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

Last change on this file since 136 was 136, checked in by ldecherf, 7 years ago

Doc
File size: 14.6 KB
Line 
1// This file is part of Eigen, a lightweight C++ template library
2// for linear algebra.
3//
4// Copyright (C) 2009, 2010 Jitse Niesen <jitse@maths.leeds.ac.uk>
5// Copyright (C) 2011 Chen-Pang He <jdh8@ms63.hinet.net>
6//
7// This Source Code Form is subject to the terms of the Mozilla
8// Public License v. 2.0. If a copy of the MPL was not distributed
9// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
10
11#ifndef EIGEN_MATRIX_EXPONENTIAL
12#define EIGEN_MATRIX_EXPONENTIAL
13
14#include "StemFunction.h"
15
16namespace Eigen {
17
18/** \ingroup MatrixFunctions_Module
19 * \brief Class for computing the matrix exponential.
20 * \tparam MatrixType type of the argument of the exponential,
21 * expected to be an instantiation of the Matrix class template.
22 */
23template <typename MatrixType>
24class MatrixExponential {
25
26 public:
27
28 /** \brief Constructor.
29 *
30 * The class stores a reference to \p M, so it should not be
31 * changed (or destroyed) before compute() is called.
32 *
33 * \param[in] M matrix whose exponential is to be computed.
34 */
35 MatrixExponential(const MatrixType &M);
36
37 /** \brief Computes the matrix exponential.
38 *
39 * \param[out] result the matrix exponential of \p M in the constructor.
40 */
41 template <typename ResultType>
42 void compute(ResultType &result);
43
44 private:
45
46 // Prevent copying
47 MatrixExponential(const MatrixExponential&);
48 MatrixExponential& operator=(const MatrixExponential&);
49
50 /** \brief Compute the (3,3)-Pad&eacute; approximant to the exponential.
51 *
52 * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Pad&eacute;
53 * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
54 *
55 * \param[in] A Argument of matrix exponential
56 */
57 void pade3(const MatrixType &A);
58
59 /** \brief Compute the (5,5)-Pad&eacute; approximant to the exponential.
60 *
61 * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Pad&eacute;
62 * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
63 *
64 * \param[in] A Argument of matrix exponential
65 */
66 void pade5(const MatrixType &A);
67
68 /** \brief Compute the (7,7)-Pad&eacute; approximant to the exponential.
69 *
70 * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Pad&eacute;
71 * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
72 *
73 * \param[in] A Argument of matrix exponential
74 */
75 void pade7(const MatrixType &A);
76
77 /** \brief Compute the (9,9)-Pad&eacute; approximant to the exponential.
78 *
79 * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Pad&eacute;
80 * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
81 *
82 * \param[in] A Argument of matrix exponential
83 */
84 void pade9(const MatrixType &A);
85
86 /** \brief Compute the (13,13)-Pad&eacute; approximant to the exponential.
87 *
88 * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Pad&eacute;
89 * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
90 *
91 * \param[in] A Argument of matrix exponential
92 */
93 void pade13(const MatrixType &A);
94
95 /** \brief Compute the (17,17)-Pad&eacute; approximant to the exponential.
96 *
97 * After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Pad&eacute;
98 * approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
99 *
100 * This function activates only if your long double is double-double or quadruple.
101 *
102 * \param[in] A Argument of matrix exponential
103 */
104 void pade17(const MatrixType &A);
105
106 /** \brief Compute Pad&eacute; approximant to the exponential.
107 *
108 * Computes \c m_U, \c m_V and \c m_squarings such that
109 * \f$ (V+U)(V-U)^{-1} \f$ is a Pad&eacute; of
110 * \f$ \exp(2^{-\mbox{squarings}}M) \f$ around \f$ M = 0 \f$. The
111 * degree of the Pad&eacute; approximant and the value of
112 * squarings are chosen such that the approximation error is no
113 * more than the round-off error.
114 *
115 * The argument of this function should correspond with the (real
116 * part of) the entries of \c m_M. It is used to select the
117 * correct implementation using overloading.
118 */
119 void computeUV(double);
120
121 /** \brief Compute Pad&eacute; approximant to the exponential.
122 *
123 * \sa computeUV(double);
124 */
125 void computeUV(float);
126
127 /** \brief Compute Pad&eacute; approximant to the exponential.
128 *
129 * \sa computeUV(double);
130 */
131 void computeUV(long double);
132
133 typedef typename internal::traits<MatrixType>::Scalar Scalar;
134 typedef typename NumTraits<Scalar>::Real RealScalar;
135 typedef typename std::complex<RealScalar> ComplexScalar;
136
137 /** \brief Reference to matrix whose exponential is to be computed. */
138 typename internal::nested<MatrixType>::type m_M;
139
140 /** \brief Odd-degree terms in numerator of Pad&eacute; approximant. */
141 MatrixType m_U;
142
143 /** \brief Even-degree terms in numerator of Pad&eacute; approximant. */
144 MatrixType m_V;
145
146 /** \brief Used for temporary storage. */
147 MatrixType m_tmp1;
148
149 /** \brief Used for temporary storage. */
150 MatrixType m_tmp2;
151
152 /** \brief Identity matrix of the same size as \c m_M. */
153 MatrixType m_Id;
154
155 /** \brief Number of squarings required in the last step. */
156 int m_squarings;
157
158 /** \brief L1 norm of m_M. */
159 RealScalar m_l1norm;
160};
161
162template <typename MatrixType>
163MatrixExponential<MatrixType>::MatrixExponential(const MatrixType &M) :
164 m_M(M),
165 m_U(M.rows(),M.cols()),
166 m_V(M.rows(),M.cols()),
167 m_tmp1(M.rows(),M.cols()),
168 m_tmp2(M.rows(),M.cols()),
169 m_Id(MatrixType::Identity(M.rows(), M.cols())),
170 m_squarings(0),
171 m_l1norm(M.cwiseAbs().colwise().sum().maxCoeff())
172{
173 /* empty body */
174}
175
176template <typename MatrixType>
177template <typename ResultType>
178void MatrixExponential<MatrixType>::compute(ResultType &result)
179{
180#if LDBL_MANT_DIG > 112 // rarely happens
181 if(sizeof(RealScalar) > 14) {
182 result = m_M.matrixFunction(StdStemFunctions<ComplexScalar>::exp);
183 return;
184 }
185#endif
186 computeUV(RealScalar());
187 m_tmp1 = m_U + m_V; // numerator of Pade approximant
188 m_tmp2 = -m_U + m_V; // denominator of Pade approximant
189 result = m_tmp2.partialPivLu().solve(m_tmp1);
190 for (int i=0; i<m_squarings; i++)
191 result *= result; // undo scaling by repeated squaring
192}
193
194template <typename MatrixType>
195EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade3(const MatrixType &A)
196{
197 const RealScalar b[] = {120., 60., 12., 1.};
198 m_tmp1.noalias() = A * A;
199 m_tmp2 = b[3]*m_tmp1 + b[1]*m_Id;
200 m_U.noalias() = A * m_tmp2;
201 m_V = b[2]*m_tmp1 + b[0]*m_Id;
202}
203
204template <typename MatrixType>
205EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade5(const MatrixType &A)
206{
207 const RealScalar b[] = {30240., 15120., 3360., 420., 30., 1.};
208 MatrixType A2 = A * A;
209 m_tmp1.noalias() = A2 * A2;
210 m_tmp2 = b[5]*m_tmp1 + b[3]*A2 + b[1]*m_Id;
211 m_U.noalias() = A * m_tmp2;
212 m_V = b[4]*m_tmp1 + b[2]*A2 + b[0]*m_Id;
213}
214
215template <typename MatrixType>
216EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade7(const MatrixType &A)
217{
218 const RealScalar b[] = {17297280., 8648640., 1995840., 277200., 25200., 1512., 56., 1.};
219 MatrixType A2 = A * A;
220 MatrixType A4 = A2 * A2;
221 m_tmp1.noalias() = A4 * A2;
222 m_tmp2 = b[7]*m_tmp1 + b[5]*A4 + b[3]*A2 + b[1]*m_Id;
223 m_U.noalias() = A * m_tmp2;
224 m_V = b[6]*m_tmp1 + b[4]*A4 + b[2]*A2 + b[0]*m_Id;
225}
226
227template <typename MatrixType>
228EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade9(const MatrixType &A)
229{
230 const RealScalar b[] = {17643225600., 8821612800., 2075673600., 302702400., 30270240.,
231 2162160., 110880., 3960., 90., 1.};
232 MatrixType A2 = A * A;
233 MatrixType A4 = A2 * A2;
234 MatrixType A6 = A4 * A2;
235 m_tmp1.noalias() = A6 * A2;
236 m_tmp2 = b[9]*m_tmp1 + b[7]*A6 + b[5]*A4 + b[3]*A2 + b[1]*m_Id;
237 m_U.noalias() = A * m_tmp2;
238 m_V = b[8]*m_tmp1 + b[6]*A6 + b[4]*A4 + b[2]*A2 + b[0]*m_Id;
239}
240
241template <typename MatrixType>
242EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade13(const MatrixType &A)
243{
244 const RealScalar b[] = {64764752532480000., 32382376266240000., 7771770303897600.,
245 1187353796428800., 129060195264000., 10559470521600., 670442572800.,
246 33522128640., 1323241920., 40840800., 960960., 16380., 182., 1.};
247 MatrixType A2 = A * A;
248 MatrixType A4 = A2 * A2;
249 m_tmp1.noalias() = A4 * A2;
250 m_V = b[13]*m_tmp1 + b[11]*A4 + b[9]*A2; // used for temporary storage
251 m_tmp2.noalias() = m_tmp1 * m_V;
252 m_tmp2 += b[7]*m_tmp1 + b[5]*A4 + b[3]*A2 + b[1]*m_Id;
253 m_U.noalias() = A * m_tmp2;
254 m_tmp2 = b[12]*m_tmp1 + b[10]*A4 + b[8]*A2;
255 m_V.noalias() = m_tmp1 * m_tmp2;
256 m_V += b[6]*m_tmp1 + b[4]*A4 + b[2]*A2 + b[0]*m_Id;
257}
258
259#if LDBL_MANT_DIG > 64
260template <typename MatrixType>
261EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade17(const MatrixType &A)
262{
263 const RealScalar b[] = {830034394580628357120000.L, 415017197290314178560000.L,
264 100610229646136770560000.L, 15720348382208870400000.L,
265 1774878043152614400000.L, 153822763739893248000.L, 10608466464820224000.L,
266 595373117923584000.L, 27563570274240000.L, 1060137318240000.L,
267 33924394183680.L, 899510451840.L, 19554575040.L, 341863200.L, 4651200.L,
268 46512.L, 306.L, 1.L};
269 MatrixType A2 = A * A;
270 MatrixType A4 = A2 * A2;
271 MatrixType A6 = A4 * A2;
272 m_tmp1.noalias() = A4 * A4;
273 m_V = b[17]*m_tmp1 + b[15]*A6 + b[13]*A4 + b[11]*A2; // used for temporary storage
274 m_tmp2.noalias() = m_tmp1 * m_V;
275 m_tmp2 += b[9]*m_tmp1 + b[7]*A6 + b[5]*A4 + b[3]*A2 + b[1]*m_Id;
276 m_U.noalias() = A * m_tmp2;
277 m_tmp2 = b[16]*m_tmp1 + b[14]*A6 + b[12]*A4 + b[10]*A2;
278 m_V.noalias() = m_tmp1 * m_tmp2;
279 m_V += b[8]*m_tmp1 + b[6]*A6 + b[4]*A4 + b[2]*A2 + b[0]*m_Id;
280}
281#endif
282
283template <typename MatrixType>
284void MatrixExponential<MatrixType>::computeUV(float)
285{
286 using std::frexp;
287 using std::pow;
288 if (m_l1norm < 4.258730016922831e-001) {
289 pade3(m_M);
290 } else if (m_l1norm < 1.880152677804762e+000) {
291 pade5(m_M);
292 } else {
293 const float maxnorm = 3.925724783138660f;
294 frexp(m_l1norm / maxnorm, &m_squarings);
295 if (m_squarings < 0) m_squarings = 0;
296 MatrixType A = m_M / Scalar(pow(2, m_squarings));
297 pade7(A);
298 }
299}
300
301template <typename MatrixType>
302void MatrixExponential<MatrixType>::computeUV(double)
303{
304 using std::frexp;
305 using std::pow;
306 if (m_l1norm < 1.495585217958292e-002) {
307 pade3(m_M);
308 } else if (m_l1norm < 2.539398330063230e-001) {
309 pade5(m_M);
310 } else if (m_l1norm < 9.504178996162932e-001) {
311 pade7(m_M);
312 } else if (m_l1norm < 2.097847961257068e+000) {
313 pade9(m_M);
314 } else {
315 const double maxnorm = 5.371920351148152;
316 frexp(m_l1norm / maxnorm, &m_squarings);
317 if (m_squarings < 0) m_squarings = 0;
318 MatrixType A = m_M / Scalar(pow(2, m_squarings));
319 pade13(A);
320 }
321}
322
323template <typename MatrixType>
324void MatrixExponential<MatrixType>::computeUV(long double)
325{
326 using std::frexp;
327 using std::pow;
328#if LDBL_MANT_DIG == 53 // double precision
329 computeUV(double());
330#elif LDBL_MANT_DIG <= 64 // extended precision
331 if (m_l1norm < 4.1968497232266989671e-003L) {
332 pade3(m_M);
333 } else if (m_l1norm < 1.1848116734693823091e-001L) {
334 pade5(m_M);
335 } else if (m_l1norm < 5.5170388480686700274e-001L) {
336 pade7(m_M);
337 } else if (m_l1norm < 1.3759868875587845383e+000L) {
338 pade9(m_M);
339 } else {
340 const long double maxnorm = 4.0246098906697353063L;
341 frexp(m_l1norm / maxnorm, &m_squarings);
342 if (m_squarings < 0) m_squarings = 0;
343 MatrixType A = m_M / Scalar(pow(2, m_squarings));
344 pade13(A);
345 }
346#elif LDBL_MANT_DIG <= 106 // double-double
347 if (m_l1norm < 3.2787892205607026992947488108213e-005L) {
348 pade3(m_M);
349 } else if (m_l1norm < 6.4467025060072760084130906076332e-003L) {
350 pade5(m_M);
351 } else if (m_l1norm < 6.8988028496595374751374122881143e-002L) {
352 pade7(m_M);
353 } else if (m_l1norm < 2.7339737518502231741495857201670e-001L) {
354 pade9(m_M);
355 } else if (m_l1norm < 1.3203382096514474905666448850278e+000L) {
356 pade13(m_M);
357 } else {
358 const long double maxnorm = 3.2579440895405400856599663723517L;
359 frexp(m_l1norm / maxnorm, &m_squarings);
360 if (m_squarings < 0) m_squarings = 0;
361 MatrixType A = m_M / pow(Scalar(2), m_squarings);
362 pade17(A);
363 }
364#elif LDBL_MANT_DIG <= 112 // quadruple precison
365 if (m_l1norm < 1.639394610288918690547467954466970e-005L) {
366 pade3(m_M);
367 } else if (m_l1norm < 4.253237712165275566025884344433009e-003L) {
368 pade5(m_M);
369 } else if (m_l1norm < 5.125804063165764409885122032933142e-002L) {
370 pade7(m_M);
371 } else if (m_l1norm < 2.170000765161155195453205651889853e-001L) {
372 pade9(m_M);
373 } else if (m_l1norm < 1.125358383453143065081397882891878e+000L) {
374 pade13(m_M);
375 } else {
376 const long double maxnorm = 2.884233277829519311757165057717815L;
377 frexp(m_l1norm / maxnorm, &m_squarings);
378 if (m_squarings < 0) m_squarings = 0;
379 MatrixType A = m_M / Scalar(pow(2, m_squarings));
380 pade17(A);
381 }
382#else
383 // this case should be handled in compute()
384 eigen_assert(false && "Bug in MatrixExponential");
385#endif // LDBL_MANT_DIG
386}
387
388/** \ingroup MatrixFunctions_Module
389 *
390 * \brief Proxy for the matrix exponential of some matrix (expression).
391 *
392 * \tparam Derived Type of the argument to the matrix exponential.
393 *
394 * This class holds the argument to the matrix exponential until it
395 * is assigned or evaluated for some other reason (so the argument
396 * should not be changed in the meantime). It is the return type of
397 * MatrixBase::exp() and most of the time this is the only way it is
398 * used.
399 */
400template<typename Derived> struct MatrixExponentialReturnValue
401: public ReturnByValue<MatrixExponentialReturnValue<Derived> >
402{
403 typedef typename Derived::Index Index;
404 public:
405 /** \brief Constructor.
406 *
407 * \param[in] src %Matrix (expression) forming the argument of the
408 * matrix exponential.
409 */
410 MatrixExponentialReturnValue(const Derived& src) : m_src(src) { }
411
412 /** \brief Compute the matrix exponential.
413 *
414 * \param[out] result the matrix exponential of \p src in the
415 * constructor.
416 */
417 template <typename ResultType>
418 inline void evalTo(ResultType& result) const
419 {
420 const typename Derived::PlainObject srcEvaluated = m_src.eval();
421 MatrixExponential<typename Derived::PlainObject> me(srcEvaluated);
422 me.compute(result);
423 }
424
425 Index rows() const { return m_src.rows(); }
426 Index cols() const { return m_src.cols(); }
427
428 protected:
429 const Derived& m_src;
430 private:
431 MatrixExponentialReturnValue& operator=(const MatrixExponentialReturnValue&);
432};
433
434namespace internal {
435template<typename Derived>
436struct traits<MatrixExponentialReturnValue<Derived> >
437{
438 typedef typename Derived::PlainObject ReturnType;
439};
440}
441
442template <typename Derived>
443const MatrixExponentialReturnValue<Derived> MatrixBase<Derived>::exp() const
444{
445 eigen_assert(rows() == cols());
446 return MatrixExponentialReturnValue<Derived>(derived());
447}
448
449} // end namespace Eigen
450
451#endif // EIGEN_MATRIX_EXPONENTIAL
Note: See TracBrowser for help on using the repository browser.