI / RealScalar(2));
250  if ((degree - degree2 <= 1) || (numberOfExtraSquareRoots == 1))
251  break;
252  ++numberOfExtraSquareRoots;
253  }
254  matrix_sqrt_triangular(T, sqrtT);
255  T = sqrtT.template triangularView<Upper>();
256  ++numberOfSquareRoots;
257  }
258 
259  matrix_log_compute_pade(result, T, degree);
260  result *= pow(RealScalar(2), numberOfSquareRoots);
261 }
262 
271 template <typename MatrixType>
272 class MatrixLogarithmAtomic
273 {
274 public:
279  MatrixType compute(const MatrixType& A);
280 };
281 
282 template <typename MatrixType>
283 MatrixType MatrixLogarithmAtomic<MatrixType>::compute(const MatrixType& A)
284 {
285  using std::log;
286  MatrixType result(A.rows(), A.rows());
287  if (A.rows() == 1)
288  result(0,0) = log(A(0,0));
289  else if (A.rows() == 2)
290  matrix_log_compute_2x2(A, result);
291  else
292  matrix_log_compute_big(A, result);
293  return result;
294 }
295 
296 } // end of namespace internal
297 
310 template<typename Derived> class MatrixLogarithmReturnValue
311 : public ReturnByValue<MatrixLogarithmReturnValue<Derived> >
312 {
313 public:
314  typedef typename Derived::Scalar Scalar;
315  typedef typename Derived::Index Index;
316 
317 protected:
318  typedef typename internal::ref_selector<Derived>::type DerivedNested;
319 
320 public:
321 
326  explicit MatrixLogarithmReturnValue(const Derived& A) : m_A(A) { }
327 
332  template <typename ResultType>
333  inline void evalTo(ResultType& result) const
334  {
335  typedef typename internal::nested_eval<Derived, 10>::type DerivedEvalType;
336  typedef typename internal::remove_all<DerivedEvalType>::type DerivedEvalTypeClean;
337  typedef internal::traits<DerivedEvalTypeClean> Traits;
338  static const int RowsAtCompileTime = Traits::RowsAtCompileTime;
339  static const int ColsAtCompileTime = Traits::ColsAtCompileTime;
340  static const int Options = DerivedEvalTypeClean::Options;
341  typedef std::complex<typename NumTraits<Scalar>::Real> ComplexScalar;
342  typedef Matrix<ComplexScalar, Dynamic, Dynamic, Options, RowsAtCompileTime, ColsAtCompileTime> DynMatrixType;
343  typedef internal::MatrixLogarithmAtomic<DynMatrixType> AtomicType;
344  AtomicType atomic;
345 
346  internal::matrix_function_compute<DerivedEvalTypeClean>::run(m_A, atomic, result);
347  }
348 
349  Index rows() const { return m_A.rows(); }
350  Index cols() const { return m_A.cols(); }
351 
352 private:
353  const DerivedNested m_A;
354 };
355 
356 namespace internal {
357  template<typename Derived>
358  struct traits<MatrixLogarithmReturnValue<Derived> >
359  {
360  typedef typename Derived::PlainObject ReturnType;
361  };
362 }
363 
364 
365 /********** MatrixBase method **********/
366 
367 
368 template <typename Derived>
369 const MatrixLogarithmReturnValue<Derived> MatrixBase<Derived>::log() const
370 {
371  eigen_assert(rows() == cols());
372  return MatrixLogarithmReturnValue<Derived>(derived());
373 }
374 
375 } // end namespace Eigen
376 
377 #endif // EIGEN_MATRIX_LOGARITHM
void evalTo(ResultType &result) const
Compute the matrix logarithm.
Definition: MatrixLogarithm.h:333
MatrixLogarithmReturnValue(const Derived &A)
Constructor.
Definition: MatrixLogarithm.h:326
Namespace containing all symbols from the Eigen library.
Definition: CXX11Meta.h:13
void matrix_sqrt_triangular(const MatrixType &arg, ResultType &result)
Compute matrix square root of triangular matrix.
Definition: MatrixSquareRoot.h:207
Proxy for the matrix logarithm of some matrix (expression).
Definition: MatrixLogarithm.h:310