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