MatrixExponential.h
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#if defined(_MSC_VER) || defined(__FreeBSD__)
19 template <typename Scalar> Scalar log2(Scalar v) { using std::log; return log(v)/log(Scalar(2)); }
20#endif
21
22
28template <typename MatrixType>
29class MatrixExponential {
30
31 public:
32
40 MatrixExponential(const MatrixType &M);
41
46 template <typename ResultType>
47 void compute(ResultType &result);
48
49 private:
50
51 // Prevent copying
52 MatrixExponential(const MatrixExponential&);
53 MatrixExponential& operator=(const MatrixExponential&);
54
62 void pade3(const MatrixType &A);
63
71 void pade5(const MatrixType &A);
72
80 void pade7(const MatrixType &A);
81
89 void pade9(const MatrixType &A);
90
98 void pade13(const MatrixType &A);
99
109 void pade17(const MatrixType &A);
110
124 void computeUV(double);
125
130 void computeUV(float);
131
136 void computeUV(long double);
137
138 typedef typename internal::traits<MatrixType>::Scalar Scalar;
139 typedef typename NumTraits<Scalar>::Real RealScalar;
140 typedef typename std::complex<RealScalar> ComplexScalar;
141
143 typename internal::nested<MatrixType>::type m_M;
144
146 MatrixType m_U;
147
149 MatrixType m_V;
150
152 MatrixType m_tmp1;
153
155 MatrixType m_tmp2;
156
158 MatrixType m_Id;
159
161 int m_squarings;
162
164 RealScalar m_l1norm;
165};
166
167template <typename MatrixType>
168MatrixExponential<MatrixType>::MatrixExponential(const MatrixType &M) :
169 m_M(M),
170 m_U(M.rows(),M.cols()),
171 m_V(M.rows(),M.cols()),
172 m_tmp1(M.rows(),M.cols()),
173 m_tmp2(M.rows(),M.cols()),
174 m_Id(MatrixType::Identity(M.rows(), M.cols())),
175 m_squarings(0),
176 m_l1norm(M.cwiseAbs().colwise().sum().maxCoeff())
177{
178 /* empty body */
179}
180
181template <typename MatrixType>
182template <typename ResultType>
183void MatrixExponential<MatrixType>::compute(ResultType &result)
184{
185#if LDBL_MANT_DIG > 112 // rarely happens
186 if(sizeof(RealScalar) > 14) {
187 result = m_M.matrixFunction(StdStemFunctions<ComplexScalar>::exp);
188 return;
189 }
190#endif
191 computeUV(RealScalar());
192 m_tmp1 = m_U + m_V; // numerator of Pade approximant
193 m_tmp2 = -m_U + m_V; // denominator of Pade approximant
194 result = m_tmp2.partialPivLu().solve(m_tmp1);
195 for (int i=0; i<m_squarings; i++)
196 result *= result; // undo scaling by repeated squaring
197}
198
199template <typename MatrixType>
200EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade3(const MatrixType &A)
201{
202 const RealScalar b[] = {120., 60., 12., 1.};
203 m_tmp1.noalias() = A * A;
204 m_tmp2 = b[3]*m_tmp1 + b[1]*m_Id;
205 m_U.noalias() = A * m_tmp2;
206 m_V = b[2]*m_tmp1 + b[0]*m_Id;
207}
208
209template <typename MatrixType>
210EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade5(const MatrixType &A)
211{
212 const RealScalar b[] = {30240., 15120., 3360., 420., 30., 1.};
213 MatrixType A2 = A * A;
214 m_tmp1.noalias() = A2 * A2;
215 m_tmp2 = b[5]*m_tmp1 + b[3]*A2 + b[1]*m_Id;
216 m_U.noalias() = A * m_tmp2;
217 m_V = b[4]*m_tmp1 + b[2]*A2 + b[0]*m_Id;
218}
219
220template <typename MatrixType>
221EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade7(const MatrixType &A)
222{
223 const RealScalar b[] = {17297280., 8648640., 1995840., 277200., 25200., 1512., 56., 1.};
224 MatrixType A2 = A * A;
225 MatrixType A4 = A2 * A2;
226 m_tmp1.noalias() = A4 * A2;
227 m_tmp2 = b[7]*m_tmp1 + b[5]*A4 + b[3]*A2 + b[1]*m_Id;
228 m_U.noalias() = A * m_tmp2;
229 m_V = b[6]*m_tmp1 + b[4]*A4 + b[2]*A2 + b[0]*m_Id;
230}
231
232template <typename MatrixType>
233EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade9(const MatrixType &A)
234{
235 const RealScalar b[] = {17643225600., 8821612800., 2075673600., 302702400., 30270240.,
236 2162160., 110880., 3960., 90., 1.};
237 MatrixType A2 = A * A;
238 MatrixType A4 = A2 * A2;
239 MatrixType A6 = A4 * A2;
240 m_tmp1.noalias() = A6 * A2;
241 m_tmp2 = b[9]*m_tmp1 + b[7]*A6 + b[5]*A4 + b[3]*A2 + b[1]*m_Id;
242 m_U.noalias() = A * m_tmp2;
243 m_V = b[8]*m_tmp1 + b[6]*A6 + b[4]*A4 + b[2]*A2 + b[0]*m_Id;
244}
245
246template <typename MatrixType>
247EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade13(const MatrixType &A)
248{
249 const RealScalar b[] = {64764752532480000., 32382376266240000., 7771770303897600.,
250 1187353796428800., 129060195264000., 10559470521600., 670442572800.,
251 33522128640., 1323241920., 40840800., 960960., 16380., 182., 1.};
252 MatrixType A2 = A * A;
253 MatrixType A4 = A2 * A2;
254 m_tmp1.noalias() = A4 * A2;
255 m_V = b[13]*m_tmp1 + b[11]*A4 + b[9]*A2; // used for temporary storage
256 m_tmp2.noalias() = m_tmp1 * m_V;
257 m_tmp2 += b[7]*m_tmp1 + b[5]*A4 + b[3]*A2 + b[1]*m_Id;
258 m_U.noalias() = A * m_tmp2;
259 m_tmp2 = b[12]*m_tmp1 + b[10]*A4 + b[8]*A2;
260 m_V.noalias() = m_tmp1 * m_tmp2;
261 m_V += b[6]*m_tmp1 + b[4]*A4 + b[2]*A2 + b[0]*m_Id;
262}
263
264#if LDBL_MANT_DIG > 64
265template <typename MatrixType>
266EIGEN_STRONG_INLINE void MatrixExponential<MatrixType>::pade17(const MatrixType &A)
267{
268 const RealScalar b[] = {830034394580628357120000.L, 415017197290314178560000.L,
269 100610229646136770560000.L, 15720348382208870400000.L,
270 1774878043152614400000.L, 153822763739893248000.L, 10608466464820224000.L,
271 595373117923584000.L, 27563570274240000.L, 1060137318240000.L,
272 33924394183680.L, 899510451840.L, 19554575040.L, 341863200.L, 4651200.L,
273 46512.L, 306.L, 1.L};
274 MatrixType A2 = A * A;
275 MatrixType A4 = A2 * A2;
276 MatrixType A6 = A4 * A2;
277 m_tmp1.noalias() = A4 * A4;
278 m_V = b[17]*m_tmp1 + b[15]*A6 + b[13]*A4 + b[11]*A2; // used for temporary storage
279 m_tmp2.noalias() = m_tmp1 * m_V;
280 m_tmp2 += b[9]*m_tmp1 + b[7]*A6 + b[5]*A4 + b[3]*A2 + b[1]*m_Id;
281 m_U.noalias() = A * m_tmp2;
282 m_tmp2 = b[16]*m_tmp1 + b[14]*A6 + b[12]*A4 + b[10]*A2;
283 m_V.noalias() = m_tmp1 * m_tmp2;
284 m_V += b[8]*m_tmp1 + b[6]*A6 + b[4]*A4 + b[2]*A2 + b[0]*m_Id;
285}
286#endif
287
288template <typename MatrixType>
289void MatrixExponential<MatrixType>::computeUV(float)
290{
291 using std::max;
292 using std::pow;
293 using std::ceil;
294 if (m_l1norm < 4.258730016922831e-001) {
295 pade3(m_M);
296 } else if (m_l1norm < 1.880152677804762e+000) {
297 pade5(m_M);
298 } else {
299 const float maxnorm = 3.925724783138660f;
300 m_squarings = (max)(0, (int)ceil(log2(m_l1norm / maxnorm)));
301 MatrixType A = m_M / pow(Scalar(2), m_squarings);
302 pade7(A);
303 }
304}
305
306template <typename MatrixType>
307void MatrixExponential<MatrixType>::computeUV(double)
308{
309 using std::max;
310 using std::pow;
311 using std::ceil;
312 if (m_l1norm < 1.495585217958292e-002) {
313 pade3(m_M);
314 } else if (m_l1norm < 2.539398330063230e-001) {
315 pade5(m_M);
316 } else if (m_l1norm < 9.504178996162932e-001) {
317 pade7(m_M);
318 } else if (m_l1norm < 2.097847961257068e+000) {
319 pade9(m_M);
320 } else {
321 const double maxnorm = 5.371920351148152;
322 m_squarings = (max)(0, (int)ceil(log2(m_l1norm / maxnorm)));
323 MatrixType A = m_M / pow(Scalar(2), m_squarings);
324 pade13(A);
325 }
326}
327
328template <typename MatrixType>
329void MatrixExponential<MatrixType>::computeUV(long double)
330{
331 using std::max;
332 using std::pow;
333 using std::ceil;
334#if LDBL_MANT_DIG == 53 // double precision
335 computeUV(double());
336#elif LDBL_MANT_DIG <= 64 // extended precision
337 if (m_l1norm < 4.1968497232266989671e-003L) {
338 pade3(m_M);
339 } else if (m_l1norm < 1.1848116734693823091e-001L) {
340 pade5(m_M);
341 } else if (m_l1norm < 5.5170388480686700274e-001L) {
342 pade7(m_M);
343 } else if (m_l1norm < 1.3759868875587845383e+000L) {
344 pade9(m_M);
345 } else {
346 const long double maxnorm = 4.0246098906697353063L;
347 m_squarings = (max)(0, (int)ceil(log2(m_l1norm / maxnorm)));
348 MatrixType A = m_M / pow(Scalar(2), m_squarings);
349 pade13(A);
350 }
351#elif LDBL_MANT_DIG <= 106 // double-double
352 if (m_l1norm < 3.2787892205607026992947488108213e-005L) {
353 pade3(m_M);
354 } else if (m_l1norm < 6.4467025060072760084130906076332e-003L) {
355 pade5(m_M);
356 } else if (m_l1norm < 6.8988028496595374751374122881143e-002L) {
357 pade7(m_M);
358 } else if (m_l1norm < 2.7339737518502231741495857201670e-001L) {
359 pade9(m_M);
360 } else if (m_l1norm < 1.3203382096514474905666448850278e+000L) {
361 pade13(m_M);
362 } else {
363 const long double maxnorm = 3.2579440895405400856599663723517L;
364 m_squarings = (max)(0, (int)ceil(log2(m_l1norm / maxnorm)));
365 MatrixType A = m_M / pow(Scalar(2), m_squarings);
366 pade17(A);
367 }
368#elif LDBL_MANT_DIG <= 112 // quadruple precison
369 if (m_l1norm < 1.639394610288918690547467954466970e-005L) {
370 pade3(m_M);
371 } else if (m_l1norm < 4.253237712165275566025884344433009e-003L) {
372 pade5(m_M);
373 } else if (m_l1norm < 5.125804063165764409885122032933142e-002L) {
374 pade7(m_M);
375 } else if (m_l1norm < 2.170000765161155195453205651889853e-001L) {
376 pade9(m_M);
377 } else if (m_l1norm < 1.125358383453143065081397882891878e+000L) {
378 pade13(m_M);
379 } else {
380 const long double maxnorm = 2.884233277829519311757165057717815L;
381 m_squarings = (max)(0, (int)ceil(log2(m_l1norm / maxnorm)));
382 MatrixType A = m_M / pow(Scalar(2), m_squarings);
383 pade17(A);
384 }
385#else
386 // this case should be handled in compute()
387 eigen_assert(false && "Bug in MatrixExponential");
388#endif // LDBL_MANT_DIG
389}
390
403template<typename Derived> struct MatrixExponentialReturnValue
404: public ReturnByValue<MatrixExponentialReturnValue<Derived> >
405{
406 typedef typename Derived::Index Index;
407 public:
413 MatrixExponentialReturnValue(const Derived& src) : m_src(src) { }
414
420 template <typename ResultType>
421 inline void evalTo(ResultType& result) const
422 {
423 const typename Derived::PlainObject srcEvaluated = m_src.eval();
424 MatrixExponential<typename Derived::PlainObject> me(srcEvaluated);
425 me.compute(result);
426 }
427
428 Index rows() const { return m_src.rows(); }
429 Index cols() const { return m_src.cols(); }
430
431 protected:
432 const Derived& m_src;
433 private:
434 MatrixExponentialReturnValue& operator=(const MatrixExponentialReturnValue&);
435};
436
437namespace internal {
438template<typename Derived>
439struct traits<MatrixExponentialReturnValue<Derived> >
440{
441 typedef typename Derived::PlainObject ReturnType;
442};
443}
444
445template <typename Derived>
446const MatrixExponentialReturnValue<Derived> MatrixBase<Derived>::exp() const
447{
448 eigen_assert(rows() == cols());
449 return MatrixExponentialReturnValue<Derived>(derived());
450}
451
452} // end namespace Eigen
453
454#endif // EIGEN_MATRIX_EXPONENTIAL
Eigen::MatrixExponential
Class for computing the matrix exponential.
Definition MatrixExponential.h:29
Eigen::MatrixExponential::compute
void compute(ResultType &result)
Computes the matrix exponential.
Definition MatrixExponential.h:183
Eigen::MatrixExponential::MatrixExponential
MatrixExponential(const MatrixType &M)
Constructor.
Definition MatrixExponential.h:168
Eigen::StdStemFunctions::exp
static Scalar exp(Scalar x, int)
The exponential function (and its derivatives).
Definition StemFunction.h:24
Eigen::MatrixExponentialReturnValue
Proxy for the matrix exponential of some matrix (expression).
Definition MatrixExponential.h:405
Eigen::MatrixExponentialReturnValue::MatrixExponentialReturnValue
MatrixExponentialReturnValue(const Derived &src)
Constructor.
Definition MatrixExponential.h:413
Eigen::MatrixExponentialReturnValue::evalTo
void evalTo(ResultType &result) const
Compute the matrix exponential.
Definition MatrixExponential.h:421
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