Eigen-unsupported  3.4.90 (git rev a4098ac676528a83cfb73d4d26ce1b42ec05f47c)
MatrixPower.h
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#include "./InternalHeaderCheck.h"
14
15namespace Eigen {
16
17template<typename MatrixType> class MatrixPower;
18
32/* TODO This class is only used by MatrixPower, so it should be nested
33 * into MatrixPower, like MatrixPower::ReturnValue. However, my
34 * compiler complained about unused template parameter in the
35 * following declaration in namespace internal.
36 *
37 * template<typename MatrixType>
38 * struct traits<MatrixPower<MatrixType>::ReturnValue>;
39 */
40template<typename MatrixType>
41class MatrixPowerParenthesesReturnValue : public ReturnByValue< MatrixPowerParenthesesReturnValue<MatrixType> >
42{
43 public:
44 typedef typename MatrixType::RealScalar RealScalar;
45
52 MatrixPowerParenthesesReturnValue(MatrixPower<MatrixType>& pow, RealScalar p) : m_pow(pow), m_p(p)
53 { }
54
60 template<typename ResultType>
61 inline void evalTo(ResultType& result) const
62 { m_pow.compute(result, m_p); }
63
64 Index rows() const { return m_pow.rows(); }
65 Index cols() const { return m_pow.cols(); }
66
67 private:
68 MatrixPower<MatrixType>& m_pow;
69 const RealScalar m_p;
70};
71
87template<typename MatrixType>
88class MatrixPowerAtomic : internal::noncopyable
89{
90 private:
91 enum {
92 RowsAtCompileTime = MatrixType::RowsAtCompileTime,
93 MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime
94 };
95 typedef typename MatrixType::Scalar Scalar;
96 typedef typename MatrixType::RealScalar RealScalar;
97 typedef std::complex<RealScalar> ComplexScalar;
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:
124 MatrixPowerAtomic(const MatrixType& T, RealScalar p);
125
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-RealScalar(degree)) / RealScalar(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-RealScalar(i/2))/RealScalar(2*i) : (m_p-RealScalar(i/2))/RealScalar(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 = RealScalar(
200 digits <= 24? 4.3386528e-1L // single precision
201 : digits <= 53? 2.789358995219730e-1L // double precision
202 : digits <= 64? 2.4471944416607995472e-1L // extended precision
203 : digits <= 106? 1.1016843812851143391275867258512e-1L // double-double
204 : 9.134603732914548552537150753385375e-2L); // quadruple precision
205 MatrixType IminusT, sqrtT, T = m_A.template triangularView<Upper>();
206 RealScalar normIminusT;
207 int degree, degree2, numberOfSquareRoots = 0;
208 bool hasExtraSquareRoot = false;
209
210 for (Index i=0; i < m_A.cols(); ++i)
211 eigen_assert(m_A(i,i) != RealScalar(0));
212
213 while (true) {
214 IminusT = MatrixType::Identity(m_A.rows(), m_A.cols()) - T;
215 normIminusT = IminusT.cwiseAbs().colwise().sum().maxCoeff();
216 if (normIminusT < maxNormForPade) {
217 degree = getPadeDegree(normIminusT);
218 degree2 = getPadeDegree(normIminusT/2);
219 if (degree - degree2 <= 1 || hasExtraSquareRoot)
220 break;
221 hasExtraSquareRoot = true;
222 }
223 matrix_sqrt_triangular(T, sqrtT);
224 T = sqrtT.template triangularView<Upper>();
225 ++numberOfSquareRoots;
226 }
227 computePade(degree, IminusT, res);
228
229 for (; numberOfSquareRoots; --numberOfSquareRoots) {
230 compute2x2(res, ldexp(m_p, -numberOfSquareRoots));
231 res = res.template triangularView<Upper>() * res;
232 }
233 compute2x2(res, m_p);
234}
235
236template<typename MatrixType>
237inline int MatrixPowerAtomic<MatrixType>::getPadeDegree(float normIminusT)
238{
239 const float maxNormForPade[] = { 2.8064004e-1f /* degree = 3 */ , 4.3386528e-1f };
240 int degree = 3;
241 for (; degree <= 4; ++degree)
242 if (normIminusT <= maxNormForPade[degree - 3])
243 break;
244 return degree;
245}
246
247template<typename MatrixType>
248inline int MatrixPowerAtomic<MatrixType>::getPadeDegree(double normIminusT)
249{
250 const double maxNormForPade[] = { 1.884160592658218e-2 /* degree = 3 */ , 6.038881904059573e-2, 1.239917516308172e-1,
251 1.999045567181744e-1, 2.789358995219730e-1 };
252 int degree = 3;
253 for (; degree <= 7; ++degree)
254 if (normIminusT <= maxNormForPade[degree - 3])
255 break;
256 return degree;
257}
258
259template<typename MatrixType>
260inline int MatrixPowerAtomic<MatrixType>::getPadeDegree(long double normIminusT)
261{
262#if LDBL_MANT_DIG == 53
263 const int maxPadeDegree = 7;
264 const double maxNormForPade[] = { 1.884160592658218e-2L /* degree = 3 */ , 6.038881904059573e-2L, 1.239917516308172e-1L,
265 1.999045567181744e-1L, 2.789358995219730e-1L };
266#elif LDBL_MANT_DIG <= 64
267 const int maxPadeDegree = 8;
268 const long double maxNormForPade[] = { 6.3854693117491799460e-3L /* degree = 3 */ , 2.6394893435456973676e-2L,
269 6.4216043030404063729e-2L, 1.1701165502926694307e-1L, 1.7904284231268670284e-1L, 2.4471944416607995472e-1L };
270#elif LDBL_MANT_DIG <= 106
271 const int maxPadeDegree = 10;
272 const double maxNormForPade[] = { 1.0007161601787493236741409687186e-4L /* degree = 3 */ ,
273 1.0007161601787493236741409687186e-3L, 4.7069769360887572939882574746264e-3L, 1.3220386624169159689406653101695e-2L,
274 2.8063482381631737920612944054906e-2L, 4.9625993951953473052385361085058e-2L, 7.7367040706027886224557538328171e-2L,
275 1.1016843812851143391275867258512e-1L };
276#else
277 const int maxPadeDegree = 10;
278 const double maxNormForPade[] = { 5.524506147036624377378713555116378e-5L /* degree = 3 */ ,
279 6.640600568157479679823602193345995e-4L, 3.227716520106894279249709728084626e-3L,
280 9.619593944683432960546978734646284e-3L, 2.134595382433742403911124458161147e-2L,
281 3.908166513900489428442993794761185e-2L, 6.266780814639442865832535460550138e-2L,
282 9.134603732914548552537150753385375e-2L };
283#endif
284 int degree = 3;
285 for (; degree <= maxPadeDegree; ++degree)
286 if (normIminusT <= maxNormForPade[degree - 3])
287 break;
288 return degree;
289}
290
291template<typename MatrixType>
292inline typename MatrixPowerAtomic<MatrixType>::ComplexScalar
293MatrixPowerAtomic<MatrixType>::computeSuperDiag(const ComplexScalar& curr, const ComplexScalar& prev, RealScalar p)
294{
295 using std::ceil;
296 using std::exp;
297 using std::log;
298 using std::sinh;
299
300 ComplexScalar logCurr = log(curr);
301 ComplexScalar logPrev = log(prev);
302 RealScalar unwindingNumber = ceil((numext::imag(logCurr - logPrev) - RealScalar(EIGEN_PI)) / RealScalar(2*EIGEN_PI));
303 ComplexScalar w = numext::log1p((curr-prev)/prev)/RealScalar(2) + ComplexScalar(0, RealScalar(EIGEN_PI)*unwindingNumber);
304 return RealScalar(2) * exp(RealScalar(0.5) * p * (logCurr + logPrev)) * sinh(p * w) / (curr - prev);
305}
306
307template<typename MatrixType>
308inline typename MatrixPowerAtomic<MatrixType>::RealScalar
309MatrixPowerAtomic<MatrixType>::computeSuperDiag(RealScalar curr, RealScalar prev, RealScalar p)
310{
311 using std::exp;
312 using std::log;
313 using std::sinh;
314
315 RealScalar w = numext::log1p((curr-prev)/prev)/RealScalar(2);
316 return 2 * exp(p * (log(curr) + log(prev)) / 2) * sinh(p * w) / (curr - prev);
317}
318
338template<typename MatrixType>
339class MatrixPower : internal::noncopyable
340{
341 private:
342 typedef typename MatrixType::Scalar Scalar;
343 typedef typename MatrixType::RealScalar RealScalar;
344
345 public:
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
368 const MatrixPowerParenthesesReturnValue<MatrixType> operator()(RealScalar p)
369 { return MatrixPowerParenthesesReturnValue<MatrixType>(*this, p); }
370
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
390 typename MatrixType::Nested m_A;
391
393 MatrixType m_tmp;
394
396 ComplexMatrix m_T, m_U;
397
399 ComplexMatrix m_fT;
400
407 RealScalar m_conditionNumber;
408
410 Index m_rank;
411
413 Index m_nulls;
414
424 void split(RealScalar& p, RealScalar& intpart);
425
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
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
610 MatrixPowerReturnValue(const Derived& A, RealScalar p) : m_A(A), m_p(p)
611 { }
612
619 template<typename ResultType>
620 inline void evalTo(ResultType& result) const
621 { MatrixPower<PlainObject>(m_A.eval()).compute(result, m_p); }
622
623 Index rows() const { return m_A.rows(); }
624 Index cols() const { return m_A.cols(); }
625
626 private:
627 const Derived& m_A;
628 const RealScalar m_p;
629};
630
644template<typename Derived>
645class MatrixComplexPowerReturnValue : public ReturnByValue< MatrixComplexPowerReturnValue<Derived> >
646{
647 public:
648 typedef typename Derived::PlainObject PlainObject;
649 typedef typename std::complex<typename Derived::RealScalar> ComplexScalar;
650
657 MatrixComplexPowerReturnValue(const Derived& A, const ComplexScalar& p) : m_A(A), m_p(p)
658 { }
659
669 template<typename ResultType>
670 inline void evalTo(ResultType& result) const
671 { result = (m_p * m_A.log()).exp(); }
672
673 Index rows() const { return m_A.rows(); }
674 Index cols() const { return m_A.cols(); }
675
676 private:
677 const Derived& m_A;
678 const ComplexScalar m_p;
679};
680
681namespace internal {
682
683template<typename MatrixPowerType>
684struct traits< MatrixPowerParenthesesReturnValue<MatrixPowerType> >
685{ typedef typename MatrixPowerType::PlainObject ReturnType; };
686
687template<typename Derived>
688struct traits< MatrixPowerReturnValue<Derived> >
689{ typedef typename Derived::PlainObject ReturnType; };
690
691template<typename Derived>
692struct traits< MatrixComplexPowerReturnValue<Derived> >
693{ typedef typename Derived::PlainObject ReturnType; };
694
695}
696
697template<typename Derived>
698const MatrixPowerReturnValue<Derived> MatrixBase<Derived>::pow(const RealScalar& p) const
699{ return MatrixPowerReturnValue<Derived>(derived(), p); }
700
701template<typename Derived>
702const MatrixComplexPowerReturnValue<Derived> MatrixBase<Derived>::pow(const std::complex<RealScalar>& p) const
703{ return MatrixComplexPowerReturnValue<Derived>(derived(), p); }
704
705} // namespace Eigen
706
707#endif // EIGEN_MATRIX_POWER
Eigen::MatrixBase::pow
const MatrixPowerReturnValue< Derived > pow(const RealScalar &p) const
Definition: MatrixPower.h:698
Eigen::MatrixComplexPowerReturnValue
Proxy for the matrix power of some matrix (expression).
Definition: MatrixPower.h:646
Eigen::MatrixComplexPowerReturnValue::MatrixComplexPowerReturnValue
MatrixComplexPowerReturnValue(const Derived &A, const ComplexScalar &p)
Constructor.
Definition: MatrixPower.h:657
Eigen::MatrixComplexPowerReturnValue::evalTo
void evalTo(ResultType &result) const
Compute the matrix power.
Definition: MatrixPower.h:670
Eigen::MatrixPowerAtomic
Class for computing matrix powers.
Definition: MatrixPower.h:89
Eigen::MatrixPowerAtomic::MatrixPowerAtomic
MatrixPowerAtomic(const MatrixType &T, RealScalar p)
Constructor.
Definition: MatrixPower.h:136
Eigen::MatrixPowerAtomic::compute
void compute(ResultType &res) const
Compute the matrix power.
Definition: MatrixPower.h:144
Eigen::MatrixPowerParenthesesReturnValue
Proxy for the matrix power of some matrix.
Definition: MatrixPower.h:42
Eigen::MatrixPowerParenthesesReturnValue::MatrixPowerParenthesesReturnValue
MatrixPowerParenthesesReturnValue(MatrixPower< MatrixType > &pow, RealScalar p)
Constructor.
Definition: MatrixPower.h:52
Eigen::MatrixPowerParenthesesReturnValue::evalTo
void evalTo(ResultType &result) const
Compute the matrix power.
Definition: MatrixPower.h:61
Eigen::MatrixPowerReturnValue
Proxy for the matrix power of some matrix (expression).
Definition: MatrixPower.h:599
Eigen::MatrixPowerReturnValue::MatrixPowerReturnValue
MatrixPowerReturnValue(const Derived &A, RealScalar p)
Constructor.
Definition: MatrixPower.h:610
Eigen::MatrixPowerReturnValue::evalTo
void evalTo(ResultType &result) const
Compute the matrix power.
Definition: MatrixPower.h:620
Eigen::MatrixPower
Class for computing matrix powers.
Definition: MatrixPower.h:340
Eigen::MatrixPower::MatrixPower
MatrixPower(const MatrixType &A)
Constructor.
Definition: MatrixPower.h:354
Eigen::MatrixPower::operator()
const MatrixPowerParenthesesReturnValue< MatrixType > operator()(RealScalar p)
Returns the matrix power.
Definition: MatrixPower.h:368
Eigen::MatrixPower::compute
void compute(ResultType &res, RealScalar p)
Compute the matrix power.
Definition: MatrixPower.h:450
Eigen::matrix_sqrt_triangular
void matrix_sqrt_triangular(const MatrixType &arg, ResultType &result)
Compute matrix square root of triangular matrix.
Definition: MatrixSquareRoot.h:206
Eigen
Namespace containing all symbols from the Eigen library.
Eigen::exp
const Eigen::CwiseUnaryOp< Eigen::internal::scalar_exp_op< typename Derived::Scalar >, const Derived > exp(const Eigen::ArrayBase< Derived > &x)
Eigen::abs
const Eigen::CwiseUnaryOp< Eigen::internal::scalar_abs_op< typename Derived::Scalar >, const Derived > abs(const Eigen::ArrayBase< Derived > &x)
Eigen::Index
EIGEN_DEFAULT_DENSE_INDEX_TYPE Index
Eigen::log
const Eigen::CwiseUnaryOp< Eigen::internal::scalar_log_op< typename Derived::Scalar >, const Derived > log(const Eigen::ArrayBase< Derived > &x)
Eigen::sinh
const Eigen::CwiseUnaryOp< Eigen::internal::scalar_sinh_op< typename Derived::Scalar >, const Derived > sinh(const Eigen::ArrayBase< Derived > &x)
Eigen::Dynamic
const int Dynamic
Eigen::floor
const Eigen::CwiseUnaryOp< Eigen::internal::scalar_floor_op< typename Derived::Scalar >, const Derived > floor(const Eigen::ArrayBase< Derived > &x)
Eigen::ceil
const Eigen::CwiseUnaryOp< Eigen::internal::scalar_ceil_op< typename Derived::Scalar >, const Derived > ceil(const Eigen::ArrayBase< Derived > &x)