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
13namespace Eigen {
14
15template<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 */
38template<typename MatrixType>
39class 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     */
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>
60    inline void evalTo(ResultType& res) const
61    { m_pow.compute(res, m_p); }
62
63    Index rows() const { return m_pow.rows(); }
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$.
81 *
82 * \note Currently this class is only used by MatrixPower. One may
83 * insist that this be nested into MatrixPower. This class is here to
84 * faciliate future development of triangular matrix functions.
85 */
86template<typename MatrixType>
87class MatrixPowerAtomic : internal::noncopyable
88{
89  private:
90    enum {
91      RowsAtCompileTime = MatrixType::RowsAtCompileTime,
92      MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime
93    };
94    typedef typename MatrixType::Scalar Scalar;
95    typedef typename MatrixType::RealScalar RealScalar;
96    typedef std::complex<RealScalar> ComplexScalar;
97    typedef typename MatrixType::Index Index;
98    typedef Block<MatrixType,Dynamic,Dynamic> ResultType;
99
100    const MatrixType& m_A;
101    RealScalar m_p;
102
103    void computePade(int degree, const MatrixType& IminusT, ResultType& res) const;
104    void compute2x2(ResultType& res, RealScalar p) const;
105    void computeBig(ResultType& res) const;
106    static int getPadeDegree(float normIminusT);
107    static int getPadeDegree(double normIminusT);
108    static int getPadeDegree(long double normIminusT);
109    static ComplexScalar computeSuperDiag(const ComplexScalar&, const ComplexScalar&, RealScalar p);
110    static RealScalar computeSuperDiag(RealScalar, RealScalar, RealScalar p);
111
112  public:
113    /**
114     * \brief Constructor.
115     *
116     * \param[in] T  the base of the matrix power.
117     * \param[in] p  the exponent of the matrix power, should be in
118     * \f$ (-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
135template<typename MatrixType>
136MatrixPowerAtomic<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
143template<typename MatrixType>
144void 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
161template<typename MatrixType>
162void 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)
175template<typename MatrixType>
176void 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
194template<typename MatrixType>
195void 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
235template<typename MatrixType>
236inline 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
246template<typename MatrixType>
247inline 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
258template<typename MatrixType>
259inline 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
290template<typename MatrixType>
291inline typename MatrixPowerAtomic<MatrixType>::ComplexScalar
292MatrixPowerAtomic<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
306template<typename MatrixType>
307inline typename MatrixPowerAtomic<MatrixType>::RealScalar
308MatrixPowerAtomic<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 */
337template<typename MatrixType>
338class 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     */
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$, where A is specified in the
366     * constructor.
367     */
368    const MatrixPowerParenthesesReturnValue<MatrixType> operator()(RealScalar p)
369    { return MatrixPowerParenthesesReturnValue<MatrixType>(*this, p); }
370
371    /**
372     * \brief Compute the matrix power.
373     *
374     * \param[in]  p    exponent, a real scalar.
375     * \param[out] res  \f$ A^p \f$ where A is specified in the
376     * constructor.
377     */
378    template<typename ResultType>
379    void compute(ResultType& res, RealScalar p);
380
381    Index rows() const { return m_A.rows(); }
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
448template<typename MatrixType>
449template<typename ResultType>
450void 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
469template<typename MatrixType>
470void 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
490template<typename MatrixType>
491void 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
528template<typename MatrixType>
529template<typename ResultType>
530void 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
551template<typename MatrixType>
552template<typename ResultType>
553void 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
568template<typename MatrixType>
569template<int Rows, int Cols, int Options, int MaxRows, int MaxCols>
570inline 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
576template<typename MatrixType>
577template<int Rows, int Cols, int Options, int MaxRows, int MaxCols>
578inline 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 */
597template<typename Derived>
598class 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     */
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>
621    inline void evalTo(ResultType& res) const
622    { MatrixPower<PlainObject>(m_A.eval()).compute(res, m_p); }
623
624    Index rows() const { return m_A.rows(); }
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 */
645template<typename Derived>
646class 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     */
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>
672    inline void evalTo(ResultType& res) const
673    { res = (m_p * m_A.log()).exp(); }
674
675    Index rows() const { return m_A.rows(); }
676    Index cols() const { return m_A.cols(); }
677
678  private:
679    const Derived& m_A;
680    const ComplexScalar m_p;
681};
682
683namespace internal {
684
685template<typename MatrixPowerType>
686struct traits< MatrixPowerParenthesesReturnValue<MatrixPowerType> >
687{ typedef typename MatrixPowerType::PlainObject ReturnType; };
688
689template<typename Derived>
690struct traits< MatrixPowerReturnValue<Derived> >
691{ typedef typename Derived::PlainObject ReturnType; };
692
693template<typename Derived>
694struct traits< MatrixComplexPowerReturnValue<Derived> >
695{ typedef typename Derived::PlainObject ReturnType; };
696
697}
698
699template<typename Derived>
700const MatrixPowerReturnValue<Derived> MatrixBase<Derived>::pow(const RealScalar& p) const
701{ return MatrixPowerReturnValue<Derived>(derived(), p); }
702
703template<typename Derived>
704const 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