xref: /aosp_15_r20/external/eigen/unsupported/Eigen/src/MatrixFunctions/MatrixExponential.h (revision bf2c37156dfe67e5dfebd6d394bad8b2ab5804d4) 
1 // This file is part of Eigen, a lightweight C++ template library
2 // for linear algebra.
3 //
4 // Copyright (C) 2009, 2010, 2013 Jitse Niesen <[email protected]>
5 // Copyright (C) 2011, 2013 Chen-Pang He <[email protected]>
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 namespace internal {
18 
19 /** \brief Scaling operator.
20  *
21  * This struct is used by CwiseUnaryOp to scale a matrix by \f$ 2^{-s} \f$.
22  */
23 template <typename RealScalar>
24 struct MatrixExponentialScalingOp
25 {
26   /** \brief Constructor.
27    *
28    * \param[in] squarings  The integer \f$ s \f$ in this document.
29    */
MatrixExponentialScalingOpMatrixExponentialScalingOp30   MatrixExponentialScalingOp(int squarings) : m_squarings(squarings) { }
31 
32 
33   /** \brief Scale a matrix coefficient.
34    *
35    * \param[in,out] x  The scalar to be scaled, becoming \f$ 2^{-s} x \f$.
36    */
operatorMatrixExponentialScalingOp37   inline const RealScalar operator() (const RealScalar& x) const
38   {
39     using std::ldexp;
40     return ldexp(x, -m_squarings);
41   }
42 
43   typedef std::complex<RealScalar> ComplexScalar;
44 
45   /** \brief Scale a matrix coefficient.
46    *
47    * \param[in,out] x  The scalar to be scaled, becoming \f$ 2^{-s} x \f$.
48    */
operatorMatrixExponentialScalingOp49   inline const ComplexScalar operator() (const ComplexScalar& x) const
50   {
51     using std::ldexp;
52     return ComplexScalar(ldexp(x.real(), -m_squarings), ldexp(x.imag(), -m_squarings));
53   }
54 
55   private:
56     int m_squarings;
57 };
58 
59 /** \brief Compute the (3,3)-Pad&eacute; approximant to the exponential.
60  *
61  *  After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Pad&eacute;
62  *  approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
63  */
64 template <typename MatA, typename MatU, typename MatV>
matrix_exp_pade3(const MatA & A,MatU & U,MatV & V)65 void matrix_exp_pade3(const MatA& A, MatU& U, MatV& V)
66 {
67   typedef typename MatA::PlainObject MatrixType;
68   typedef typename NumTraits<typename traits<MatA>::Scalar>::Real RealScalar;
69   const RealScalar b[] = {120.L, 60.L, 12.L, 1.L};
70   const MatrixType A2 = A * A;
71   const MatrixType tmp = b[3] * A2 + b[1] * MatrixType::Identity(A.rows(), A.cols());
72   U.noalias() = A * tmp;
73   V = b[2] * A2 + b[0] * MatrixType::Identity(A.rows(), A.cols());
74 }
75 
76 /** \brief Compute the (5,5)-Pad&eacute; approximant to the exponential.
77  *
78  *  After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Pad&eacute;
79  *  approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
80  */
81 template <typename MatA, typename MatU, typename MatV>
matrix_exp_pade5(const MatA & A,MatU & U,MatV & V)82 void matrix_exp_pade5(const MatA& A, MatU& U, MatV& V)
83 {
84   typedef typename MatA::PlainObject MatrixType;
85   typedef typename NumTraits<typename traits<MatrixType>::Scalar>::Real RealScalar;
86   const RealScalar b[] = {30240.L, 15120.L, 3360.L, 420.L, 30.L, 1.L};
87   const MatrixType A2 = A * A;
88   const MatrixType A4 = A2 * A2;
89   const MatrixType tmp = b[5] * A4 + b[3] * A2 + b[1] * MatrixType::Identity(A.rows(), A.cols());
90   U.noalias() = A * tmp;
91   V = b[4] * A4 + b[2] * A2 + b[0] * MatrixType::Identity(A.rows(), A.cols());
92 }
93 
94 /** \brief Compute the (7,7)-Pad&eacute; approximant to the exponential.
95  *
96  *  After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Pad&eacute;
97  *  approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
98  */
99 template <typename MatA, typename MatU, typename MatV>
matrix_exp_pade7(const MatA & A,MatU & U,MatV & V)100 void matrix_exp_pade7(const MatA& A, MatU& U, MatV& V)
101 {
102   typedef typename MatA::PlainObject MatrixType;
103   typedef typename NumTraits<typename traits<MatrixType>::Scalar>::Real RealScalar;
104   const RealScalar b[] = {17297280.L, 8648640.L, 1995840.L, 277200.L, 25200.L, 1512.L, 56.L, 1.L};
105   const MatrixType A2 = A * A;
106   const MatrixType A4 = A2 * A2;
107   const MatrixType A6 = A4 * A2;
108   const MatrixType tmp = b[7] * A6 + b[5] * A4 + b[3] * A2
109     + b[1] * MatrixType::Identity(A.rows(), A.cols());
110   U.noalias() = A * tmp;
111   V = b[6] * A6 + b[4] * A4 + b[2] * A2 + b[0] * MatrixType::Identity(A.rows(), A.cols());
112 
113 }
114 
115 /** \brief Compute the (9,9)-Pad&eacute; approximant to the exponential.
116  *
117  *  After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Pad&eacute;
118  *  approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
119  */
120 template <typename MatA, typename MatU, typename MatV>
matrix_exp_pade9(const MatA & A,MatU & U,MatV & V)121 void matrix_exp_pade9(const MatA& A, MatU& U, MatV& V)
122 {
123   typedef typename MatA::PlainObject MatrixType;
124   typedef typename NumTraits<typename traits<MatrixType>::Scalar>::Real RealScalar;
125   const RealScalar b[] = {17643225600.L, 8821612800.L, 2075673600.L, 302702400.L, 30270240.L,
126                           2162160.L, 110880.L, 3960.L, 90.L, 1.L};
127   const MatrixType A2 = A * A;
128   const MatrixType A4 = A2 * A2;
129   const MatrixType A6 = A4 * A2;
130   const MatrixType A8 = A6 * A2;
131   const MatrixType tmp = b[9] * A8 + b[7] * A6 + b[5] * A4 + b[3] * A2
132     + b[1] * MatrixType::Identity(A.rows(), A.cols());
133   U.noalias() = A * tmp;
134   V = b[8] * A8 + b[6] * A6 + b[4] * A4 + b[2] * A2 + b[0] * MatrixType::Identity(A.rows(), A.cols());
135 }
136 
137 /** \brief Compute the (13,13)-Pad&eacute; approximant to the exponential.
138  *
139  *  After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Pad&eacute;
140  *  approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
141  */
142 template <typename MatA, typename MatU, typename MatV>
matrix_exp_pade13(const MatA & A,MatU & U,MatV & V)143 void matrix_exp_pade13(const MatA& A, MatU& U, MatV& V)
144 {
145   typedef typename MatA::PlainObject MatrixType;
146   typedef typename NumTraits<typename traits<MatrixType>::Scalar>::Real RealScalar;
147   const RealScalar b[] = {64764752532480000.L, 32382376266240000.L, 7771770303897600.L,
148                           1187353796428800.L, 129060195264000.L, 10559470521600.L, 670442572800.L,
149                           33522128640.L, 1323241920.L, 40840800.L, 960960.L, 16380.L, 182.L, 1.L};
150   const MatrixType A2 = A * A;
151   const MatrixType A4 = A2 * A2;
152   const MatrixType A6 = A4 * A2;
153   V = b[13] * A6 + b[11] * A4 + b[9] * A2; // used for temporary storage
154   MatrixType tmp = A6 * V;
155   tmp += b[7] * A6 + b[5] * A4 + b[3] * A2 + b[1] * MatrixType::Identity(A.rows(), A.cols());
156   U.noalias() = A * tmp;
157   tmp = b[12] * A6 + b[10] * A4 + b[8] * A2;
158   V.noalias() = A6 * tmp;
159   V += b[6] * A6 + b[4] * A4 + b[2] * A2 + b[0] * MatrixType::Identity(A.rows(), A.cols());
160 }
161 
162 /** \brief Compute the (17,17)-Pad&eacute; approximant to the exponential.
163  *
164  *  After exit, \f$ (V+U)(V-U)^{-1} \f$ is the Pad&eacute;
165  *  approximant of \f$ \exp(A) \f$ around \f$ A = 0 \f$.
166  *
167  *  This function activates only if your long double is double-double or quadruple.
168  */
169 #if LDBL_MANT_DIG > 64
170 template <typename MatA, typename MatU, typename MatV>
matrix_exp_pade17(const MatA & A,MatU & U,MatV & V)171 void matrix_exp_pade17(const MatA& A, MatU& U, MatV& V)
172 {
173   typedef typename MatA::PlainObject MatrixType;
174   typedef typename NumTraits<typename traits<MatrixType>::Scalar>::Real RealScalar;
175   const RealScalar b[] = {830034394580628357120000.L, 415017197290314178560000.L,
176                           100610229646136770560000.L, 15720348382208870400000.L,
177                           1774878043152614400000.L, 153822763739893248000.L, 10608466464820224000.L,
178                           595373117923584000.L, 27563570274240000.L, 1060137318240000.L,
179                           33924394183680.L, 899510451840.L, 19554575040.L, 341863200.L, 4651200.L,
180                           46512.L, 306.L, 1.L};
181   const MatrixType A2 = A * A;
182   const MatrixType A4 = A2 * A2;
183   const MatrixType A6 = A4 * A2;
184   const MatrixType A8 = A4 * A4;
185   V = b[17] * A8 + b[15] * A6 + b[13] * A4 + b[11] * A2; // used for temporary storage
186   MatrixType tmp = A8 * V;
187   tmp += b[9] * A8 + b[7] * A6 + b[5] * A4 + b[3] * A2
188     + b[1] * MatrixType::Identity(A.rows(), A.cols());
189   U.noalias() = A * tmp;
190   tmp = b[16] * A8 + b[14] * A6 + b[12] * A4 + b[10] * A2;
191   V.noalias() = tmp * A8;
192   V += b[8] * A8 + b[6] * A6 + b[4] * A4 + b[2] * A2
193     + b[0] * MatrixType::Identity(A.rows(), A.cols());
194 }
195 #endif
196 
197 template <typename MatrixType, typename RealScalar = typename NumTraits<typename traits<MatrixType>::Scalar>::Real>
198 struct matrix_exp_computeUV
199 {
200   /** \brief Compute Pad&eacute; approximant to the exponential.
201     *
202     * Computes \c U, \c V and \c squarings such that \f$ (V+U)(V-U)^{-1} \f$ is a Pad&eacute;
203     * approximant of \f$ \exp(2^{-\mbox{squarings}}M) \f$ around \f$ M = 0 \f$, where \f$ M \f$
204     * denotes the matrix \c arg. The degree of the Pad&eacute; approximant and the value of squarings
205     * are chosen such that the approximation error is no more than the round-off error.
206     */
207   static void run(const MatrixType& arg, MatrixType& U, MatrixType& V, int& squarings);
208 };
209 
210 template <typename MatrixType>
211 struct matrix_exp_computeUV<MatrixType, float>
212 {
213   template <typename ArgType>
214   static void run(const ArgType& arg, MatrixType& U, MatrixType& V, int& squarings)
215   {
216     using std::frexp;
217     using std::pow;
218     const float l1norm = arg.cwiseAbs().colwise().sum().maxCoeff();
219     squarings = 0;
220     if (l1norm < 4.258730016922831e-001f) {
221       matrix_exp_pade3(arg, U, V);
222     } else if (l1norm < 1.880152677804762e+000f) {
223       matrix_exp_pade5(arg, U, V);
224     } else {
225       const float maxnorm = 3.925724783138660f;
226       frexp(l1norm / maxnorm, &squarings);
227       if (squarings < 0) squarings = 0;
228       MatrixType A = arg.unaryExpr(MatrixExponentialScalingOp<float>(squarings));
229       matrix_exp_pade7(A, U, V);
230     }
231   }
232 };
233 
234 template <typename MatrixType>
235 struct matrix_exp_computeUV<MatrixType, double>
236 {
237   typedef typename NumTraits<typename traits<MatrixType>::Scalar>::Real RealScalar;
238   template <typename ArgType>
239   static void run(const ArgType& arg, MatrixType& U, MatrixType& V, int& squarings)
240   {
241     using std::frexp;
242     using std::pow;
243     const RealScalar l1norm = arg.cwiseAbs().colwise().sum().maxCoeff();
244     squarings = 0;
245     if (l1norm < 1.495585217958292e-002) {
246       matrix_exp_pade3(arg, U, V);
247     } else if (l1norm < 2.539398330063230e-001) {
248       matrix_exp_pade5(arg, U, V);
249     } else if (l1norm < 9.504178996162932e-001) {
250       matrix_exp_pade7(arg, U, V);
251     } else if (l1norm < 2.097847961257068e+000) {
252       matrix_exp_pade9(arg, U, V);
253     } else {
254       const RealScalar maxnorm = 5.371920351148152;
255       frexp(l1norm / maxnorm, &squarings);
256       if (squarings < 0) squarings = 0;
257       MatrixType A = arg.unaryExpr(MatrixExponentialScalingOp<RealScalar>(squarings));
258       matrix_exp_pade13(A, U, V);
259     }
260   }
261 };
262 
263 template <typename MatrixType>
264 struct matrix_exp_computeUV<MatrixType, long double>
265 {
266   template <typename ArgType>
267   static void run(const ArgType& arg, MatrixType& U, MatrixType& V, int& squarings)
268   {
269 #if   LDBL_MANT_DIG == 53   // double precision
270     matrix_exp_computeUV<MatrixType, double>::run(arg, U, V, squarings);
271 
272 #else
273 
274     using std::frexp;
275     using std::pow;
276     const long double l1norm = arg.cwiseAbs().colwise().sum().maxCoeff();
277     squarings = 0;
278 
279 #if LDBL_MANT_DIG <= 64   // extended precision
280 
281     if (l1norm < 4.1968497232266989671e-003L) {
282       matrix_exp_pade3(arg, U, V);
283     } else if (l1norm < 1.1848116734693823091e-001L) {
284       matrix_exp_pade5(arg, U, V);
285     } else if (l1norm < 5.5170388480686700274e-001L) {
286       matrix_exp_pade7(arg, U, V);
287     } else if (l1norm < 1.3759868875587845383e+000L) {
288       matrix_exp_pade9(arg, U, V);
289     } else {
290       const long double maxnorm = 4.0246098906697353063L;
291       frexp(l1norm / maxnorm, &squarings);
292       if (squarings < 0) squarings = 0;
293       MatrixType A = arg.unaryExpr(MatrixExponentialScalingOp<long double>(squarings));
294       matrix_exp_pade13(A, U, V);
295     }
296 
297 #elif LDBL_MANT_DIG <= 106  // double-double
298 
299     if (l1norm < 3.2787892205607026992947488108213e-005L) {
300       matrix_exp_pade3(arg, U, V);
301     } else if (l1norm < 6.4467025060072760084130906076332e-003L) {
302       matrix_exp_pade5(arg, U, V);
303     } else if (l1norm < 6.8988028496595374751374122881143e-002L) {
304       matrix_exp_pade7(arg, U, V);
305     } else if (l1norm < 2.7339737518502231741495857201670e-001L) {
306       matrix_exp_pade9(arg, U, V);
307     } else if (l1norm < 1.3203382096514474905666448850278e+000L) {
308       matrix_exp_pade13(arg, U, V);
309     } else {
310       const long double maxnorm = 3.2579440895405400856599663723517L;
311       frexp(l1norm / maxnorm, &squarings);
312       if (squarings < 0) squarings = 0;
313       MatrixType A = arg.unaryExpr(MatrixExponentialScalingOp<long double>(squarings));
314       matrix_exp_pade17(A, U, V);
315     }
316 
317 #elif LDBL_MANT_DIG <= 113  // quadruple precision
318 
319     if (l1norm < 1.639394610288918690547467954466970e-005L) {
320       matrix_exp_pade3(arg, U, V);
321     } else if (l1norm < 4.253237712165275566025884344433009e-003L) {
322       matrix_exp_pade5(arg, U, V);
323     } else if (l1norm < 5.125804063165764409885122032933142e-002L) {
324       matrix_exp_pade7(arg, U, V);
325     } else if (l1norm < 2.170000765161155195453205651889853e-001L) {
326       matrix_exp_pade9(arg, U, V);
327     } else if (l1norm < 1.125358383453143065081397882891878e+000L) {
328       matrix_exp_pade13(arg, U, V);
329     } else {
330       const long double maxnorm = 2.884233277829519311757165057717815L;
331       frexp(l1norm / maxnorm, &squarings);
332       if (squarings < 0) squarings = 0;
333       MatrixType A = arg.unaryExpr(MatrixExponentialScalingOp<long double>(squarings));
334       matrix_exp_pade17(A, U, V);
335     }
336 
337 #else
338 
339     // this case should be handled in compute()
340     eigen_assert(false && "Bug in MatrixExponential");
341 
342 #endif
343 #endif  // LDBL_MANT_DIG
344   }
345 };
346 
347 template<typename T> struct is_exp_known_type : false_type {};
348 template<> struct is_exp_known_type<float> : true_type {};
349 template<> struct is_exp_known_type<double> : true_type {};
350 #if LDBL_MANT_DIG <= 113
351 template<> struct is_exp_known_type<long double> : true_type {};
352 #endif
353 
354 template <typename ArgType, typename ResultType>
355 void matrix_exp_compute(const ArgType& arg, ResultType &result, true_type) // natively supported scalar type
356 {
357   typedef typename ArgType::PlainObject MatrixType;
358   MatrixType U, V;
359   int squarings;
360   matrix_exp_computeUV<MatrixType>::run(arg, U, V, squarings); // Pade approximant is (U+V) / (-U+V)
361   MatrixType numer = U + V;
362   MatrixType denom = -U + V;
363   result = denom.partialPivLu().solve(numer);
364   for (int i=0; i<squarings; i++)
365     result *= result;   // undo scaling by repeated squaring
366 }
367 
368 
369 /* Computes the matrix exponential
370  *
371  * \param arg    argument of matrix exponential (should be plain object)
372  * \param result variable in which result will be stored
373  */
374 template <typename ArgType, typename ResultType>
375 void matrix_exp_compute(const ArgType& arg, ResultType &result, false_type) // default
376 {
377   typedef typename ArgType::PlainObject MatrixType;
378   typedef typename traits<MatrixType>::Scalar Scalar;
379   typedef typename NumTraits<Scalar>::Real RealScalar;
380   typedef typename std::complex<RealScalar> ComplexScalar;
381   result = arg.matrixFunction(internal::stem_function_exp<ComplexScalar>);
382 }
383 
384 } // end namespace Eigen::internal
385 
386 /** \ingroup MatrixFunctions_Module
387   *
388   * \brief Proxy for the matrix exponential of some matrix (expression).
389   *
390   * \tparam Derived  Type of the argument to the matrix exponential.
391   *
392   * This class holds the argument to the matrix exponential until it is assigned or evaluated for
393   * some other reason (so the argument should not be changed in the meantime). It is the return type
394   * of MatrixBase::exp() and most of the time this is the only way it is used.
395   */
396 template<typename Derived> struct MatrixExponentialReturnValue
397 : public ReturnByValue<MatrixExponentialReturnValue<Derived> >
398 {
399   public:
400     /** \brief Constructor.
401       *
402       * \param src %Matrix (expression) forming the argument of the matrix exponential.
403       */
404     MatrixExponentialReturnValue(const Derived& src) : m_src(src) { }
405 
406     /** \brief Compute the matrix exponential.
407       *
408       * \param result the matrix exponential of \p src in the constructor.
409       */
410     template <typename ResultType>
411     inline void evalTo(ResultType& result) const
412     {
413       const typename internal::nested_eval<Derived, 10>::type tmp(m_src);
414       internal::matrix_exp_compute(tmp, result, internal::is_exp_known_type<typename Derived::RealScalar>());
415     }
416 
417     Index rows() const { return m_src.rows(); }
418     Index cols() const { return m_src.cols(); }
419 
420   protected:
421     const typename internal::ref_selector<Derived>::type m_src;
422 };
423 
424 namespace internal {
425 template<typename Derived>
426 struct traits<MatrixExponentialReturnValue<Derived> >
427 {
428   typedef typename Derived::PlainObject ReturnType;
429 };
430 }
431 
432 template <typename Derived>
433 const MatrixExponentialReturnValue<Derived> MatrixBase<Derived>::exp() const
434 {
435   eigen_assert(rows() == cols());
436   return MatrixExponentialReturnValue<Derived>(derived());
437 }
438 
439 } // end namespace Eigen
440 
441 #endif // EIGEN_MATRIX_EXPONENTIAL
442