11 #ifndef EIGEN_MATRIX_EXPONENTIAL
12 #define EIGEN_MATRIX_EXPONENTIAL
14 #include "StemFunction.h"
18 #if defined(_MSC_VER) || defined(__FreeBSD__)
19 template <
typename Scalar> Scalar log2(Scalar v) {
using std::log;
return log(v)/log(Scalar(2)); }
28 template <
typename MatrixType>
46 template <
typename ResultType>
47 void compute(ResultType &result);
62 void pade3(
const MatrixType &A);
71 void pade5(
const MatrixType &A);
80 void pade7(
const MatrixType &A);
89 void pade9(
const MatrixType &A);
98 void pade13(
const MatrixType &A);
109 void pade17(
const MatrixType &A);
124 void computeUV(
double);
130 void computeUV(
float);
136 void computeUV(
long double);
138 typedef typename internal::traits<MatrixType>::Scalar Scalar;
140 typedef typename std::complex<RealScalar> ComplexScalar;
143 typename internal::nested<MatrixType>::type m_M;
167 template <
typename MatrixType>
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())),
176 m_l1norm(M.cwiseAbs().colwise().sum().maxCoeff())
181 template <
typename MatrixType>
182 template <
typename ResultType>
185 #if LDBL_MANT_DIG > 112 // rarely happens
186 if(
sizeof(RealScalar) > 14) {
191 computeUV(RealScalar());
194 result = m_tmp2.partialPivLu().solve(m_tmp1);
195 for (
int i=0; i<m_squarings; i++)
199 template <
typename MatrixType>
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;
209 template <
typename MatrixType>
210 EIGEN_STRONG_INLINE
void MatrixExponential<MatrixType>::pade5(
const MatrixType &A)
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;
220 template <
typename MatrixType>
221 EIGEN_STRONG_INLINE
void MatrixExponential<MatrixType>::pade7(
const MatrixType &A)
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;
232 template <
typename MatrixType>
233 EIGEN_STRONG_INLINE
void MatrixExponential<MatrixType>::pade9(
const MatrixType &A)
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;
246 template <
typename MatrixType>
247 EIGEN_STRONG_INLINE
void MatrixExponential<MatrixType>::pade13(
const MatrixType &A)
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;
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;
264 #if LDBL_MANT_DIG > 64
265 template <
typename MatrixType>
266 EIGEN_STRONG_INLINE
void MatrixExponential<MatrixType>::pade17(
const MatrixType &A)
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;
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;
288 template <
typename MatrixType>
289 void MatrixExponential<MatrixType>::computeUV(
float)
294 if (m_l1norm < 4.258730016922831e-001) {
296 }
else if (m_l1norm < 1.880152677804762e+000) {
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);
306 template <
typename MatrixType>
307 void MatrixExponential<MatrixType>::computeUV(
double)
312 if (m_l1norm < 1.495585217958292e-002) {
314 }
else if (m_l1norm < 2.539398330063230e-001) {
316 }
else if (m_l1norm < 9.504178996162932e-001) {
318 }
else if (m_l1norm < 2.097847961257068e+000) {
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);
328 template <
typename MatrixType>
329 void MatrixExponential<MatrixType>::computeUV(
long double)
334 #if LDBL_MANT_DIG == 53 // double precision
336 #elif LDBL_MANT_DIG <= 64 // extended precision
337 if (m_l1norm < 4.1968497232266989671e-003L) {
339 }
else if (m_l1norm < 1.1848116734693823091e-001L) {
341 }
else if (m_l1norm < 5.5170388480686700274e-001L) {
343 }
else if (m_l1norm < 1.3759868875587845383e+000L) {
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);
351 #elif LDBL_MANT_DIG <= 106 // double-double
352 if (m_l1norm < 3.2787892205607026992947488108213e-005L) {
354 }
else if (m_l1norm < 6.4467025060072760084130906076332e-003L) {
356 }
else if (m_l1norm < 6.8988028496595374751374122881143e-002L) {
358 }
else if (m_l1norm < 2.7339737518502231741495857201670e-001L) {
360 }
else if (m_l1norm < 1.3203382096514474905666448850278e+000L) {
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);
368 #elif LDBL_MANT_DIG <= 112 // quadruple precison
369 if (m_l1norm < 1.639394610288918690547467954466970e-005L) {
371 }
else if (m_l1norm < 4.253237712165275566025884344433009e-003L) {
373 }
else if (m_l1norm < 5.125804063165764409885122032933142e-002L) {
375 }
else if (m_l1norm < 2.170000765161155195453205651889853e-001L) {
377 }
else if (m_l1norm < 1.125358383453143065081397882891878e+000L) {
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);
387 eigen_assert(
false &&
"Bug in MatrixExponential");
388 #endif // LDBL_MANT_DIG
404 :
public ReturnByValue<MatrixExponentialReturnValue<Derived> >
406 typedef typename Derived::Index Index;
420 template <
typename ResultType>
421 inline void evalTo(ResultType& result)
const
423 const typename Derived::PlainObject srcEvaluated = m_src.eval();
428 Index rows()
const {
return m_src.rows(); }
429 Index cols()
const {
return m_src.cols(); }
432 const Derived& m_src;
438 template<
typename Derived>
439 struct traits<MatrixExponentialReturnValue<Derived> >
441 typedef typename Derived::PlainObject ReturnType;
445 template <
typename Derived>
446 const MatrixExponentialReturnValue<Derived> MatrixBase<Derived>::exp()
const
448 eigen_assert(rows() == cols());
449 return MatrixExponentialReturnValue<Derived>(derived());
454 #endif // EIGEN_MATRIX_EXPONENTIAL