Eigen-unsupported  3.4.90 (git rev a4098ac676528a83cfb73d4d26ce1b42ec05f47c)
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
15
16namespace Eigen {
17
18namespace internal {
19
20template <typename Scalar>
22{
23 static const int value = 3;
24};
25
26template <typename Scalar>
28{
29 typedef typename NumTraits<Scalar>::Real RealScalar;
30 static const int value = std::numeric_limits<RealScalar>::digits<= 24? 5: // single precision
31 std::numeric_limits<RealScalar>::digits<= 53? 7: // double precision
32 std::numeric_limits<RealScalar>::digits<= 64? 8: // extended precision
33 std::numeric_limits<RealScalar>::digits<=106? 10: // double-double
35};
36
38template <typename MatrixType>
39void matrix_log_compute_2x2(const MatrixType& A, MatrixType& result)
40{
41 typedef typename MatrixType::Scalar Scalar;
42 typedef typename MatrixType::RealScalar RealScalar;
43 using std::abs;
44 using std::ceil;
45 using std::imag;
46 using std::log;
47
48 Scalar logA00 = log(A(0,0));
49 Scalar logA11 = log(A(1,1));
50
51 result(0,0) = logA00;
52 result(1,0) = Scalar(0);
53 result(1,1) = logA11;
54
55 Scalar y = A(1,1) - A(0,0);
56 if (y==Scalar(0))
57 {
58 result(0,1) = A(0,1) / A(0,0);
59 }
60 else if ((abs(A(0,0)) < RealScalar(0.5)*abs(A(1,1))) || (abs(A(0,0)) > 2*abs(A(1,1))))
61 {
62 result(0,1) = A(0,1) * (logA11 - logA00) / y;
63 }
64 else
65 {
66 // computation in previous branch is inaccurate if A(1,1) \approx A(0,0)
67 RealScalar unwindingNumber = ceil((imag(logA11 - logA00) - RealScalar(EIGEN_PI)) / RealScalar(2*EIGEN_PI));
68 result(0,1) = A(0,1) * (numext::log1p(y/A(0,0)) + Scalar(0,RealScalar(2*EIGEN_PI)*unwindingNumber)) / y;
69 }
70}
71
72/* \brief Get suitable degree for Pade approximation. (specialized for RealScalar = float) */
74{
75 const float maxNormForPade[] = { 2.5111573934555054e-1 /* degree = 3 */ , 4.0535837411880493e-1,
76 5.3149729967117310e-1 };
80 for (; degree <= maxPadeDegree; ++degree)
82 break;
83 return degree;
84}
85
86/* \brief Get suitable degree for Pade approximation. (specialized for RealScalar = double) */
88{
89 const double maxNormForPade[] = { 1.6206284795015624e-2 /* degree = 3 */ , 5.3873532631381171e-2,
90 1.1352802267628681e-1, 1.8662860613541288e-1, 2.642960831111435e-1 };
94 for (; degree <= maxPadeDegree; ++degree)
96 break;
97 return degree;
98}
99
100/* \brief Get suitable degree for Pade approximation. (specialized for RealScalar = long double) */
102{
103#if LDBL_MANT_DIG == 53 // double precision
104 const long double maxNormForPade[] = { 1.6206284795015624e-2L /* degree = 3 */ , 5.3873532631381171e-2L,
105 1.1352802267628681e-1L, 1.8662860613541288e-1L, 2.642960831111435e-1L };
106#elif LDBL_MANT_DIG <= 64 // extended precision
107 const long double maxNormForPade[] = { 5.48256690357782863103e-3L /* degree = 3 */, 2.34559162387971167321e-2L,
108 5.84603923897347449857e-2L, 1.08486423756725170223e-1L, 1.68385767881294446649e-1L,
109 2.32777776523703892094e-1L };
110#elif LDBL_MANT_DIG <= 106 // double-double
111 const long double maxNormForPade[] = { 8.58970550342939562202529664318890e-5L /* degree = 3 */,
112 9.34074328446359654039446552677759e-4L, 4.26117194647672175773064114582860e-3L,
113 1.21546224740281848743149666560464e-2L, 2.61100544998339436713088248557444e-2L,
114 4.66170074627052749243018566390567e-2L, 7.32585144444135027565872014932387e-2L,
115 1.05026503471351080481093652651105e-1L };
117 const long double maxNormForPade[] = { 4.7419931187193005048501568167858103e-5L /* degree = 3 */,
118 5.8853168473544560470387769480192666e-4L, 2.9216120366601315391789493628113520e-3L,
119 8.8415758124319434347116734705174308e-3L, 1.9850836029449446668518049562565291e-2L,
120 3.6688019729653446926585242192447447e-2L, 5.9290962294020186998954055264528393e-2L,
121 8.6998436081634343903250580992127677e-2L, 1.1880960220216759245467951592883642e-1L };
122#endif
126 for (; degree <= maxPadeDegree; ++degree)
128 break;
129 return degree;
130}
131
132/* \brief Compute Pade approximation to matrix logarithm */
133template <typename MatrixType>
134void matrix_log_compute_pade(MatrixType& result, const MatrixType& T, int degree)
135{
136 typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
137 const int minPadeDegree = 3;
138 const int maxPadeDegree = 11;
140 // FIXME this creates float-conversion-warnings if these are enabled.
141 // Either manually convert each value, or disable the warning locally
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) {
219 result += weight * (MatrixType::Identity(T.rows(), T.rows()) + node * TminusI)
220 .template triangularView<Upper>().solve(TminusI);
221 }
222}
223
226template <typename MatrixType>
227void 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
239 const RealScalar maxNormForPade = RealScalar(
240 maxPadeDegree<= 5? 5.3149729967117310e-1L: // single precision
241 maxPadeDegree<= 7? 2.6429608311114350e-1L: // double precision
242 maxPadeDegree<= 8? 2.32777776523703892094e-1L: // extended precision
245
246 while (true) {
247 RealScalar normTminusI = (T - MatrixType::Identity(T.rows(), T.rows())).cwiseAbs().colwise().sum().maxCoeff();
248 if (normTminusI < maxNormForPade) {
250 int degree2 = matrix_log_get_pade_degree(normTminusI / RealScalar(2));
251 if ((degree - degree2 <= 1) || (numberOfExtraSquareRoots == 1))
252 break;
253 ++numberOfExtraSquareRoots;
254 }
255 matrix_sqrt_triangular(T, sqrtT);
256 T = sqrtT.template triangularView<Upper>();
257 ++numberOfSquareRoots;
258 }
259
261 result *= pow(RealScalar(2), RealScalar(numberOfSquareRoots)); // TODO replace by bitshift if possible
262}
263
272template <typename MatrixType>
273class MatrixLogarithmAtomic
274{
275public:
280 MatrixType compute(const MatrixType& A);
281};
282
283template <typename MatrixType>
284MatrixType MatrixLogarithmAtomic<MatrixType>::compute(const MatrixType& A)
285{
286 using std::log;
287 MatrixType result(A.rows(), A.rows());
288 if (A.rows() == 1)
289 result(0,0) = log(A(0,0));
290 else if (A.rows() == 2)
291 matrix_log_compute_2x2(A, result);
292 else
293 matrix_log_compute_big(A, result);
294 return result;
295}
296
297} // end of namespace internal
298
311template<typename Derived> class MatrixLogarithmReturnValue
312: public ReturnByValue<MatrixLogarithmReturnValue<Derived> >
313{
314public:
315 typedef typename Derived::Scalar Scalar;
316 typedef typename Derived::Index Index;
317
318protected:
319 typedef typename internal::ref_selector<Derived>::type DerivedNested;
320
321public:
322
327 explicit MatrixLogarithmReturnValue(const Derived& A) : m_A(A) { }
328
333 template <typename ResultType>
334 inline void evalTo(ResultType& result) const
335 {
336 typedef typename internal::nested_eval<Derived, 10>::type DerivedEvalType;
337 typedef typename internal::remove_all<DerivedEvalType>::type DerivedEvalTypeClean;
338 typedef internal::traits<DerivedEvalTypeClean> Traits;
339 typedef std::complex<typename NumTraits<Scalar>::Real> ComplexScalar;
341 typedef internal::MatrixLogarithmAtomic<DynMatrixType> AtomicType;
342 AtomicType atomic;
343
344 internal::matrix_function_compute<typename DerivedEvalTypeClean::PlainObject>::run(m_A, atomic, result);
345 }
346
347 Index rows() const { return m_A.rows(); }
348 Index cols() const { return m_A.cols(); }
349
350private:
351 const DerivedNested m_A;
352};
353
354namespace internal {
355 template<typename Derived>
356 struct traits<MatrixLogarithmReturnValue<Derived> >
357 {
358 typedef typename Derived::PlainObject ReturnType;
359 };
360}
361
362
363/********** MatrixBase method **********/
364
365
366template <typename Derived>
367const MatrixLogarithmReturnValue<Derived> MatrixBase<Derived>::log() const
368{
369 eigen_assert(rows() == cols());
370 return MatrixLogarithmReturnValue<Derived>(derived());
371}
372
373} // end namespace Eigen
374
375#endif // EIGEN_MATRIX_LOGARITHM
const MatrixLogarithmReturnValue< Derived > log() const
Definition: MatrixLogarithm.h:367
Proxy for the matrix logarithm of some matrix (expression).
Definition: MatrixLogarithm.h:313
void evalTo(ResultType &result) const
Compute the matrix logarithm.
Definition: MatrixLogarithm.h:334
MatrixLogarithmReturnValue(const Derived &A)
Constructor.
Definition: MatrixLogarithm.h:327
void matrix_sqrt_triangular(const MatrixType &arg, ResultType &result)
Compute matrix square root of triangular matrix.
Definition: MatrixSquareRoot.h:206
Namespace containing all symbols from the Eigen library.
const Eigen::CwiseUnaryOp< Eigen::internal::scalar_abs_op< typename Derived::Scalar >, const Derived > abs(const Eigen::ArrayBase< Derived > &x)
EIGEN_DEFAULT_DENSE_INDEX_TYPE Index
const Eigen::CwiseUnaryOp< Eigen::internal::scalar_imag_op< typename Derived::Scalar >, const Derived > imag(const Eigen::ArrayBase< Derived > &x)
const Eigen::CwiseUnaryOp< Eigen::internal::scalar_log_op< typename Derived::Scalar >, const Derived > log(const Eigen::ArrayBase< Derived > &x)
const Eigen::CwiseUnaryOp< Eigen::internal::scalar_ceil_op< typename Derived::Scalar >, const Derived > ceil(const Eigen::ArrayBase< Derived > &x)