26 #ifndef EIGEN_MATRIX_EXPONENTIAL
27 #define EIGEN_MATRIX_EXPONENTIAL
29 #include "StemFunction.h"
33 #if defined(_MSC_VER) || defined(__FreeBSD__)
34 template <
typename Scalar> Scalar log2(Scalar v) {
using std::log;
return log(v)/
log(Scalar(2)); }
43 template <
typename MatrixType>
61 template <
typename ResultType>
62 void compute(ResultType &result);
77 void pade3(
const MatrixType &A);
86 void pade5(
const MatrixType &A);
95 void pade7(
const MatrixType &A);
104 void pade9(
const MatrixType &A);
113 void pade13(
const MatrixType &A);
124 void pade17(
const MatrixType &A);
139 void computeUV(
double);
145 void computeUV(
float);
151 void computeUV(
long double);
153 typedef typename internal::traits<MatrixType>::Scalar Scalar;
154 typedef typename NumTraits<Scalar>::Real RealScalar;
155 typedef typename std::complex<RealScalar> ComplexScalar;
158 typename internal::nested<MatrixType>::type m_M;
182 template <
typename MatrixType>
185 m_U(M.rows(),M.cols()),
186 m_V(M.rows(),M.cols()),
187 m_tmp1(M.rows(),M.cols()),
188 m_tmp2(M.rows(),M.cols()),
189 m_Id(MatrixType::Identity(M.rows(), M.cols())),
191 m_l1norm(M.cwiseAbs().colwise().sum().maxCoeff())
196 template <
typename MatrixType>
197 template <
typename ResultType>
200 #if LDBL_MANT_DIG > 112 // rarely happens
201 if(
sizeof(RealScalar) > 14) {
206 computeUV(RealScalar());
209 result = m_tmp2.partialPivLu().solve(m_tmp1);
210 for (
int i=0; i<m_squarings; i++)
214 template <
typename MatrixType>
217 const RealScalar b[] = {120., 60., 12., 1.};
218 m_tmp1.noalias() = A * A;
219 m_tmp2 = b[3]*m_tmp1 + b[1]*m_Id;
220 m_U.noalias() = A * m_tmp2;
221 m_V = b[2]*m_tmp1 + b[0]*m_Id;
224 template <
typename MatrixType>
225 EIGEN_STRONG_INLINE
void MatrixExponential<MatrixType>::pade5(
const MatrixType &A)
227 const RealScalar b[] = {30240., 15120., 3360., 420., 30., 1.};
228 MatrixType A2 = A * A;
229 m_tmp1.noalias() = A2 * A2;
230 m_tmp2 = b[5]*m_tmp1 + b[3]*A2 + b[1]*m_Id;
231 m_U.noalias() = A * m_tmp2;
232 m_V = b[4]*m_tmp1 + b[2]*A2 + b[0]*m_Id;
235 template <
typename MatrixType>
236 EIGEN_STRONG_INLINE
void MatrixExponential<MatrixType>::pade7(
const MatrixType &A)
238 const RealScalar b[] = {17297280., 8648640., 1995840., 277200., 25200., 1512., 56., 1.};
239 MatrixType A2 = A * A;
240 MatrixType A4 = A2 * A2;
241 m_tmp1.noalias() = A4 * A2;
242 m_tmp2 = b[7]*m_tmp1 + b[5]*A4 + b[3]*A2 + b[1]*m_Id;
243 m_U.noalias() = A * m_tmp2;
244 m_V = b[6]*m_tmp1 + b[4]*A4 + b[2]*A2 + b[0]*m_Id;
247 template <
typename MatrixType>
248 EIGEN_STRONG_INLINE
void MatrixExponential<MatrixType>::pade9(
const MatrixType &A)
250 const RealScalar b[] = {17643225600., 8821612800., 2075673600., 302702400., 30270240.,
251 2162160., 110880., 3960., 90., 1.};
252 MatrixType A2 = A * A;
253 MatrixType A4 = A2 * A2;
254 MatrixType A6 = A4 * A2;
255 m_tmp1.noalias() = A6 * A2;
256 m_tmp2 = b[9]*m_tmp1 + b[7]*A6 + b[5]*A4 + b[3]*A2 + b[1]*m_Id;
257 m_U.noalias() = A * m_tmp2;
258 m_V = b[8]*m_tmp1 + b[6]*A6 + b[4]*A4 + b[2]*A2 + b[0]*m_Id;
261 template <
typename MatrixType>
262 EIGEN_STRONG_INLINE
void MatrixExponential<MatrixType>::pade13(
const MatrixType &A)
264 const RealScalar b[] = {64764752532480000., 32382376266240000., 7771770303897600.,
265 1187353796428800., 129060195264000., 10559470521600., 670442572800.,
266 33522128640., 1323241920., 40840800., 960960., 16380., 182., 1.};
267 MatrixType A2 = A * A;
268 MatrixType A4 = A2 * A2;
269 m_tmp1.noalias() = A4 * A2;
270 m_V = b[13]*m_tmp1 + b[11]*A4 + b[9]*A2;
271 m_tmp2.noalias() = m_tmp1 * m_V;
272 m_tmp2 += b[7]*m_tmp1 + b[5]*A4 + b[3]*A2 + b[1]*m_Id;
273 m_U.noalias() = A * m_tmp2;
274 m_tmp2 = b[12]*m_tmp1 + b[10]*A4 + b[8]*A2;
275 m_V.noalias() = m_tmp1 * m_tmp2;
276 m_V += b[6]*m_tmp1 + b[4]*A4 + b[2]*A2 + b[0]*m_Id;
279 #if LDBL_MANT_DIG > 64
280 template <
typename MatrixType>
281 EIGEN_STRONG_INLINE
void MatrixExponential<MatrixType>::pade17(
const MatrixType &A)
283 const RealScalar b[] = {830034394580628357120000.L, 415017197290314178560000.L,
284 100610229646136770560000.L, 15720348382208870400000.L,
285 1774878043152614400000.L, 153822763739893248000.L, 10608466464820224000.L,
286 595373117923584000.L, 27563570274240000.L, 1060137318240000.L,
287 33924394183680.L, 899510451840.L, 19554575040.L, 341863200.L, 4651200.L,
288 46512.L, 306.L, 1.L};
289 MatrixType A2 = A * A;
290 MatrixType A4 = A2 * A2;
291 MatrixType A6 = A4 * A2;
292 m_tmp1.noalias() = A4 * A4;
293 m_V = b[17]*m_tmp1 + b[15]*A6 + b[13]*A4 + b[11]*A2;
294 m_tmp2.noalias() = m_tmp1 * m_V;
295 m_tmp2 += b[9]*m_tmp1 + b[7]*A6 + b[5]*A4 + b[3]*A2 + b[1]*m_Id;
296 m_U.noalias() = A * m_tmp2;
297 m_tmp2 = b[16]*m_tmp1 + b[14]*A6 + b[12]*A4 + b[10]*A2;
298 m_V.noalias() = m_tmp1 * m_tmp2;
299 m_V += b[8]*m_tmp1 + b[6]*A6 + b[4]*A4 + b[2]*A2 + b[0]*m_Id;
303 template <
typename MatrixType>
304 void MatrixExponential<MatrixType>::computeUV(
float)
309 if (m_l1norm < 4.258730016922831e-001) {
311 }
else if (m_l1norm < 1.880152677804762e+000) {
314 const float maxnorm = 3.925724783138660f;
315 m_squarings = (max)(0, (
int)ceil(log2(m_l1norm / maxnorm)));
316 MatrixType A = m_M /
pow(Scalar(2), m_squarings);
321 template <
typename MatrixType>
322 void MatrixExponential<MatrixType>::computeUV(
double)
327 if (m_l1norm < 1.495585217958292e-002) {
329 }
else if (m_l1norm < 2.539398330063230e-001) {
331 }
else if (m_l1norm < 9.504178996162932e-001) {
333 }
else if (m_l1norm < 2.097847961257068e+000) {
336 const double maxnorm = 5.371920351148152;
337 m_squarings = (max)(0, (
int)ceil(log2(m_l1norm / maxnorm)));
338 MatrixType A = m_M /
pow(Scalar(2), m_squarings);
343 template <
typename MatrixType>
344 void MatrixExponential<MatrixType>::computeUV(
long double)
349 #if LDBL_MANT_DIG == 53 // double precision
351 #elif LDBL_MANT_DIG <= 64 // extended precision
352 if (m_l1norm < 4.1968497232266989671e-003L) {
354 }
else if (m_l1norm < 1.1848116734693823091e-001L) {
356 }
else if (m_l1norm < 5.5170388480686700274e-001L) {
358 }
else if (m_l1norm < 1.3759868875587845383e+000L) {
361 const long double maxnorm = 4.0246098906697353063L;
362 m_squarings = (max)(0, (
int)ceil(log2(m_l1norm / maxnorm)));
363 MatrixType A = m_M /
pow(Scalar(2), m_squarings);
366 #elif LDBL_MANT_DIG <= 106 // double-double
367 if (m_l1norm < 3.2787892205607026992947488108213e-005L) {
369 }
else if (m_l1norm < 6.4467025060072760084130906076332e-003L) {
371 }
else if (m_l1norm < 6.8988028496595374751374122881143e-002L) {
373 }
else if (m_l1norm < 2.7339737518502231741495857201670e-001L) {
375 }
else if (m_l1norm < 1.3203382096514474905666448850278e+000L) {
378 const long double maxnorm = 3.2579440895405400856599663723517L;
379 m_squarings = (max)(0, (
int)ceil(log2(m_l1norm / maxnorm)));
380 MatrixType A = m_M /
pow(Scalar(2), m_squarings);
383 #elif LDBL_MANT_DIG <= 112 // quadruple precison
384 if (m_l1norm < 1.639394610288918690547467954466970e-005L) {
386 }
else if (m_l1norm < 4.253237712165275566025884344433009e-003L) {
388 }
else if (m_l1norm < 5.125804063165764409885122032933142e-002L) {
390 }
else if (m_l1norm < 2.170000765161155195453205651889853e-001L) {
392 }
else if (m_l1norm < 1.125358383453143065081397882891878e+000L) {
395 const long double maxnorm = 2.884233277829519311757165057717815L;
396 m_squarings = (max)(0, (
int)ceil(log2(m_l1norm / maxnorm)));
397 MatrixType A = m_M /
pow(Scalar(2), m_squarings);
402 eigen_assert(
false &&
"Bug in MatrixExponential");
403 #endif // LDBL_MANT_DIG
419 :
public ReturnByValue<MatrixExponentialReturnValue<Derived> >
421 typedef typename Derived::Index Index;
435 template <
typename ResultType>
436 inline void evalTo(ResultType& result)
const
438 const typename Derived::PlainObject srcEvaluated = m_src.eval();
443 Index rows()
const {
return m_src.rows(); }
444 Index cols()
const {
return m_src.cols(); }
447 const Derived& m_src;
453 template<
typename Derived>
454 struct traits<MatrixExponentialReturnValue<Derived> >
456 typedef typename Derived::PlainObject ReturnType;
460 template <
typename Derived>
461 const MatrixExponentialReturnValue<Derived> MatrixBase<Derived>::exp()
const
463 eigen_assert(rows() == cols());
464 return MatrixExponentialReturnValue<Derived>(derived());
469 #endif // EIGEN_MATRIX_EXPONENTIAL