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
18namespace Eigen {
19
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
42 MatrixLogarithmAtomic() { }
43
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
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
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
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
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 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 */
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
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
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;
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
Eigen::MatrixFunction
Class for computing matrix functions.
Definition MatrixFunction.h:38
Eigen::MatrixFunction::compute
void compute(ResultType &result)
Compute the matrix function.
Eigen::MatrixLogarithmAtomic
Helper class for computing matrix logarithm of atomic matrices.
Definition MatrixLogarithm.h:32
Eigen::MatrixLogarithmAtomic::compute
MatrixType compute(const MatrixType &A)
Compute matrix logarithm of atomic matrix.
Definition MatrixLogarithm.h:82
Eigen::MatrixLogarithmAtomic::MatrixLogarithmAtomic
MatrixLogarithmAtomic()
Constructor.
Definition MatrixLogarithm.h:42
Eigen::MatrixLogarithmReturnValue
Proxy for the matrix logarithm of some matrix (expression).
Definition MatrixLogarithm.h:431
Eigen::MatrixLogarithmReturnValue::MatrixLogarithmReturnValue
MatrixLogarithmReturnValue(const Derived &A)
Constructor.
Definition MatrixLogarithm.h:441
Eigen::MatrixLogarithmReturnValue::evalTo
void evalTo(ResultType &result) const
Compute the matrix logarithm.
Definition MatrixLogarithm.h:448
Eigen::MatrixSquareRootTriangular
Class for computing matrix square roots of upper triangular matrices.
Definition MatrixSquareRoot.h:265
Eigen::MatrixSquareRootTriangular::compute
void compute(ResultType &result)
Compute the matrix square root.
Definition MatrixSquareRoot.h:290
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