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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, 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 namespace Eigen {
15 
16 namespace internal {
17 
18 template <typename Scalar>
19 struct matrix_log_min_pade_degree
20 {
21  static const int value = 3;
22 };
23 
24 template <typename Scalar>
25 struct matrix_log_max_pade_degree
26 {
27  typedef typename NumTraits<Scalar>::Real RealScalar;
28  static const int value = std::numeric_limits<RealScalar>::digits<= 24? 5: // single precision
29  std::numeric_limits<RealScalar>::digits<= 53? 7: // double precision
30  std::numeric_limits<RealScalar>::digits<= 64? 8: // extended precision
31  std::numeric_limits<RealScalar>::digits<=106? 10: // double-double
32  11; // quadruple precision
33 };
34 
36 template <typename MatrixType>
37 void matrix_log_compute_2x2(const MatrixType& A, MatrixType& result)
38 {
39  typedef typename MatrixType::Scalar Scalar;
40  typedef typename MatrixType::RealScalar RealScalar;
41  using std::abs;
42  using std::ceil;
43  using std::imag;
44  using std::log;
45 
46  Scalar logA00 = log(A(0,0));
47  Scalar logA11 = log(A(1,1));
48 
49  result(0,0) = logA00;
50  result(1,0) = Scalar(0);
51  result(1,1) = logA11;
52 
53  Scalar y = A(1,1) - A(0,0);
54  if (y==Scalar(0))
55  {
56  result(0,1) = A(0,1) / A(0,0);
57  }
58  else if ((abs(A(0,0)) < RealScalar(0.5)*abs(A(1,1))) || (abs(A(0,0)) > 2*abs(A(1,1))))
59  {
60  result(0,1) = A(0,1) * (logA11 - logA00) / y;
61  }
62  else
63  {
64  // computation in previous branch is inaccurate if A(1,1) \approx A(0,0)
65  int unwindingNumber = static_cast<int>(ceil((imag(logA11 - logA00) - RealScalar(EIGEN_PI)) / RealScalar(2*EIGEN_PI)));
66  result(0,1) = A(0,1) * (numext::log1p(y/A(0,0)) + Scalar(0,2*EIGEN_PI*unwindingNumber)) / y;
67  }
68 }
69 
70 /* \brief Get suitable degree for Pade approximation. (specialized for RealScalar = float) */
71 inline int matrix_log_get_pade_degree(float normTminusI)
72 {
73  const float maxNormForPade[] = { 2.5111573934555054e-1 /* degree = 3 */ , 4.0535837411880493e-1,
74  5.3149729967117310e-1 };
75  const int minPadeDegree = matrix_log_min_pade_degree<float>::value;
76  const int maxPadeDegree = matrix_log_max_pade_degree<float>::value;
77  int degree = minPadeDegree;
78  for (; degree <= maxPadeDegree; ++degree)
79  if (normTminusI <= maxNormForPade[degree - minPadeDegree])
80  break;
81  return degree;
82 }
83 
84 /* \brief Get suitable degree for Pade approximation. (specialized for RealScalar = double) */
85 inline int matrix_log_get_pade_degree(double normTminusI)
86 {
87  const double maxNormForPade[] = { 1.6206284795015624e-2 /* degree = 3 */ , 5.3873532631381171e-2,
88  1.1352802267628681e-1, 1.8662860613541288e-1, 2.642960831111435e-1 };
89  const int minPadeDegree = matrix_log_min_pade_degree<double>::value;
90  const int maxPadeDegree = matrix_log_max_pade_degree<double>::value;
91  int degree = minPadeDegree;
92  for (; degree <= maxPadeDegree; ++degree)
93  if (normTminusI <= maxNormForPade[degree - minPadeDegree])
94  break;
95  return degree;
96 }
97 
98 /* \brief Get suitable degree for Pade approximation. (specialized for RealScalar = long double) */
99 inline int matrix_log_get_pade_degree(long double normTminusI)
100 {
101 #if LDBL_MANT_DIG == 53 // double precision
102  const long double maxNormForPade[] = { 1.6206284795015624e-2L /* degree = 3 */ , 5.3873532631381171e-2L,
103  1.1352802267628681e-1L, 1.8662860613541288e-1L, 2.642960831111435e-1L };
104 #elif LDBL_MANT_DIG <= 64 // extended precision
105  const long double maxNormForPade[] = { 5.48256690357782863103e-3L /* degree = 3 */, 2.34559162387971167321e-2L,
106  5.84603923897347449857e-2L, 1.08486423756725170223e-1L, 1.68385767881294446649e-1L,
107  2.32777776523703892094e-1L };
108 #elif LDBL_MANT_DIG <= 106 // double-double
109  const long double maxNormForPade[] = { 8.58970550342939562202529664318890e-5L /* degree = 3 */,
110  9.34074328446359654039446552677759e-4L, 4.26117194647672175773064114582860e-3L,
111  1.21546224740281848743149666560464e-2L, 2.61100544998339436713088248557444e-2L,
112  4.66170074627052749243018566390567e-2L, 7.32585144444135027565872014932387e-2L,
113  1.05026503471351080481093652651105e-1L };
114 #else // quadruple precision
115  const long double maxNormForPade[] = { 4.7419931187193005048501568167858103e-5L /* degree = 3 */,
116  5.8853168473544560470387769480192666e-4L, 2.9216120366601315391789493628113520e-3L,
117  8.8415758124319434347116734705174308e-3L, 1.9850836029449446668518049562565291e-2L,
118  3.6688019729653446926585242192447447e-2L, 5.9290962294020186998954055264528393e-2L,
119  8.6998436081634343903250580992127677e-2L, 1.1880960220216759245467951592883642e-1L };
120 #endif
121  const int minPadeDegree = matrix_log_min_pade_degree<long double>::value;
122  const int maxPadeDegree = matrix_log_max_pade_degree<long double>::value;
123  int degree = minPadeDegree;
124  for (; degree <= maxPadeDegree; ++degree)
125  if (normTminusI <= maxNormForPade[degree - minPadeDegree])
126  break;
127  return degree;
128 }
129 
130 /* \brief Compute Pade approximation to matrix logarithm */
131 template <typename MatrixType>
132 void matrix_log_compute_pade(MatrixType& result, const MatrixType& T, int degree)
133 {
134  typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
135  const int minPadeDegree = 3;
136  const int maxPadeDegree = 11;
137  assert(degree >= minPadeDegree && degree <= maxPadeDegree);
138 
139  const RealScalar nodes[][maxPadeDegree] = {
140  { 0.1127016653792583114820734600217600L, 0.5000000000000000000000000000000000L, // degree 3
141  0.8872983346207416885179265399782400L },
142  { 0.0694318442029737123880267555535953L, 0.3300094782075718675986671204483777L, // degree 4
143  0.6699905217924281324013328795516223L, 0.9305681557970262876119732444464048L },
144  { 0.0469100770306680036011865608503035L, 0.2307653449471584544818427896498956L, // degree 5
145  0.5000000000000000000000000000000000L, 0.7692346550528415455181572103501044L,
146  0.9530899229693319963988134391496965L },
147  { 0.0337652428984239860938492227530027L, 0.1693953067668677431693002024900473L, // degree 6
148  0.3806904069584015456847491391596440L, 0.6193095930415984543152508608403560L,
149  0.8306046932331322568306997975099527L, 0.9662347571015760139061507772469973L },
150  { 0.0254460438286207377369051579760744L, 0.1292344072003027800680676133596058L, // degree 7
151  0.2970774243113014165466967939615193L, 0.5000000000000000000000000000000000L,
152  0.7029225756886985834533032060384807L, 0.8707655927996972199319323866403942L,
153  0.9745539561713792622630948420239256L },
154  { 0.0198550717512318841582195657152635L, 0.1016667612931866302042230317620848L, // degree 8
155  0.2372337950418355070911304754053768L, 0.4082826787521750975302619288199080L,
156  0.5917173212478249024697380711800920L, 0.7627662049581644929088695245946232L,
157  0.8983332387068133697957769682379152L, 0.9801449282487681158417804342847365L },
158  { 0.0159198802461869550822118985481636L, 0.0819844463366821028502851059651326L, // degree 9
159  0.1933142836497048013456489803292629L, 0.3378732882980955354807309926783317L,
160  0.5000000000000000000000000000000000L, 0.6621267117019044645192690073216683L,
161  0.8066857163502951986543510196707371L, 0.9180155536633178971497148940348674L,
162  0.9840801197538130449177881014518364L },
163  { 0.0130467357414141399610179939577740L, 0.0674683166555077446339516557882535L, // degree 10
164  0.1602952158504877968828363174425632L, 0.2833023029353764046003670284171079L,
165  0.4255628305091843945575869994351400L, 0.5744371694908156054424130005648600L,
166  0.7166976970646235953996329715828921L, 0.8397047841495122031171636825574368L,
167  0.9325316833444922553660483442117465L, 0.9869532642585858600389820060422260L },
168  { 0.0108856709269715035980309994385713L, 0.0564687001159523504624211153480364L, // degree 11
169  0.1349239972129753379532918739844233L, 0.2404519353965940920371371652706952L,
170  0.3652284220238275138342340072995692L, 0.5000000000000000000000000000000000L,
171  0.6347715779761724861657659927004308L, 0.7595480646034059079628628347293048L,
172  0.8650760027870246620467081260155767L, 0.9435312998840476495375788846519636L,
173  0.9891143290730284964019690005614287L } };
174 
175  const RealScalar weights[][maxPadeDegree] = {
176  { 0.2777777777777777777777777777777778L, 0.4444444444444444444444444444444444L, // degree 3
177  0.2777777777777777777777777777777778L },
178  { 0.1739274225687269286865319746109997L, 0.3260725774312730713134680253890003L, // degree 4
179  0.3260725774312730713134680253890003L, 0.1739274225687269286865319746109997L },
180  { 0.1184634425280945437571320203599587L, 0.2393143352496832340206457574178191L, // degree 5
181  0.2844444444444444444444444444444444L, 0.2393143352496832340206457574178191L,
182  0.1184634425280945437571320203599587L },
183  { 0.0856622461895851725201480710863665L, 0.1803807865240693037849167569188581L, // degree 6
184  0.2339569672863455236949351719947755L, 0.2339569672863455236949351719947755L,
185  0.1803807865240693037849167569188581L, 0.0856622461895851725201480710863665L },
186  { 0.0647424830844348466353057163395410L, 0.1398526957446383339507338857118898L, // degree 7
187  0.1909150252525594724751848877444876L, 0.2089795918367346938775510204081633L,
188  0.1909150252525594724751848877444876L, 0.1398526957446383339507338857118898L,
189  0.0647424830844348466353057163395410L },
190  { 0.0506142681451881295762656771549811L, 0.1111905172266872352721779972131204L, // degree 8
191  0.1568533229389436436689811009933007L, 0.1813418916891809914825752246385978L,
192  0.1813418916891809914825752246385978L, 0.1568533229389436436689811009933007L,
193  0.1111905172266872352721779972131204L, 0.0506142681451881295762656771549811L },
194  { 0.0406371941807872059859460790552618L, 0.0903240803474287020292360156214564L, // degree 9
195  0.1303053482014677311593714347093164L, 0.1561735385200014200343152032922218L,
196  0.1651196775006298815822625346434870L, 0.1561735385200014200343152032922218L,
197  0.1303053482014677311593714347093164L, 0.0903240803474287020292360156214564L,
198  0.0406371941807872059859460790552618L },
199  { 0.0333356721543440687967844049466659L, 0.0747256745752902965728881698288487L, // degree 10
200  0.1095431812579910219977674671140816L, 0.1346333596549981775456134607847347L,
201  0.1477621123573764350869464973256692L, 0.1477621123573764350869464973256692L,
202  0.1346333596549981775456134607847347L, 0.1095431812579910219977674671140816L,
203  0.0747256745752902965728881698288487L, 0.0333356721543440687967844049466659L },
204  { 0.0278342835580868332413768602212743L, 0.0627901847324523123173471496119701L, // degree 11
205  0.0931451054638671257130488207158280L, 0.1165968822959952399592618524215876L,
206  0.1314022722551233310903444349452546L, 0.1364625433889503153572417641681711L,
207  0.1314022722551233310903444349452546L, 0.1165968822959952399592618524215876L,
208  0.0931451054638671257130488207158280L, 0.0627901847324523123173471496119701L,
209  0.0278342835580868332413768602212743L } };
210 
211  MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
212  result.setZero(T.rows(), T.rows());
213  for (int k = 0; k < degree; ++k) {
214  RealScalar weight = weights[degree-minPadeDegree][k];
215  RealScalar node = nodes[degree-minPadeDegree][k];
216  result += weight * (MatrixType::Identity(T.rows(), T.rows()) + node * TminusI)
217  .template triangularView<Upper>().solve(TminusI);
218  }
219 }
220 
223 template <typename MatrixType>
224 void matrix_log_compute_big(const MatrixType& A, MatrixType& result)
225 {
226  typedef typename MatrixType::Scalar Scalar;
227  typedef typename NumTraits<Scalar>::Real RealScalar;
228  using std::pow;
229 
230  int numberOfSquareRoots = 0;
231  int numberOfExtraSquareRoots = 0;
232  int degree;
233  MatrixType T = A, sqrtT;
234 
235  int maxPadeDegree = matrix_log_max_pade_degree<Scalar>::value;
236  const RealScalar maxNormForPade = maxPadeDegree<= 5? 5.3149729967117310e-1L: // single precision
237  maxPadeDegree<= 7? 2.6429608311114350e-1L: // double precision
238  maxPadeDegree<= 8? 2.32777776523703892094e-1L: // extended precision
239  maxPadeDegree<=10? 1.05026503471351080481093652651105e-1L: // double-double
240  1.1880960220216759245467951592883642e-1L; // quadruple precision
241 
242  while (true) {
243  RealScalar normTminusI = (T - MatrixType::Identity(T.rows(), T.rows())).cwiseAbs().colwise().sum().maxCoeff();
244  if (normTminusI < maxNormForPade) {
245  degree = matrix_log_get_pade_degree(normTminusI);
246  int degree2 = matrix_log_get_pade_degree(normTminusI / RealScalar(2));
247  if ((degree - degree2 <= 1) || (numberOfExtraSquareRoots == 1))
248  break;
249  ++numberOfExtraSquareRoots;
250  }
251  matrix_sqrt_triangular(T, sqrtT);
252  T = sqrtT.template triangularView<Upper>();
253  ++numberOfSquareRoots;
254  }
255 
256  matrix_log_compute_pade(result, T, degree);
257  result *= pow(RealScalar(2), numberOfSquareRoots);
258 }
259 
268 template <typename MatrixType>
269 class MatrixLogarithmAtomic
270 {
271 public:
276  MatrixType compute(const MatrixType& A);
277 };
278 
279 template <typename MatrixType>
280 MatrixType MatrixLogarithmAtomic<MatrixType>::compute(const MatrixType& A)
281 {
282  using std::log;
283  MatrixType result(A.rows(), A.rows());
284  if (A.rows() == 1)
285  result(0,0) = log(A(0,0));
286  else if (A.rows() == 2)
287  matrix_log_compute_2x2(A, result);
288  else
289  matrix_log_compute_big(A, result);
290  return result;
291 }
292 
293 } // end of namespace internal
294 
307 template<typename Derived> class MatrixLogarithmReturnValue
308 : public ReturnByValue<MatrixLogarithmReturnValue<Derived> >
309 {
310 public:
311  typedef typename Derived::Scalar Scalar;
312  typedef typename Derived::Index Index;
313 
314 protected:
315  typedef typename internal::ref_selector<Derived>::type DerivedNested;
316 
317 public:
318 
323  explicit MatrixLogarithmReturnValue(const Derived& A) : m_A(A) { }
324 
329  template <typename ResultType>
330  inline void evalTo(ResultType& result) const
331  {
332  typedef typename internal::nested_eval<Derived, 10>::type DerivedEvalType;
333  typedef typename internal::remove_all<DerivedEvalType>::type DerivedEvalTypeClean;
334  typedef internal::traits<DerivedEvalTypeClean> Traits;
335  static const int RowsAtCompileTime = Traits::RowsAtCompileTime;
336  static const int ColsAtCompileTime = Traits::ColsAtCompileTime;
337  typedef std::complex<typename NumTraits<Scalar>::Real> ComplexScalar;
339  typedef internal::MatrixLogarithmAtomic<DynMatrixType> AtomicType;
340  AtomicType atomic;
341 
342  internal::matrix_function_compute<typename DerivedEvalTypeClean::PlainObject>::run(m_A, atomic, result);
343  }
344 
345  Index rows() const { return m_A.rows(); }
346  Index cols() const { return m_A.cols(); }
347 
348 private:
349  const DerivedNested m_A;
350 };
351 
352 namespace internal {
353  template<typename Derived>
354  struct traits<MatrixLogarithmReturnValue<Derived> >
355  {
356  typedef typename Derived::PlainObject ReturnType;
357  };
358 }
359 
360 
361 /********** MatrixBase method **********/
362 
363 
364 template <typename Derived>
365 const MatrixLogarithmReturnValue<Derived> MatrixBase<Derived>::log() const
366 {
367  eigen_assert(rows() == cols());
368  return MatrixLogarithmReturnValue<Derived>(derived());
369 }
370 
371 } // end namespace Eigen
372 
373 #endif // EIGEN_MATRIX_LOGARITHM
Eigen
Namespace containing all symbols from the Eigen library.
Eigen::MatrixLogarithmReturnValue
Proxy for the matrix logarithm of some matrix (expression).
Definition: MatrixLogarithm.h:307
Eigen::ceil
const Eigen::CwiseUnaryOp< Eigen::internal::scalar_ceil_op< typename Derived::Scalar >, const Derived > ceil(const Eigen::ArrayBase< Derived > &x)
Eigen::MatrixLogarithmReturnValue::evalTo
void evalTo(ResultType &result) const
Compute the matrix logarithm.
Definition: MatrixLogarithm.h:330
Eigen::MatrixLogarithmReturnValue::MatrixLogarithmReturnValue
MatrixLogarithmReturnValue(const Derived &A)
Constructor.
Definition: MatrixLogarithm.h:323
Eigen::abs
const Eigen::CwiseUnaryOp< Eigen::internal::scalar_abs_op< typename Derived::Scalar >, const Derived > abs(const Eigen::ArrayBase< Derived > &x)
Eigen::matrix_sqrt_triangular
void matrix_sqrt_triangular(const MatrixType &arg, ResultType &result)
Compute matrix square root of triangular matrix.
Definition: MatrixSquareRoot.h:204
Eigen::log
const Eigen::CwiseUnaryOp< Eigen::internal::scalar_log_op< typename Derived::Scalar >, const Derived > log(const Eigen::ArrayBase< Derived > &x)
Eigen::MatrixBase::log
const MatrixLogarithmReturnValue< Derived > log() const
Definition: MatrixLogarithm.h:365
Eigen::imag
const Eigen::CwiseUnaryOp< Eigen::internal::scalar_imag_op< typename Derived::Scalar >, const Derived > imag(const Eigen::ArrayBase< Derived > &x)
Eigen::Matrix
Eigen::Index
EIGEN_DEFAULT_DENSE_INDEX_TYPE Index