source: pacpussensors/trunk/Vislab/lib3dv/eigen/unsupported/Eigen/src/MatrixFunctions/MatrixPower.h@ 136

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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
17template<typename MatrixType>
18class MatrixPowerRetval : public ReturnByValue< MatrixPowerRetval<MatrixType> >
19{
20 public:
21 typedef typename MatrixType::RealScalar RealScalar;
22 typedef typename MatrixType::Index Index;
23
24 MatrixPowerRetval(MatrixPower<MatrixType>& pow, RealScalar p) : m_pow(pow), m_p(p)
25 { }
26
27 template<typename ResultType>
28 inline void evalTo(ResultType& res) const
29 { m_pow.compute(res, m_p); }
30
31 Index rows() const { return m_pow.rows(); }
32 Index cols() const { return m_pow.cols(); }
33
34 private:
35 MatrixPower<MatrixType>& m_pow;
36 const RealScalar m_p;
37 MatrixPowerRetval& operator=(const MatrixPowerRetval&);
38};
39
40template<typename MatrixType>
41class MatrixPowerAtomic
42{
43 private:
44 enum {
45 RowsAtCompileTime = MatrixType::RowsAtCompileTime,
46 MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime
47 };
48 typedef typename MatrixType::Scalar Scalar;
49 typedef typename MatrixType::RealScalar RealScalar;
50 typedef std::complex<RealScalar> ComplexScalar;
51 typedef typename MatrixType::Index Index;
52 typedef Array<Scalar, RowsAtCompileTime, 1, ColMajor, MaxRowsAtCompileTime> ArrayType;
53
54 const MatrixType& m_A;
55 RealScalar m_p;
56
57 void computePade(int degree, const MatrixType& IminusT, MatrixType& res) const;
58 void compute2x2(MatrixType& res, RealScalar p) const;
59 void computeBig(MatrixType& res) const;
60 static int getPadeDegree(float normIminusT);
61 static int getPadeDegree(double normIminusT);
62 static int getPadeDegree(long double normIminusT);
63 static ComplexScalar computeSuperDiag(const ComplexScalar&, const ComplexScalar&, RealScalar p);
64 static RealScalar computeSuperDiag(RealScalar, RealScalar, RealScalar p);
65
66 public:
67 MatrixPowerAtomic(const MatrixType& T, RealScalar p);
68 void compute(MatrixType& res) const;
69};
70
71template<typename MatrixType>
72MatrixPowerAtomic<MatrixType>::MatrixPowerAtomic(const MatrixType& T, RealScalar p) :
73 m_A(T), m_p(p)
74{ eigen_assert(T.rows() == T.cols()); }
75
76template<typename MatrixType>
77void MatrixPowerAtomic<MatrixType>::compute(MatrixType& res) const
78{
79 res.resizeLike(m_A);
80 switch (m_A.rows()) {
81 case 0:
82 break;
83 case 1:
84 res(0,0) = std::pow(m_A(0,0), m_p);
85 break;
86 case 2:
87 compute2x2(res, m_p);
88 break;
89 default:
90 computeBig(res);
91 }
92}
93
94template<typename MatrixType>
95void MatrixPowerAtomic<MatrixType>::computePade(int degree, const MatrixType& IminusT, MatrixType& res) const
96{
97 int i = degree<<1;
98 res = (m_p-degree) / ((i-1)<<1) * IminusT;
99 for (--i; i; --i) {
100 res = (MatrixType::Identity(IminusT.rows(), IminusT.cols()) + res).template triangularView<Upper>()
101 .solve((i==1 ? -m_p : i&1 ? (-m_p-(i>>1))/(i<<1) : (m_p-(i>>1))/((i-1)<<1)) * IminusT).eval();
102 }
103 res += MatrixType::Identity(IminusT.rows(), IminusT.cols());
104}
105
106// This function assumes that res has the correct size (see bug 614)
107template<typename MatrixType>
108void MatrixPowerAtomic<MatrixType>::compute2x2(MatrixType& res, RealScalar p) const
109{
110 using std::abs;
111 using std::pow;
112
113 res.coeffRef(0,0) = pow(m_A.coeff(0,0), p);
114
115 for (Index i=1; i < m_A.cols(); ++i) {
116 res.coeffRef(i,i) = pow(m_A.coeff(i,i), p);
117 if (m_A.coeff(i-1,i-1) == m_A.coeff(i,i))
118 res.coeffRef(i-1,i) = p * pow(m_A.coeff(i,i), p-1);
119 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)))
120 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));
121 else
122 res.coeffRef(i-1,i) = computeSuperDiag(m_A.coeff(i,i), m_A.coeff(i-1,i-1), p);
123 res.coeffRef(i-1,i) *= m_A.coeff(i-1,i);
124 }
125}
126
127template<typename MatrixType>
128void MatrixPowerAtomic<MatrixType>::computeBig(MatrixType& res) const
129{
130 const int digits = std::numeric_limits<RealScalar>::digits;
131 const RealScalar maxNormForPade = digits <= 24? 4.3386528e-1f: // sigle precision
132 digits <= 53? 2.789358995219730e-1: // double precision
133 digits <= 64? 2.4471944416607995472e-1L: // extended precision
134 digits <= 106? 1.1016843812851143391275867258512e-1L: // double-double
135 9.134603732914548552537150753385375e-2L; // quadruple precision
136 MatrixType IminusT, sqrtT, T = m_A.template triangularView<Upper>();
137 RealScalar normIminusT;
138 int degree, degree2, numberOfSquareRoots = 0;
139 bool hasExtraSquareRoot = false;
140
141 /* FIXME
142 * For singular T, norm(I - T) >= 1 but maxNormForPade < 1, leads to infinite
143 * loop. We should move 0 eigenvalues to bottom right corner. We need not
144 * worry about tiny values (e.g. 1e-300) because they will reach 1 if
145 * repetitively sqrt'ed.
146 *
147 * If the 0 eigenvalues are semisimple, they can form a 0 matrix at the
148 * bottom right corner.
149 *
150 * [ T A ]^p [ T^p (T^-1 T^p A) ]
151 * [ ] = [ ]
152 * [ 0 0 ] [ 0 0 ]
153 */
154 for (Index i=0; i < m_A.cols(); ++i)
155 eigen_assert(m_A(i,i) != RealScalar(0));
156
157 while (true) {
158 IminusT = MatrixType::Identity(m_A.rows(), m_A.cols()) - T;
159 normIminusT = IminusT.cwiseAbs().colwise().sum().maxCoeff();
160 if (normIminusT < maxNormForPade) {
161 degree = getPadeDegree(normIminusT);
162 degree2 = getPadeDegree(normIminusT/2);
163 if (degree - degree2 <= 1 || hasExtraSquareRoot)
164 break;
165 hasExtraSquareRoot = true;
166 }
167 MatrixSquareRootTriangular<MatrixType>(T).compute(sqrtT);
168 T = sqrtT.template triangularView<Upper>();
169 ++numberOfSquareRoots;
170 }
171 computePade(degree, IminusT, res);
172
173 for (; numberOfSquareRoots; --numberOfSquareRoots) {
174 compute2x2(res, std::ldexp(m_p, -numberOfSquareRoots));
175 res = res.template triangularView<Upper>() * res;
176 }
177 compute2x2(res, m_p);
178}
179
180template<typename MatrixType>
181inline int MatrixPowerAtomic<MatrixType>::getPadeDegree(float normIminusT)
182{
183 const float maxNormForPade[] = { 2.8064004e-1f /* degree = 3 */ , 4.3386528e-1f };
184 int degree = 3;
185 for (; degree <= 4; ++degree)
186 if (normIminusT <= maxNormForPade[degree - 3])
187 break;
188 return degree;
189}
190
191template<typename MatrixType>
192inline int MatrixPowerAtomic<MatrixType>::getPadeDegree(double normIminusT)
193{
194 const double maxNormForPade[] = { 1.884160592658218e-2 /* degree = 3 */ , 6.038881904059573e-2, 1.239917516308172e-1,
195 1.999045567181744e-1, 2.789358995219730e-1 };
196 int degree = 3;
197 for (; degree <= 7; ++degree)
198 if (normIminusT <= maxNormForPade[degree - 3])
199 break;
200 return degree;
201}
202
203template<typename MatrixType>
204inline int MatrixPowerAtomic<MatrixType>::getPadeDegree(long double normIminusT)
205{
206#if LDBL_MANT_DIG == 53
207 const int maxPadeDegree = 7;
208 const double maxNormForPade[] = { 1.884160592658218e-2L /* degree = 3 */ , 6.038881904059573e-2L, 1.239917516308172e-1L,
209 1.999045567181744e-1L, 2.789358995219730e-1L };
210#elif LDBL_MANT_DIG <= 64
211 const int maxPadeDegree = 8;
212 const double maxNormForPade[] = { 6.3854693117491799460e-3L /* degree = 3 */ , 2.6394893435456973676e-2L,
213 6.4216043030404063729e-2L, 1.1701165502926694307e-1L, 1.7904284231268670284e-1L, 2.4471944416607995472e-1L };
214#elif LDBL_MANT_DIG <= 106
215 const int maxPadeDegree = 10;
216 const double maxNormForPade[] = { 1.0007161601787493236741409687186e-4L /* degree = 3 */ ,
217 1.0007161601787493236741409687186e-3L, 4.7069769360887572939882574746264e-3L, 1.3220386624169159689406653101695e-2L,
218 2.8063482381631737920612944054906e-2L, 4.9625993951953473052385361085058e-2L, 7.7367040706027886224557538328171e-2L,
219 1.1016843812851143391275867258512e-1L };
220#else
221 const int maxPadeDegree = 10;
222 const double maxNormForPade[] = { 5.524506147036624377378713555116378e-5L /* degree = 3 */ ,
223 6.640600568157479679823602193345995e-4L, 3.227716520106894279249709728084626e-3L,
224 9.619593944683432960546978734646284e-3L, 2.134595382433742403911124458161147e-2L,
225 3.908166513900489428442993794761185e-2L, 6.266780814639442865832535460550138e-2L,
226 9.134603732914548552537150753385375e-2L };
227#endif
228 int degree = 3;
229 for (; degree <= maxPadeDegree; ++degree)
230 if (normIminusT <= maxNormForPade[degree - 3])
231 break;
232 return degree;
233}
234
235template<typename MatrixType>
236inline typename MatrixPowerAtomic<MatrixType>::ComplexScalar
237MatrixPowerAtomic<MatrixType>::computeSuperDiag(const ComplexScalar& curr, const ComplexScalar& prev, RealScalar p)
238{
239 ComplexScalar logCurr = std::log(curr);
240 ComplexScalar logPrev = std::log(prev);
241 int unwindingNumber = std::ceil((numext::imag(logCurr - logPrev) - M_PI) / (2*M_PI));
242 ComplexScalar w = numext::atanh2(curr - prev, curr + prev) + ComplexScalar(0, M_PI*unwindingNumber);
243 return RealScalar(2) * std::exp(RealScalar(0.5) * p * (logCurr + logPrev)) * std::sinh(p * w) / (curr - prev);
244}
245
246template<typename MatrixType>
247inline typename MatrixPowerAtomic<MatrixType>::RealScalar
248MatrixPowerAtomic<MatrixType>::computeSuperDiag(RealScalar curr, RealScalar prev, RealScalar p)
249{
250 RealScalar w = numext::atanh2(curr - prev, curr + prev);
251 return 2 * std::exp(p * (std::log(curr) + std::log(prev)) / 2) * std::sinh(p * w) / (curr - prev);
252}
253
254/**
255 * \ingroup MatrixFunctions_Module
256 *
257 * \brief Class for computing matrix powers.
258 *
259 * \tparam MatrixType type of the base, expected to be an instantiation
260 * of the Matrix class template.
261 *
262 * This class is capable of computing real/complex matrices raised to
263 * an arbitrary real power. Meanwhile, it saves the result of Schur
264 * decomposition if an non-integral power has even been calculated.
265 * Therefore, if you want to compute multiple (>= 2) matrix powers
266 * for the same matrix, using the class directly is more efficient than
267 * calling MatrixBase::pow().
268 *
269 * Example:
270 * \include MatrixPower_optimal.cpp
271 * Output: \verbinclude MatrixPower_optimal.out
272 */
273template<typename MatrixType>
274class MatrixPower
275{
276 private:
277 enum {
278 RowsAtCompileTime = MatrixType::RowsAtCompileTime,
279 ColsAtCompileTime = MatrixType::ColsAtCompileTime,
280 MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
281 MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
282 };
283 typedef typename MatrixType::Scalar Scalar;
284 typedef typename MatrixType::RealScalar RealScalar;
285 typedef typename MatrixType::Index Index;
286
287 public:
288 /**
289 * \brief Constructor.
290 *
291 * \param[in] A the base of the matrix power.
292 *
293 * The class stores a reference to A, so it should not be changed
294 * (or destroyed) before evaluation.
295 */
296 explicit MatrixPower(const MatrixType& A) : m_A(A), m_conditionNumber(0)
297 { eigen_assert(A.rows() == A.cols()); }
298
299 /**
300 * \brief Returns the matrix power.
301 *
302 * \param[in] p exponent, a real scalar.
303 * \return The expression \f$ A^p \f$, where A is specified in the
304 * constructor.
305 */
306 const MatrixPowerRetval<MatrixType> operator()(RealScalar p)
307 { return MatrixPowerRetval<MatrixType>(*this, p); }
308
309 /**
310 * \brief Compute the matrix power.
311 *
312 * \param[in] p exponent, a real scalar.
313 * \param[out] res \f$ A^p \f$ where A is specified in the
314 * constructor.
315 */
316 template<typename ResultType>
317 void compute(ResultType& res, RealScalar p);
318
319 Index rows() const { return m_A.rows(); }
320 Index cols() const { return m_A.cols(); }
321
322 private:
323 typedef std::complex<RealScalar> ComplexScalar;
324 typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, MatrixType::Options,
325 MaxRowsAtCompileTime, MaxColsAtCompileTime> ComplexMatrix;
326
327 typename MatrixType::Nested m_A;
328 MatrixType m_tmp;
329 ComplexMatrix m_T, m_U, m_fT;
330 RealScalar m_conditionNumber;
331
332 RealScalar modfAndInit(RealScalar, RealScalar*);
333
334 template<typename ResultType>
335 void computeIntPower(ResultType&, RealScalar);
336
337 template<typename ResultType>
338 void computeFracPower(ResultType&, RealScalar);
339
340 template<int Rows, int Cols, int Options, int MaxRows, int MaxCols>
341 static void revertSchur(
342 Matrix<ComplexScalar, Rows, Cols, Options, MaxRows, MaxCols>& res,
343 const ComplexMatrix& T,
344 const ComplexMatrix& U);
345
346 template<int Rows, int Cols, int Options, int MaxRows, int MaxCols>
347 static void revertSchur(
348 Matrix<RealScalar, Rows, Cols, Options, MaxRows, MaxCols>& res,
349 const ComplexMatrix& T,
350 const ComplexMatrix& U);
351};
352
353template<typename MatrixType>
354template<typename ResultType>
355void MatrixPower<MatrixType>::compute(ResultType& res, RealScalar p)
356{
357 switch (cols()) {
358 case 0:
359 break;
360 case 1:
361 res(0,0) = std::pow(m_A.coeff(0,0), p);
362 break;
363 default:
364 RealScalar intpart, x = modfAndInit(p, &intpart);
365 computeIntPower(res, intpart);
366 computeFracPower(res, x);
367 }
368}
369
370template<typename MatrixType>
371typename MatrixPower<MatrixType>::RealScalar
372MatrixPower<MatrixType>::modfAndInit(RealScalar x, RealScalar* intpart)
373{
374 typedef Array<RealScalar, RowsAtCompileTime, 1, ColMajor, MaxRowsAtCompileTime> RealArray;
375
376 *intpart = std::floor(x);
377 RealScalar res = x - *intpart;
378
379 if (!m_conditionNumber && res) {
380 const ComplexSchur<MatrixType> schurOfA(m_A);
381 m_T = schurOfA.matrixT();
382 m_U = schurOfA.matrixU();
383
384 const RealArray absTdiag = m_T.diagonal().array().abs();
385 m_conditionNumber = absTdiag.maxCoeff() / absTdiag.minCoeff();
386 }
387
388 if (res>RealScalar(0.5) && res>(1-res)*std::pow(m_conditionNumber, res)) {
389 --res;
390 ++*intpart;
391 }
392 return res;
393}
394
395template<typename MatrixType>
396template<typename ResultType>
397void MatrixPower<MatrixType>::computeIntPower(ResultType& res, RealScalar p)
398{
399 RealScalar pp = std::abs(p);
400
401 if (p<0) m_tmp = m_A.inverse();
402 else m_tmp = m_A;
403
404 res = MatrixType::Identity(rows(), cols());
405 while (pp >= 1) {
406 if (std::fmod(pp, 2) >= 1)
407 res = m_tmp * res;
408 m_tmp *= m_tmp;
409 pp /= 2;
410 }
411}
412
413template<typename MatrixType>
414template<typename ResultType>
415void MatrixPower<MatrixType>::computeFracPower(ResultType& res, RealScalar p)
416{
417 if (p) {
418 eigen_assert(m_conditionNumber);
419 MatrixPowerAtomic<ComplexMatrix>(m_T, p).compute(m_fT);
420 revertSchur(m_tmp, m_fT, m_U);
421 res = m_tmp * res;
422 }
423}
424
425template<typename MatrixType>
426template<int Rows, int Cols, int Options, int MaxRows, int MaxCols>
427inline void MatrixPower<MatrixType>::revertSchur(
428 Matrix<ComplexScalar, Rows, Cols, Options, MaxRows, MaxCols>& res,
429 const ComplexMatrix& T,
430 const ComplexMatrix& U)
431{ res.noalias() = U * (T.template triangularView<Upper>() * U.adjoint()); }
432
433template<typename MatrixType>
434template<int Rows, int Cols, int Options, int MaxRows, int MaxCols>
435inline void MatrixPower<MatrixType>::revertSchur(
436 Matrix<RealScalar, Rows, Cols, Options, MaxRows, MaxCols>& res,
437 const ComplexMatrix& T,
438 const ComplexMatrix& U)
439{ res.noalias() = (U * (T.template triangularView<Upper>() * U.adjoint())).real(); }
440
441/**
442 * \ingroup MatrixFunctions_Module
443 *
444 * \brief Proxy for the matrix power of some matrix (expression).
445 *
446 * \tparam Derived type of the base, a matrix (expression).
447 *
448 * This class holds the arguments to the matrix power until it is
449 * assigned or evaluated for some other reason (so the argument
450 * should not be changed in the meantime). It is the return type of
451 * MatrixBase::pow() and related functions and most of the
452 * time this is the only way it is used.
453 */
454template<typename Derived>
455class MatrixPowerReturnValue : public ReturnByValue< MatrixPowerReturnValue<Derived> >
456{
457 public:
458 typedef typename Derived::PlainObject PlainObject;
459 typedef typename Derived::RealScalar RealScalar;
460 typedef typename Derived::Index Index;
461
462 /**
463 * \brief Constructor.
464 *
465 * \param[in] A %Matrix (expression), the base of the matrix power.
466 * \param[in] p scalar, the exponent of the matrix power.
467 */
468 MatrixPowerReturnValue(const Derived& A, RealScalar p) : m_A(A), m_p(p)
469 { }
470
471 /**
472 * \brief Compute the matrix power.
473 *
474 * \param[out] result \f$ A^p \f$ where \p A and \p p are as in the
475 * constructor.
476 */
477 template<typename ResultType>
478 inline void evalTo(ResultType& res) const
479 { MatrixPower<PlainObject>(m_A.eval()).compute(res, m_p); }
480
481 Index rows() const { return m_A.rows(); }
482 Index cols() const { return m_A.cols(); }
483
484 private:
485 const Derived& m_A;
486 const RealScalar m_p;
487 MatrixPowerReturnValue& operator=(const MatrixPowerReturnValue&);
488};
489
490namespace internal {
491
492template<typename MatrixPowerType>
493struct traits< MatrixPowerRetval<MatrixPowerType> >
494{ typedef typename MatrixPowerType::PlainObject ReturnType; };
495
496template<typename Derived>
497struct traits< MatrixPowerReturnValue<Derived> >
498{ typedef typename Derived::PlainObject ReturnType; };
499
500}
501
502template<typename Derived>
503const MatrixPowerReturnValue<Derived> MatrixBase<Derived>::pow(const RealScalar& p) const
504{ return MatrixPowerReturnValue<Derived>(derived(), p); }
505
506} // namespace Eigen
507
508#endif // EIGEN_MATRIX_POWER
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