1 // This file is part of Eigen, a lightweight C++ template library
2 // for linear algebra.
3 //
4 // Copyright (C) 2012, 2013 Chen-Pang He <jdh8@ms63.hinet.net>
5 //
6 // This Source Code Form is subject to the terms of the Mozilla
7 // Public License v. 2.0. If a copy of the MPL was not distributed
8 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
9
10 #ifndef EIGEN_MATRIX_POWER
11 #define EIGEN_MATRIX_POWER
12
13 namespace Eigen {
14
15 template<typename MatrixType> class MatrixPower;
16
17 /**
18 * \ingroup MatrixFunctions_Module
19 *
20 * \brief Proxy for the matrix power of some matrix.
21 *
22 * \tparam MatrixType type of the base, a matrix.
23 *
24 * This class holds the arguments to the matrix power until it is
25 * assigned or evaluated for some other reason (so the argument
26 * should not be changed in the meantime). It is the return type of
27 * MatrixPower::operator() and related functions and most of the
28 * time this is the only way it is used.
29 */
30 /* TODO This class is only used by MatrixPower, so it should be nested
31 * into MatrixPower, like MatrixPower::ReturnValue. However, my
32 * compiler complained about unused template parameter in the
33 * following declaration in namespace internal.
34 *
35 * template<typename MatrixType>
36 * struct traits<MatrixPower<MatrixType>::ReturnValue>;
37 */
38 template<typename MatrixType>
39 class MatrixPowerParenthesesReturnValue : public ReturnByValue< MatrixPowerParenthesesReturnValue<MatrixType> >
40 {
41 public:
42 typedef typename MatrixType::RealScalar RealScalar;
43 typedef typename MatrixType::Index Index;
44
45 /**
46 * \brief Constructor.
47 *
48 * \param[in] pow %MatrixPower storing the base.
49 * \param[in] p scalar, the exponent of the matrix power.
50 */
MatrixPowerParenthesesReturnValue(MatrixPower<MatrixType> & pow,RealScalar p)51 MatrixPowerParenthesesReturnValue(MatrixPower<MatrixType>& pow, RealScalar p) : m_pow(pow), m_p(p)
52 { }
53
54 /**
55 * \brief Compute the matrix power.
56 *
57 * \param[out] result
58 */
59 template<typename ResultType>
evalTo(ResultType & res)60 inline void evalTo(ResultType& res) const
61 { m_pow.compute(res, m_p); }
62
rows()63 Index rows() const { return m_pow.rows(); }
cols()64 Index cols() const { return m_pow.cols(); }
65
66 private:
67 MatrixPower<MatrixType>& m_pow;
68 const RealScalar m_p;
69 };
70
71 /**
72 * \ingroup MatrixFunctions_Module
73 *
74 * \brief Class for computing matrix powers.
75 *
76 * \tparam MatrixType type of the base, expected to be an instantiation
77 * of the Matrix class template.
78 *
79 * This class is capable of computing triangular real/complex matrices
80 * raised to a power in the interval \f$ (-1, 1) \f$.
81 *
82 * \note Currently this class is only used by MatrixPower. One may
83 * insist that this be nested into MatrixPower. This class is here to
84 * faciliate future development of triangular matrix functions.
85 */
86 template<typename MatrixType>
87 class MatrixPowerAtomic : internal::noncopyable
88 {
89 private:
90 enum {
91 RowsAtCompileTime = MatrixType::RowsAtCompileTime,
92 MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime
93 };
94 typedef typename MatrixType::Scalar Scalar;
95 typedef typename MatrixType::RealScalar RealScalar;
96 typedef std::complex<RealScalar> ComplexScalar;
97 typedef typename MatrixType::Index Index;
98 typedef Block<MatrixType,Dynamic,Dynamic> ResultType;
99
100 const MatrixType& m_A;
101 RealScalar m_p;
102
103 void computePade(int degree, const MatrixType& IminusT, ResultType& res) const;
104 void compute2x2(ResultType& res, RealScalar p) const;
105 void computeBig(ResultType& res) const;
106 static int getPadeDegree(float normIminusT);
107 static int getPadeDegree(double normIminusT);
108 static int getPadeDegree(long double normIminusT);
109 static ComplexScalar computeSuperDiag(const ComplexScalar&, const ComplexScalar&, RealScalar p);
110 static RealScalar computeSuperDiag(RealScalar, RealScalar, RealScalar p);
111
112 public:
113 /**
114 * \brief Constructor.
115 *
116 * \param[in] T the base of the matrix power.
117 * \param[in] p the exponent of the matrix power, should be in
118 * \f$ (-1, 1) \f$.
119 *
120 * The class stores a reference to T, so it should not be changed
121 * (or destroyed) before evaluation. Only the upper triangular
122 * part of T is read.
123 */
124 MatrixPowerAtomic(const MatrixType& T, RealScalar p);
125
126 /**
127 * \brief Compute the matrix power.
128 *
129 * \param[out] res \f$ A^p \f$ where A and p are specified in the
130 * constructor.
131 */
132 void compute(ResultType& res) const;
133 };
134
135 template<typename MatrixType>
MatrixPowerAtomic(const MatrixType & T,RealScalar p)136 MatrixPowerAtomic<MatrixType>::MatrixPowerAtomic(const MatrixType& T, RealScalar p) :
137 m_A(T), m_p(p)
138 {
139 eigen_assert(T.rows() == T.cols());
140 eigen_assert(p > -1 && p < 1);
141 }
142
143 template<typename MatrixType>
compute(ResultType & res)144 void MatrixPowerAtomic<MatrixType>::compute(ResultType& res) const
145 {
146 using std::pow;
147 switch (m_A.rows()) {
148 case 0:
149 break;
150 case 1:
151 res(0,0) = pow(m_A(0,0), m_p);
152 break;
153 case 2:
154 compute2x2(res, m_p);
155 break;
156 default:
157 computeBig(res);
158 }
159 }
160
161 template<typename MatrixType>
computePade(int degree,const MatrixType & IminusT,ResultType & res)162 void MatrixPowerAtomic<MatrixType>::computePade(int degree, const MatrixType& IminusT, ResultType& res) const
163 {
164 int i = 2*degree;
165 res = (m_p-degree) / (2*i-2) * IminusT;
166
167 for (--i; i; --i) {
168 res = (MatrixType::Identity(IminusT.rows(), IminusT.cols()) + res).template triangularView<Upper>()
169 .solve((i==1 ? -m_p : i&1 ? (-m_p-i/2)/(2*i) : (m_p-i/2)/(2*i-2)) * IminusT).eval();
170 }
171 res += MatrixType::Identity(IminusT.rows(), IminusT.cols());
172 }
173
174 // This function assumes that res has the correct size (see bug 614)
175 template<typename MatrixType>
compute2x2(ResultType & res,RealScalar p)176 void MatrixPowerAtomic<MatrixType>::compute2x2(ResultType& res, RealScalar p) const
177 {
178 using std::abs;
179 using std::pow;
180 res.coeffRef(0,0) = pow(m_A.coeff(0,0), p);
181
182 for (Index i=1; i < m_A.cols(); ++i) {
183 res.coeffRef(i,i) = pow(m_A.coeff(i,i), p);
184 if (m_A.coeff(i-1,i-1) == m_A.coeff(i,i))
185 res.coeffRef(i-1,i) = p * pow(m_A.coeff(i,i), p-1);
186 else if (2*abs(m_A.coeff(i-1,i-1)) < abs(m_A.coeff(i,i)) || 2*abs(m_A.coeff(i,i)) < abs(m_A.coeff(i-1,i-1)))
187 res.coeffRef(i-1,i) = (res.coeff(i,i)-res.coeff(i-1,i-1)) / (m_A.coeff(i,i)-m_A.coeff(i-1,i-1));
188 else
189 res.coeffRef(i-1,i) = computeSuperDiag(m_A.coeff(i,i), m_A.coeff(i-1,i-1), p);
190 res.coeffRef(i-1,i) *= m_A.coeff(i-1,i);
191 }
192 }
193
194 template<typename MatrixType>
computeBig(ResultType & res)195 void MatrixPowerAtomic<MatrixType>::computeBig(ResultType& res) const
196 {
197 using std::ldexp;
198 const int digits = std::numeric_limits<RealScalar>::digits;
199 const RealScalar maxNormForPade = digits <= 24? 4.3386528e-1L // single precision
200 : digits <= 53? 2.789358995219730e-1L // double precision
201 : digits <= 64? 2.4471944416607995472e-1L // extended precision
202 : digits <= 106? 1.1016843812851143391275867258512e-1L // double-double
203 : 9.134603732914548552537150753385375e-2L; // quadruple precision
204 MatrixType IminusT, sqrtT, T = m_A.template triangularView<Upper>();
205 RealScalar normIminusT;
206 int degree, degree2, numberOfSquareRoots = 0;
207 bool hasExtraSquareRoot = false;
208
209 for (Index i=0; i < m_A.cols(); ++i)
210 eigen_assert(m_A(i,i) != RealScalar(0));
211
212 while (true) {
213 IminusT = MatrixType::Identity(m_A.rows(), m_A.cols()) - T;
214 normIminusT = IminusT.cwiseAbs().colwise().sum().maxCoeff();
215 if (normIminusT < maxNormForPade) {
216 degree = getPadeDegree(normIminusT);
217 degree2 = getPadeDegree(normIminusT/2);
218 if (degree - degree2 <= 1 || hasExtraSquareRoot)
219 break;
220 hasExtraSquareRoot = true;
221 }
222 matrix_sqrt_triangular(T, sqrtT);
223 T = sqrtT.template triangularView<Upper>();
224 ++numberOfSquareRoots;
225 }
226 computePade(degree, IminusT, res);
227
228 for (; numberOfSquareRoots; --numberOfSquareRoots) {
229 compute2x2(res, ldexp(m_p, -numberOfSquareRoots));
230 res = res.template triangularView<Upper>() * res;
231 }
232 compute2x2(res, m_p);
233 }
234
235 template<typename MatrixType>
getPadeDegree(float normIminusT)236 inline int MatrixPowerAtomic<MatrixType>::getPadeDegree(float normIminusT)
237 {
238 const float maxNormForPade[] = { 2.8064004e-1f /* degree = 3 */ , 4.3386528e-1f };
239 int degree = 3;
240 for (; degree <= 4; ++degree)
241 if (normIminusT <= maxNormForPade[degree - 3])
242 break;
243 return degree;
244 }
245
246 template<typename MatrixType>
getPadeDegree(double normIminusT)247 inline int MatrixPowerAtomic<MatrixType>::getPadeDegree(double normIminusT)
248 {
249 const double maxNormForPade[] = { 1.884160592658218e-2 /* degree = 3 */ , 6.038881904059573e-2, 1.239917516308172e-1,
250 1.999045567181744e-1, 2.789358995219730e-1 };
251 int degree = 3;
252 for (; degree <= 7; ++degree)
253 if (normIminusT <= maxNormForPade[degree - 3])
254 break;
255 return degree;
256 }
257
258 template<typename MatrixType>
getPadeDegree(long double normIminusT)259 inline int MatrixPowerAtomic<MatrixType>::getPadeDegree(long double normIminusT)
260 {
261 #if LDBL_MANT_DIG == 53
262 const int maxPadeDegree = 7;
263 const double maxNormForPade[] = { 1.884160592658218e-2L /* degree = 3 */ , 6.038881904059573e-2L, 1.239917516308172e-1L,
264 1.999045567181744e-1L, 2.789358995219730e-1L };
265 #elif LDBL_MANT_DIG <= 64
266 const int maxPadeDegree = 8;
267 const long double maxNormForPade[] = { 6.3854693117491799460e-3L /* degree = 3 */ , 2.6394893435456973676e-2L,
268 6.4216043030404063729e-2L, 1.1701165502926694307e-1L, 1.7904284231268670284e-1L, 2.4471944416607995472e-1L };
269 #elif LDBL_MANT_DIG <= 106
270 const int maxPadeDegree = 10;
271 const double maxNormForPade[] = { 1.0007161601787493236741409687186e-4L /* degree = 3 */ ,
272 1.0007161601787493236741409687186e-3L, 4.7069769360887572939882574746264e-3L, 1.3220386624169159689406653101695e-2L,
273 2.8063482381631737920612944054906e-2L, 4.9625993951953473052385361085058e-2L, 7.7367040706027886224557538328171e-2L,
274 1.1016843812851143391275867258512e-1L };
275 #else
276 const int maxPadeDegree = 10;
277 const double maxNormForPade[] = { 5.524506147036624377378713555116378e-5L /* degree = 3 */ ,
278 6.640600568157479679823602193345995e-4L, 3.227716520106894279249709728084626e-3L,
279 9.619593944683432960546978734646284e-3L, 2.134595382433742403911124458161147e-2L,
280 3.908166513900489428442993794761185e-2L, 6.266780814639442865832535460550138e-2L,
281 9.134603732914548552537150753385375e-2L };
282 #endif
283 int degree = 3;
284 for (; degree <= maxPadeDegree; ++degree)
285 if (normIminusT <= maxNormForPade[degree - 3])
286 break;
287 return degree;
288 }
289
290 template<typename MatrixType>
291 inline typename MatrixPowerAtomic<MatrixType>::ComplexScalar
computeSuperDiag(const ComplexScalar & curr,const ComplexScalar & prev,RealScalar p)292 MatrixPowerAtomic<MatrixType>::computeSuperDiag(const ComplexScalar& curr, const ComplexScalar& prev, RealScalar p)
293 {
294 using std::ceil;
295 using std::exp;
296 using std::log;
297 using std::sinh;
298
299 ComplexScalar logCurr = log(curr);
300 ComplexScalar logPrev = log(prev);
301 int unwindingNumber = ceil((numext::imag(logCurr - logPrev) - RealScalar(EIGEN_PI)) / RealScalar(2*EIGEN_PI));
302 ComplexScalar w = numext::log1p((curr-prev)/prev)/RealScalar(2) + ComplexScalar(0, EIGEN_PI*unwindingNumber);
303 return RealScalar(2) * exp(RealScalar(0.5) * p * (logCurr + logPrev)) * sinh(p * w) / (curr - prev);
304 }
305
306 template<typename MatrixType>
307 inline typename MatrixPowerAtomic<MatrixType>::RealScalar
computeSuperDiag(RealScalar curr,RealScalar prev,RealScalar p)308 MatrixPowerAtomic<MatrixType>::computeSuperDiag(RealScalar curr, RealScalar prev, RealScalar p)
309 {
310 using std::exp;
311 using std::log;
312 using std::sinh;
313
314 RealScalar w = numext::log1p((curr-prev)/prev)/RealScalar(2);
315 return 2 * exp(p * (log(curr) + log(prev)) / 2) * sinh(p * w) / (curr - prev);
316 }
317
318 /**
319 * \ingroup MatrixFunctions_Module
320 *
321 * \brief Class for computing matrix powers.
322 *
323 * \tparam MatrixType type of the base, expected to be an instantiation
324 * of the Matrix class template.
325 *
326 * This class is capable of computing real/complex matrices raised to
327 * an arbitrary real power. Meanwhile, it saves the result of Schur
328 * decomposition if an non-integral power has even been calculated.
329 * Therefore, if you want to compute multiple (>= 2) matrix powers
330 * for the same matrix, using the class directly is more efficient than
331 * calling MatrixBase::pow().
332 *
333 * Example:
334 * \include MatrixPower_optimal.cpp
335 * Output: \verbinclude MatrixPower_optimal.out
336 */
337 template<typename MatrixType>
338 class MatrixPower : internal::noncopyable
339 {
340 private:
341 typedef typename MatrixType::Scalar Scalar;
342 typedef typename MatrixType::RealScalar RealScalar;
343 typedef typename MatrixType::Index Index;
344
345 public:
346 /**
347 * \brief Constructor.
348 *
349 * \param[in] A the base of the matrix power.
350 *
351 * The class stores a reference to A, so it should not be changed
352 * (or destroyed) before evaluation.
353 */
MatrixPower(const MatrixType & A)354 explicit MatrixPower(const MatrixType& A) :
355 m_A(A),
356 m_conditionNumber(0),
357 m_rank(A.cols()),
358 m_nulls(0)
359 { eigen_assert(A.rows() == A.cols()); }
360
361 /**
362 * \brief Returns the matrix power.
363 *
364 * \param[in] p exponent, a real scalar.
365 * \return The expression \f$ A^p \f$, where A is specified in the
366 * constructor.
367 */
operator()368 const MatrixPowerParenthesesReturnValue<MatrixType> operator()(RealScalar p)
369 { return MatrixPowerParenthesesReturnValue<MatrixType>(*this, p); }
370
371 /**
372 * \brief Compute the matrix power.
373 *
374 * \param[in] p exponent, a real scalar.
375 * \param[out] res \f$ A^p \f$ where A is specified in the
376 * constructor.
377 */
378 template<typename ResultType>
379 void compute(ResultType& res, RealScalar p);
380
rows()381 Index rows() const { return m_A.rows(); }
cols()382 Index cols() const { return m_A.cols(); }
383
384 private:
385 typedef std::complex<RealScalar> ComplexScalar;
386 typedef Matrix<ComplexScalar, Dynamic, Dynamic, 0,
387 MatrixType::RowsAtCompileTime, MatrixType::ColsAtCompileTime> ComplexMatrix;
388
389 /** \brief Reference to the base of matrix power. */
390 typename MatrixType::Nested m_A;
391
392 /** \brief Temporary storage. */
393 MatrixType m_tmp;
394
395 /** \brief Store the result of Schur decomposition. */
396 ComplexMatrix m_T, m_U;
397
398 /** \brief Store fractional power of m_T. */
399 ComplexMatrix m_fT;
400
401 /**
402 * \brief Condition number of m_A.
403 *
404 * It is initialized as 0 to avoid performing unnecessary Schur
405 * decomposition, which is the bottleneck.
406 */
407 RealScalar m_conditionNumber;
408
409 /** \brief Rank of m_A. */
410 Index m_rank;
411
412 /** \brief Rank deficiency of m_A. */
413 Index m_nulls;
414
415 /**
416 * \brief Split p into integral part and fractional part.
417 *
418 * \param[in] p The exponent.
419 * \param[out] p The fractional part ranging in \f$ (-1, 1) \f$.
420 * \param[out] intpart The integral part.
421 *
422 * Only if the fractional part is nonzero, it calls initialize().
423 */
424 void split(RealScalar& p, RealScalar& intpart);
425
426 /** \brief Perform Schur decomposition for fractional power. */
427 void initialize();
428
429 template<typename ResultType>
430 void computeIntPower(ResultType& res, RealScalar p);
431
432 template<typename ResultType>
433 void computeFracPower(ResultType& res, RealScalar p);
434
435 template<int Rows, int Cols, int Options, int MaxRows, int MaxCols>
436 static void revertSchur(
437 Matrix<ComplexScalar, Rows, Cols, Options, MaxRows, MaxCols>& res,
438 const ComplexMatrix& T,
439 const ComplexMatrix& U);
440
441 template<int Rows, int Cols, int Options, int MaxRows, int MaxCols>
442 static void revertSchur(
443 Matrix<RealScalar, Rows, Cols, Options, MaxRows, MaxCols>& res,
444 const ComplexMatrix& T,
445 const ComplexMatrix& U);
446 };
447
448 template<typename MatrixType>
449 template<typename ResultType>
compute(ResultType & res,RealScalar p)450 void MatrixPower<MatrixType>::compute(ResultType& res, RealScalar p)
451 {
452 using std::pow;
453 switch (cols()) {
454 case 0:
455 break;
456 case 1:
457 res(0,0) = pow(m_A.coeff(0,0), p);
458 break;
459 default:
460 RealScalar intpart;
461 split(p, intpart);
462
463 res = MatrixType::Identity(rows(), cols());
464 computeIntPower(res, intpart);
465 if (p) computeFracPower(res, p);
466 }
467 }
468
469 template<typename MatrixType>
split(RealScalar & p,RealScalar & intpart)470 void MatrixPower<MatrixType>::split(RealScalar& p, RealScalar& intpart)
471 {
472 using std::floor;
473 using std::pow;
474
475 intpart = floor(p);
476 p -= intpart;
477
478 // Perform Schur decomposition if it is not yet performed and the power is
479 // not an integer.
480 if (!m_conditionNumber && p)
481 initialize();
482
483 // Choose the more stable of intpart = floor(p) and intpart = ceil(p).
484 if (p > RealScalar(0.5) && p > (1-p) * pow(m_conditionNumber, p)) {
485 --p;
486 ++intpart;
487 }
488 }
489
490 template<typename MatrixType>
initialize()491 void MatrixPower<MatrixType>::initialize()
492 {
493 const ComplexSchur<MatrixType> schurOfA(m_A);
494 JacobiRotation<ComplexScalar> rot;
495 ComplexScalar eigenvalue;
496
497 m_fT.resizeLike(m_A);
498 m_T = schurOfA.matrixT();
499 m_U = schurOfA.matrixU();
500 m_conditionNumber = m_T.diagonal().array().abs().maxCoeff() / m_T.diagonal().array().abs().minCoeff();
501
502 // Move zero eigenvalues to the bottom right corner.
503 for (Index i = cols()-1; i>=0; --i) {
504 if (m_rank <= 2)
505 return;
506 if (m_T.coeff(i,i) == RealScalar(0)) {
507 for (Index j=i+1; j < m_rank; ++j) {
508 eigenvalue = m_T.coeff(j,j);
509 rot.makeGivens(m_T.coeff(j-1,j), eigenvalue);
510 m_T.applyOnTheRight(j-1, j, rot);
511 m_T.applyOnTheLeft(j-1, j, rot.adjoint());
512 m_T.coeffRef(j-1,j-1) = eigenvalue;
513 m_T.coeffRef(j,j) = RealScalar(0);
514 m_U.applyOnTheRight(j-1, j, rot);
515 }
516 --m_rank;
517 }
518 }
519
520 m_nulls = rows() - m_rank;
521 if (m_nulls) {
522 eigen_assert(m_T.bottomRightCorner(m_nulls, m_nulls).isZero()
523 && "Base of matrix power should be invertible or with a semisimple zero eigenvalue.");
524 m_fT.bottomRows(m_nulls).fill(RealScalar(0));
525 }
526 }
527
528 template<typename MatrixType>
529 template<typename ResultType>
computeIntPower(ResultType & res,RealScalar p)530 void MatrixPower<MatrixType>::computeIntPower(ResultType& res, RealScalar p)
531 {
532 using std::abs;
533 using std::fmod;
534 RealScalar pp = abs(p);
535
536 if (p<0)
537 m_tmp = m_A.inverse();
538 else
539 m_tmp = m_A;
540
541 while (true) {
542 if (fmod(pp, 2) >= 1)
543 res = m_tmp * res;
544 pp /= 2;
545 if (pp < 1)
546 break;
547 m_tmp *= m_tmp;
548 }
549 }
550
551 template<typename MatrixType>
552 template<typename ResultType>
computeFracPower(ResultType & res,RealScalar p)553 void MatrixPower<MatrixType>::computeFracPower(ResultType& res, RealScalar p)
554 {
555 Block<ComplexMatrix,Dynamic,Dynamic> blockTp(m_fT, 0, 0, m_rank, m_rank);
556 eigen_assert(m_conditionNumber);
557 eigen_assert(m_rank + m_nulls == rows());
558
559 MatrixPowerAtomic<ComplexMatrix>(m_T.topLeftCorner(m_rank, m_rank), p).compute(blockTp);
560 if (m_nulls) {
561 m_fT.topRightCorner(m_rank, m_nulls) = m_T.topLeftCorner(m_rank, m_rank).template triangularView<Upper>()
562 .solve(blockTp * m_T.topRightCorner(m_rank, m_nulls));
563 }
564 revertSchur(m_tmp, m_fT, m_U);
565 res = m_tmp * res;
566 }
567
568 template<typename MatrixType>
569 template<int Rows, int Cols, int Options, int MaxRows, int MaxCols>
revertSchur(Matrix<ComplexScalar,Rows,Cols,Options,MaxRows,MaxCols> & res,const ComplexMatrix & T,const ComplexMatrix & U)570 inline void MatrixPower<MatrixType>::revertSchur(
571 Matrix<ComplexScalar, Rows, Cols, Options, MaxRows, MaxCols>& res,
572 const ComplexMatrix& T,
573 const ComplexMatrix& U)
574 { res.noalias() = U * (T.template triangularView<Upper>() * U.adjoint()); }
575
576 template<typename MatrixType>
577 template<int Rows, int Cols, int Options, int MaxRows, int MaxCols>
revertSchur(Matrix<RealScalar,Rows,Cols,Options,MaxRows,MaxCols> & res,const ComplexMatrix & T,const ComplexMatrix & U)578 inline void MatrixPower<MatrixType>::revertSchur(
579 Matrix<RealScalar, Rows, Cols, Options, MaxRows, MaxCols>& res,
580 const ComplexMatrix& T,
581 const ComplexMatrix& U)
582 { res.noalias() = (U * (T.template triangularView<Upper>() * U.adjoint())).real(); }
583
584 /**
585 * \ingroup MatrixFunctions_Module
586 *
587 * \brief Proxy for the matrix power of some matrix (expression).
588 *
589 * \tparam Derived type of the base, a matrix (expression).
590 *
591 * This class holds the arguments to the matrix power until it is
592 * assigned or evaluated for some other reason (so the argument
593 * should not be changed in the meantime). It is the return type of
594 * MatrixBase::pow() and related functions and most of the
595 * time this is the only way it is used.
596 */
597 template<typename Derived>
598 class MatrixPowerReturnValue : public ReturnByValue< MatrixPowerReturnValue<Derived> >
599 {
600 public:
601 typedef typename Derived::PlainObject PlainObject;
602 typedef typename Derived::RealScalar RealScalar;
603 typedef typename Derived::Index Index;
604
605 /**
606 * \brief Constructor.
607 *
608 * \param[in] A %Matrix (expression), the base of the matrix power.
609 * \param[in] p real scalar, the exponent of the matrix power.
610 */
MatrixPowerReturnValue(const Derived & A,RealScalar p)611 MatrixPowerReturnValue(const Derived& A, RealScalar p) : m_A(A), m_p(p)
612 { }
613
614 /**
615 * \brief Compute the matrix power.
616 *
617 * \param[out] result \f$ A^p \f$ where \p A and \p p are as in the
618 * constructor.
619 */
620 template<typename ResultType>
evalTo(ResultType & res)621 inline void evalTo(ResultType& res) const
622 { MatrixPower<PlainObject>(m_A.eval()).compute(res, m_p); }
623
rows()624 Index rows() const { return m_A.rows(); }
cols()625 Index cols() const { return m_A.cols(); }
626
627 private:
628 const Derived& m_A;
629 const RealScalar m_p;
630 };
631
632 /**
633 * \ingroup MatrixFunctions_Module
634 *
635 * \brief Proxy for the matrix power of some matrix (expression).
636 *
637 * \tparam Derived type of the base, a matrix (expression).
638 *
639 * This class holds the arguments to the matrix power until it is
640 * assigned or evaluated for some other reason (so the argument
641 * should not be changed in the meantime). It is the return type of
642 * MatrixBase::pow() and related functions and most of the
643 * time this is the only way it is used.
644 */
645 template<typename Derived>
646 class MatrixComplexPowerReturnValue : public ReturnByValue< MatrixComplexPowerReturnValue<Derived> >
647 {
648 public:
649 typedef typename Derived::PlainObject PlainObject;
650 typedef typename std::complex<typename Derived::RealScalar> ComplexScalar;
651 typedef typename Derived::Index Index;
652
653 /**
654 * \brief Constructor.
655 *
656 * \param[in] A %Matrix (expression), the base of the matrix power.
657 * \param[in] p complex scalar, the exponent of the matrix power.
658 */
MatrixComplexPowerReturnValue(const Derived & A,const ComplexScalar & p)659 MatrixComplexPowerReturnValue(const Derived& A, const ComplexScalar& p) : m_A(A), m_p(p)
660 { }
661
662 /**
663 * \brief Compute the matrix power.
664 *
665 * Because \p p is complex, \f$ A^p \f$ is simply evaluated as \f$
666 * \exp(p \log(A)) \f$.
667 *
668 * \param[out] result \f$ A^p \f$ where \p A and \p p are as in the
669 * constructor.
670 */
671 template<typename ResultType>
evalTo(ResultType & res)672 inline void evalTo(ResultType& res) const
673 { res = (m_p * m_A.log()).exp(); }
674
rows()675 Index rows() const { return m_A.rows(); }
cols()676 Index cols() const { return m_A.cols(); }
677
678 private:
679 const Derived& m_A;
680 const ComplexScalar m_p;
681 };
682
683 namespace internal {
684
685 template<typename MatrixPowerType>
686 struct traits< MatrixPowerParenthesesReturnValue<MatrixPowerType> >
687 { typedef typename MatrixPowerType::PlainObject ReturnType; };
688
689 template<typename Derived>
690 struct traits< MatrixPowerReturnValue<Derived> >
691 { typedef typename Derived::PlainObject ReturnType; };
692
693 template<typename Derived>
694 struct traits< MatrixComplexPowerReturnValue<Derived> >
695 { typedef typename Derived::PlainObject ReturnType; };
696
697 }
698
699 template<typename Derived>
700 const MatrixPowerReturnValue<Derived> MatrixBase<Derived>::pow(const RealScalar& p) const
701 { return MatrixPowerReturnValue<Derived>(derived(), p); }
702
703 template<typename Derived>
704 const MatrixComplexPowerReturnValue<Derived> MatrixBase<Derived>::pow(const std::complex<RealScalar>& p) const
705 { return MatrixComplexPowerReturnValue<Derived>(derived(), p); }
706
707 } // namespace Eigen
708
709 #endif // EIGEN_MATRIX_POWER
710