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 
16 namespace 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 
28 template <typename MatrixType>
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
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 
167 template <typename MatrixType>
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 
181 template <typename MatrixType>
182 template <typename ResultType>
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 
199 template <typename MatrixType>
200 EIGEN_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 
209 template <typename MatrixType>
210 EIGEN_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 
220 template <typename MatrixType>
221 EIGEN_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 
232 template <typename MatrixType>
233 EIGEN_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 
246 template <typename MatrixType>
247 EIGEN_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
265 template <typename MatrixType>
266 EIGEN_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 
288 template <typename MatrixType>
289 void 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 
306 template <typename MatrixType>
307 void 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 
328 template <typename MatrixType>
329 void 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 
403 template<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();
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:
435 };
436 
437 namespace internal {
438 template<typename Derived>
439 struct traits<MatrixExponentialReturnValue<Derived> >
440 {
441  typedef typename Derived::PlainObject ReturnType;
442 };
443 }
444 
445 template <typename Derived>
446 const 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