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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
13namespace Eigen {
14
15template<typename MatrixType> class MatrixPower;
16
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
50 MatrixPowerParenthesesReturnValue(MatrixPower<MatrixType>& pow, RealScalar p) : m_pow(pow), m_p(p)
51 { }
52
58 template<typename ResultType>
59 inline void evalTo(ResultType& result) const
60 { m_pow.compute(result, m_p); }
61
62 Index rows() const { return m_pow.rows(); }
63 Index cols() const { return m_pow.cols(); }
64
65 private:
66 MatrixPower<MatrixType>& m_pow;
67 const RealScalar m_p;
68};
69
85template<typename MatrixType>
86class 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;
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:
122 MatrixPowerAtomic(const MatrixType& T, RealScalar p);
123
130 void compute(ResultType& res) const;
131};
132
133template<typename MatrixType>
134MatrixPowerAtomic<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
141template<typename MatrixType>
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
159template<typename MatrixType>
160void 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)
173template<typename MatrixType>
174void 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
192template<typename MatrixType>
193void 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
234template<typename MatrixType>
235inline 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
245template<typename MatrixType>
246inline 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
257template<typename MatrixType>
258inline 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
289template<typename MatrixType>
290inline typename MatrixPowerAtomic<MatrixType>::ComplexScalar
291MatrixPowerAtomic<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
305template<typename MatrixType>
306inline typename MatrixPowerAtomic<MatrixType>::RealScalar
307MatrixPowerAtomic<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
336template<typename MatrixType>
337class MatrixPower : internal::noncopyable
338{
339 private:
340 typedef typename MatrixType::Scalar Scalar;
341 typedef typename MatrixType::RealScalar RealScalar;
342
343 public:
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
368
376 template<typename ResultType>
377 void compute(ResultType& res, RealScalar p);
378
379 Index rows() const { return m_A.rows(); }
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
388 typename MatrixType::Nested m_A;
389
391 MatrixType m_tmp;
392
394 ComplexMatrix m_T, m_U;
395
397 ComplexMatrix m_fT;
398
405 RealScalar m_conditionNumber;
406
408 Index m_rank;
409
411 Index m_nulls;
412
422 void split(RealScalar& p, RealScalar& intpart);
423
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
446template<typename MatrixType>
447template<typename ResultType>
448void 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
467template<typename MatrixType>
468void 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
488template<typename MatrixType>
489void 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
526template<typename MatrixType>
527template<typename ResultType>
528void 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
549template<typename MatrixType>
550template<typename ResultType>
551void 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
566template<typename MatrixType>
567template<int Rows, int Cols, int Options, int MaxRows, int MaxCols>
568inline 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
574template<typename MatrixType>
575template<int Rows, int Cols, int Options, int MaxRows, int MaxCols>
576inline 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
595template<typename Derived>
596class MatrixPowerReturnValue : public ReturnByValue< MatrixPowerReturnValue<Derived> >
597{
598 public:
599 typedef typename Derived::PlainObject PlainObject;
600 typedef typename Derived::RealScalar RealScalar;
601
608 MatrixPowerReturnValue(const Derived& A, RealScalar p) : m_A(A), m_p(p)
609 { }
610
617 template<typename ResultType>
618 inline void evalTo(ResultType& result) const
619 { MatrixPower<PlainObject>(m_A.eval()).compute(result, m_p); }
620
621 Index rows() const { return m_A.rows(); }
622 Index cols() const { return m_A.cols(); }
623
624 private:
625 const Derived& m_A;
626 const RealScalar m_p;
627};
628
642template<typename Derived>
643class MatrixComplexPowerReturnValue : public ReturnByValue< MatrixComplexPowerReturnValue<Derived> >
644{
645 public:
646 typedef typename Derived::PlainObject PlainObject;
647 typedef typename std::complex<typename Derived::RealScalar> ComplexScalar;
648
655 MatrixComplexPowerReturnValue(const Derived& A, const ComplexScalar& p) : m_A(A), m_p(p)
656 { }
657
667 template<typename ResultType>
668 inline void evalTo(ResultType& result) const
669 { result = (m_p * m_A.log()).exp(); }
670
671 Index rows() const { return m_A.rows(); }
672 Index cols() const { return m_A.cols(); }
673
674 private:
675 const Derived& m_A;
676 const ComplexScalar m_p;
677};
678
679namespace internal {
680
681template<typename MatrixPowerType>
682struct traits< MatrixPowerParenthesesReturnValue<MatrixPowerType> >
683{ typedef typename MatrixPowerType::PlainObject ReturnType; };
684
685template<typename Derived>
686struct traits< MatrixPowerReturnValue<Derived> >
687{ typedef typename Derived::PlainObject ReturnType; };
688
689template<typename Derived>
690struct traits< MatrixComplexPowerReturnValue<Derived> >
691{ typedef typename Derived::PlainObject ReturnType; };
692
693}
694
695template<typename Derived>
696const MatrixPowerReturnValue<Derived> MatrixBase<Derived>::pow(const RealScalar& p) const
697{ return MatrixPowerReturnValue<Derived>(derived(), p); }
698
699template<typename Derived>
700const 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
const MatrixPowerReturnValue< Derived > pow(const RealScalar &p) const
Definition: MatrixPower.h:696
Proxy for the matrix power of some matrix (expression).
Definition: MatrixPower.h:644
MatrixComplexPowerReturnValue(const Derived &A, const ComplexScalar &p)
Constructor.
Definition: MatrixPower.h:655
void evalTo(ResultType &result) const
Compute the matrix power.
Definition: MatrixPower.h:668
Class for computing matrix powers.
Definition: MatrixPower.h:87
MatrixPowerAtomic(const MatrixType &T, RealScalar p)
Constructor.
Definition: MatrixPower.h:134
void compute(ResultType &res) const
Compute the matrix power.
Definition: MatrixPower.h:142
Proxy for the matrix power of some matrix.
Definition: MatrixPower.h:40
MatrixPowerParenthesesReturnValue(MatrixPower< MatrixType > &pow, RealScalar p)
Constructor.
Definition: MatrixPower.h:50
void evalTo(ResultType &result) const
Compute the matrix power.
Definition: MatrixPower.h:59
Proxy for the matrix power of some matrix (expression).
Definition: MatrixPower.h:597
MatrixPowerReturnValue(const Derived &A, RealScalar p)
Constructor.
Definition: MatrixPower.h:608
void evalTo(ResultType &result) const
Compute the matrix power.
Definition: MatrixPower.h:618
Class for computing matrix powers.
Definition: MatrixPower.h:338
MatrixPower(const MatrixType &A)
Constructor.
Definition: MatrixPower.h:352
const MatrixPowerParenthesesReturnValue< MatrixType > operator()(RealScalar p)
Returns the matrix power.
Definition: MatrixPower.h:366
void compute(ResultType &res, RealScalar p)
Compute the matrix power.
Definition: MatrixPower.h:448
void matrix_sqrt_triangular(const MatrixType &arg, ResultType &result)
Compute matrix square root of triangular matrix.
Definition: MatrixSquareRoot.h:204
Namespace containing all symbols from the Eigen library.
const Eigen::CwiseUnaryOp< Eigen::internal::scalar_exp_op< typename Derived::Scalar >, const Derived > exp(const Eigen::ArrayBase< Derived > &x)
const Eigen::CwiseUnaryOp< Eigen::internal::scalar_abs_op< typename Derived::Scalar >, const Derived > abs(const Eigen::ArrayBase< Derived > &x)
EIGEN_DEFAULT_DENSE_INDEX_TYPE Index
const Eigen::CwiseUnaryOp< Eigen::internal::scalar_log_op< typename Derived::Scalar >, const Derived > log(const Eigen::ArrayBase< Derived > &x)
const Eigen::CwiseUnaryOp< Eigen::internal::scalar_sinh_op< typename Derived::Scalar >, const Derived > sinh(const Eigen::ArrayBase< Derived > &x)
const int Dynamic
const Eigen::CwiseUnaryOp< Eigen::internal::scalar_floor_op< typename Derived::Scalar >, const Derived > floor(const Eigen::ArrayBase< Derived > &x)
const Eigen::CwiseUnaryOp< Eigen::internal::scalar_ceil_op< typename Derived::Scalar >, const Derived > ceil(const Eigen::ArrayBase< Derived > &x)