26 #ifndef EIGEN_MATRIX_LOGARITHM
27 #define EIGEN_MATRIX_LOGARITHM
30 #define M_PI 3.141592653589793238462643383279503L
45 template <
typename MatrixType>
50 typedef typename MatrixType::Scalar Scalar;
52 typedef typename NumTraits<Scalar>::Real RealScalar;
63 MatrixType
compute(
const MatrixType& A);
67 void compute2x2(
const MatrixType& A, MatrixType& result);
68 void computeBig(
const MatrixType& A, MatrixType& result);
69 static Scalar atanh(Scalar x);
70 int getPadeDegree(
float normTminusI);
71 int getPadeDegree(
double normTminusI);
72 int getPadeDegree(
long double normTminusI);
73 void computePade(MatrixType& result,
const MatrixType& T,
int degree);
74 void computePade3(MatrixType& result,
const MatrixType& T);
75 void computePade4(MatrixType& result,
const MatrixType& T);
76 void computePade5(MatrixType& result,
const MatrixType& T);
77 void computePade6(MatrixType& result,
const MatrixType& T);
78 void computePade7(MatrixType& result,
const MatrixType& T);
79 void computePade8(MatrixType& result,
const MatrixType& T);
80 void computePade9(MatrixType& result,
const MatrixType& T);
81 void computePade10(MatrixType& result,
const MatrixType& T);
82 void computePade11(MatrixType& result,
const MatrixType& T);
84 static const int minPadeDegree = 3;
85 static const int maxPadeDegree = std::numeric_limits<RealScalar>::digits<= 24? 5:
86 std::numeric_limits<RealScalar>::digits<= 53? 7:
87 std::numeric_limits<RealScalar>::digits<= 64? 8:
88 std::numeric_limits<RealScalar>::digits<=106? 10: 11;
96 template <
typename MatrixType>
100 MatrixType result(A.rows(), A.rows());
102 result(0,0) =
log(A(0,0));
103 else if (A.rows() == 2)
104 compute2x2(A, result);
106 computeBig(A, result);
111 template <
typename MatrixType>
116 if (
abs(x) >
sqrt(NumTraits<Scalar>::epsilon()))
117 return Scalar(0.5) *
log((Scalar(1) + x) / (Scalar(1) - x));
119 return x + x*x*x / Scalar(3);
123 template <
typename MatrixType>
124 void MatrixLogarithmAtomic<MatrixType>::compute2x2(
const MatrixType& A, MatrixType& result)
131 Scalar logA00 =
log(A(0,0));
132 Scalar logA11 =
log(A(1,1));
134 result(0,0) = logA00;
135 result(1,0) = Scalar(0);
136 result(1,1) = logA11;
138 if (A(0,0) == A(1,1)) {
139 result(0,1) = A(0,1) / A(0,0);
140 }
else if ((
abs(A(0,0)) < 0.5*
abs(A(1,1))) || (
abs(A(0,0)) > 2*
abs(A(1,1)))) {
141 result(0,1) = A(0,1) * (logA11 - logA00) / (A(1,1) - A(0,0));
144 int unwindingNumber =
static_cast<int>(ceil((
imag(logA11 - logA00) - M_PI) / (2*M_PI)));
145 Scalar z = (A(1,1) - A(0,0)) / (A(1,1) + A(0,0));
146 result(0,1) = A(0,1) * (Scalar(2) * atanh(z) + Scalar(0,2*M_PI*unwindingNumber)) / (A(1,1) - A(0,0));
152 template <
typename MatrixType>
153 void MatrixLogarithmAtomic<MatrixType>::computeBig(
const MatrixType& A, MatrixType& result)
155 int numberOfSquareRoots = 0;
156 int numberOfExtraSquareRoots = 0;
159 const RealScalar maxNormForPade = maxPadeDegree<= 5? 5.3149729967117310e-1:
160 maxPadeDegree<= 7? 2.6429608311114350e-1:
161 maxPadeDegree<= 8? 2.32777776523703892094e-1L:
162 maxPadeDegree<=10? 1.05026503471351080481093652651105e-1L:
163 1.1880960220216759245467951592883642e-1L;
166 RealScalar normTminusI = (T - MatrixType::Identity(T.rows(), T.rows())).cwiseAbs().colwise().sum().maxCoeff();
167 if (normTminusI < maxNormForPade) {
168 degree = getPadeDegree(normTminusI);
169 int degree2 = getPadeDegree(normTminusI / RealScalar(2));
170 if ((degree - degree2 <= 1) || (numberOfExtraSquareRoots == 1))
172 ++numberOfExtraSquareRoots;
175 MatrixSquareRootTriangular<MatrixType>(T).compute(sqrtT);
177 ++numberOfSquareRoots;
180 computePade(result, T, degree);
181 result *=
pow(RealScalar(2), numberOfSquareRoots);
185 template <
typename MatrixType>
186 int MatrixLogarithmAtomic<MatrixType>::getPadeDegree(
float normTminusI)
188 const float maxNormForPade[] = { 2.5111573934555054e-1 , 4.0535837411880493e-1,
189 5.3149729967117310e-1 };
190 for (
int degree = 3; degree <= maxPadeDegree; ++degree)
191 if (normTminusI <= maxNormForPade[degree - minPadeDegree])
197 template <
typename MatrixType>
198 int MatrixLogarithmAtomic<MatrixType>::getPadeDegree(
double normTminusI)
200 const double maxNormForPade[] = { 1.6206284795015624e-2 , 5.3873532631381171e-2,
201 1.1352802267628681e-1, 1.8662860613541288e-1, 2.642960831111435e-1 };
202 for (
int degree = 3; degree <= maxPadeDegree; ++degree)
203 if (normTminusI <= maxNormForPade[degree - minPadeDegree])
209 template <
typename MatrixType>
210 int MatrixLogarithmAtomic<MatrixType>::getPadeDegree(
long double normTminusI)
212 #if LDBL_MANT_DIG == 53 // double precision
213 const double maxNormForPade[] = { 1.6206284795015624e-2 , 5.3873532631381171e-2,
214 1.1352802267628681e-1, 1.8662860613541288e-1, 2.642960831111435e-1 };
215 #elif LDBL_MANT_DIG <= 64 // extended precision
216 const double maxNormForPade[] = { 5.48256690357782863103e-3 , 2.34559162387971167321e-2,
217 5.84603923897347449857e-2, 1.08486423756725170223e-1, 1.68385767881294446649e-1,
218 2.32777776523703892094e-1 };
219 #elif LDBL_MANT_DIG <= 106 // double-double
220 const double maxNormForPade[] = { 8.58970550342939562202529664318890e-5 ,
221 9.34074328446359654039446552677759e-4, 4.26117194647672175773064114582860e-3,
222 1.21546224740281848743149666560464e-2, 2.61100544998339436713088248557444e-2,
223 4.66170074627052749243018566390567e-2, 7.32585144444135027565872014932387e-2,
224 1.05026503471351080481093652651105e-1 };
225 #else // quadruple precision
226 const double maxNormForPade[] = { 4.7419931187193005048501568167858103e-5 ,
227 5.8853168473544560470387769480192666e-4, 2.9216120366601315391789493628113520e-3,
228 8.8415758124319434347116734705174308e-3, 1.9850836029449446668518049562565291e-2,
229 3.6688019729653446926585242192447447e-2, 5.9290962294020186998954055264528393e-2,
230 8.6998436081634343903250580992127677e-2, 1.1880960220216759245467951592883642e-1 };
232 for (
int degree = 3; degree <= maxPadeDegree; ++degree)
233 if (normTminusI <= maxNormForPade[degree - minPadeDegree])
239 template <
typename MatrixType>
240 void MatrixLogarithmAtomic<MatrixType>::computePade(MatrixType& result,
const MatrixType& T,
int degree)
243 case 3: computePade3(result, T);
break;
244 case 4: computePade4(result, T);
break;
245 case 5: computePade5(result, T);
break;
246 case 6: computePade6(result, T);
break;
247 case 7: computePade7(result, T);
break;
248 case 8: computePade8(result, T);
break;
249 case 9: computePade9(result, T);
break;
250 case 10: computePade10(result, T);
break;
251 case 11: computePade11(result, T);
break;
252 default: assert(
false);
256 template <
typename MatrixType>
257 void MatrixLogarithmAtomic<MatrixType>::computePade3(MatrixType& result,
const MatrixType& T)
259 const int degree = 3;
260 const RealScalar nodes[] = { 0.1127016653792583114820734600217600L, 0.5000000000000000000000000000000000L,
261 0.8872983346207416885179265399782400L };
262 const RealScalar weights[] = { 0.2777777777777777777777777777777778L, 0.4444444444444444444444444444444444L,
263 0.2777777777777777777777777777777778L };
264 assert(degree <= maxPadeDegree);
265 MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
266 result.setZero(T.rows(), T.rows());
267 for (
int k = 0; k < degree; ++k)
268 result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
269 .template triangularView<Upper>().solve(TminusI);
272 template <
typename MatrixType>
273 void MatrixLogarithmAtomic<MatrixType>::computePade4(MatrixType& result,
const MatrixType& T)
275 const int degree = 4;
276 const RealScalar nodes[] = { 0.0694318442029737123880267555535953L, 0.3300094782075718675986671204483777L,
277 0.6699905217924281324013328795516223L, 0.9305681557970262876119732444464048L };
278 const RealScalar weights[] = { 0.1739274225687269286865319746109997L, 0.3260725774312730713134680253890003L,
279 0.3260725774312730713134680253890003L, 0.1739274225687269286865319746109997L };
280 assert(degree <= maxPadeDegree);
281 MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
282 result.setZero(T.rows(), T.rows());
283 for (
int k = 0; k < degree; ++k)
284 result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
285 .template triangularView<Upper>().solve(TminusI);
288 template <
typename MatrixType>
289 void MatrixLogarithmAtomic<MatrixType>::computePade5(MatrixType& result,
const MatrixType& T)
291 const int degree = 5;
292 const RealScalar nodes[] = { 0.0469100770306680036011865608503035L, 0.2307653449471584544818427896498956L,
293 0.5000000000000000000000000000000000L, 0.7692346550528415455181572103501044L,
294 0.9530899229693319963988134391496965L };
295 const RealScalar weights[] = { 0.1184634425280945437571320203599587L, 0.2393143352496832340206457574178191L,
296 0.2844444444444444444444444444444444L, 0.2393143352496832340206457574178191L,
297 0.1184634425280945437571320203599587L };
298 assert(degree <= maxPadeDegree);
299 MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
300 result.setZero(T.rows(), T.rows());
301 for (
int k = 0; k < degree; ++k)
302 result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
303 .template triangularView<Upper>().solve(TminusI);
306 template <
typename MatrixType>
307 void MatrixLogarithmAtomic<MatrixType>::computePade6(MatrixType& result,
const MatrixType& T)
309 const int degree = 6;
310 const RealScalar nodes[] = { 0.0337652428984239860938492227530027L, 0.1693953067668677431693002024900473L,
311 0.3806904069584015456847491391596440L, 0.6193095930415984543152508608403560L,
312 0.8306046932331322568306997975099527L, 0.9662347571015760139061507772469973L };
313 const RealScalar weights[] = { 0.0856622461895851725201480710863665L, 0.1803807865240693037849167569188581L,
314 0.2339569672863455236949351719947755L, 0.2339569672863455236949351719947755L,
315 0.1803807865240693037849167569188581L, 0.0856622461895851725201480710863665L };
316 assert(degree <= maxPadeDegree);
317 MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
318 result.setZero(T.rows(), T.rows());
319 for (
int k = 0; k < degree; ++k)
320 result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
321 .template triangularView<Upper>().solve(TminusI);
324 template <
typename MatrixType>
325 void MatrixLogarithmAtomic<MatrixType>::computePade7(MatrixType& result,
const MatrixType& T)
327 const int degree = 7;
328 const RealScalar nodes[] = { 0.0254460438286207377369051579760744L, 0.1292344072003027800680676133596058L,
329 0.2970774243113014165466967939615193L, 0.5000000000000000000000000000000000L,
330 0.7029225756886985834533032060384807L, 0.8707655927996972199319323866403942L,
331 0.9745539561713792622630948420239256L };
332 const RealScalar weights[] = { 0.0647424830844348466353057163395410L, 0.1398526957446383339507338857118898L,
333 0.1909150252525594724751848877444876L, 0.2089795918367346938775510204081633L,
334 0.1909150252525594724751848877444876L, 0.1398526957446383339507338857118898L,
335 0.0647424830844348466353057163395410L };
336 assert(degree <= maxPadeDegree);
337 MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
338 result.setZero(T.rows(), T.rows());
339 for (
int k = 0; k < degree; ++k)
340 result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
341 .template triangularView<Upper>().solve(TminusI);
344 template <
typename MatrixType>
345 void MatrixLogarithmAtomic<MatrixType>::computePade8(MatrixType& result,
const MatrixType& T)
347 const int degree = 8;
348 const RealScalar nodes[] = { 0.0198550717512318841582195657152635L, 0.1016667612931866302042230317620848L,
349 0.2372337950418355070911304754053768L, 0.4082826787521750975302619288199080L,
350 0.5917173212478249024697380711800920L, 0.7627662049581644929088695245946232L,
351 0.8983332387068133697957769682379152L, 0.9801449282487681158417804342847365L };
352 const RealScalar weights[] = { 0.0506142681451881295762656771549811L, 0.1111905172266872352721779972131204L,
353 0.1568533229389436436689811009933007L, 0.1813418916891809914825752246385978L,
354 0.1813418916891809914825752246385978L, 0.1568533229389436436689811009933007L,
355 0.1111905172266872352721779972131204L, 0.0506142681451881295762656771549811L };
356 assert(degree <= maxPadeDegree);
357 MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
358 result.setZero(T.rows(), T.rows());
359 for (
int k = 0; k < degree; ++k)
360 result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
361 .template triangularView<Upper>().solve(TminusI);
364 template <
typename MatrixType>
365 void MatrixLogarithmAtomic<MatrixType>::computePade9(MatrixType& result,
const MatrixType& T)
367 const int degree = 9;
368 const RealScalar nodes[] = { 0.0159198802461869550822118985481636L, 0.0819844463366821028502851059651326L,
369 0.1933142836497048013456489803292629L, 0.3378732882980955354807309926783317L,
370 0.5000000000000000000000000000000000L, 0.6621267117019044645192690073216683L,
371 0.8066857163502951986543510196707371L, 0.9180155536633178971497148940348674L,
372 0.9840801197538130449177881014518364L };
373 const RealScalar weights[] = { 0.0406371941807872059859460790552618L, 0.0903240803474287020292360156214564L,
374 0.1303053482014677311593714347093164L, 0.1561735385200014200343152032922218L,
375 0.1651196775006298815822625346434870L, 0.1561735385200014200343152032922218L,
376 0.1303053482014677311593714347093164L, 0.0903240803474287020292360156214564L,
377 0.0406371941807872059859460790552618L };
378 assert(degree <= maxPadeDegree);
379 MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
380 result.setZero(T.rows(), T.rows());
381 for (
int k = 0; k < degree; ++k)
382 result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
383 .template triangularView<Upper>().solve(TminusI);
386 template <
typename MatrixType>
387 void MatrixLogarithmAtomic<MatrixType>::computePade10(MatrixType& result,
const MatrixType& T)
389 const int degree = 10;
390 const RealScalar nodes[] = { 0.0130467357414141399610179939577740L, 0.0674683166555077446339516557882535L,
391 0.1602952158504877968828363174425632L, 0.2833023029353764046003670284171079L,
392 0.4255628305091843945575869994351400L, 0.5744371694908156054424130005648600L,
393 0.7166976970646235953996329715828921L, 0.8397047841495122031171636825574368L,
394 0.9325316833444922553660483442117465L, 0.9869532642585858600389820060422260L };
395 const RealScalar weights[] = { 0.0333356721543440687967844049466659L, 0.0747256745752902965728881698288487L,
396 0.1095431812579910219977674671140816L, 0.1346333596549981775456134607847347L,
397 0.1477621123573764350869464973256692L, 0.1477621123573764350869464973256692L,
398 0.1346333596549981775456134607847347L, 0.1095431812579910219977674671140816L,
399 0.0747256745752902965728881698288487L, 0.0333356721543440687967844049466659L };
400 assert(degree <= maxPadeDegree);
401 MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
402 result.setZero(T.rows(), T.rows());
403 for (
int k = 0; k < degree; ++k)
404 result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
405 .template triangularView<Upper>().solve(TminusI);
408 template <
typename MatrixType>
409 void MatrixLogarithmAtomic<MatrixType>::computePade11(MatrixType& result,
const MatrixType& T)
411 const int degree = 11;
412 const RealScalar nodes[] = { 0.0108856709269715035980309994385713L, 0.0564687001159523504624211153480364L,
413 0.1349239972129753379532918739844233L, 0.2404519353965940920371371652706952L,
414 0.3652284220238275138342340072995692L, 0.5000000000000000000000000000000000L,
415 0.6347715779761724861657659927004308L, 0.7595480646034059079628628347293048L,
416 0.8650760027870246620467081260155767L, 0.9435312998840476495375788846519636L,
417 0.9891143290730284964019690005614287L };
418 const RealScalar weights[] = { 0.0278342835580868332413768602212743L, 0.0627901847324523123173471496119701L,
419 0.0931451054638671257130488207158280L, 0.1165968822959952399592618524215876L,
420 0.1314022722551233310903444349452546L, 0.1364625433889503153572417641681711L,
421 0.1314022722551233310903444349452546L, 0.1165968822959952399592618524215876L,
422 0.0931451054638671257130488207158280L, 0.0627901847324523123173471496119701L,
423 0.0278342835580868332413768602212743L };
424 assert(degree <= maxPadeDegree);
425 MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
426 result.setZero(T.rows(), T.rows());
427 for (
int k = 0; k < degree; ++k)
428 result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
429 .template triangularView<Upper>().solve(TminusI);
445 :
public ReturnByValue<MatrixLogarithmReturnValue<Derived> >
449 typedef typename Derived::Scalar Scalar;
450 typedef typename Derived::Index Index;
462 template <
typename ResultType>
463 inline void evalTo(ResultType& result)
const
465 typedef typename Derived::PlainObject PlainObject;
466 typedef internal::traits<PlainObject> Traits;
467 static const int RowsAtCompileTime = Traits::RowsAtCompileTime;
468 static const int ColsAtCompileTime = Traits::ColsAtCompileTime;
469 static const int Options = PlainObject::Options;
470 typedef std::complex<typename NumTraits<Scalar>::Real> ComplexScalar;
471 typedef Matrix<ComplexScalar, Dynamic, Dynamic, Options, RowsAtCompileTime, ColsAtCompileTime> DynMatrixType;
475 const PlainObject Aevaluated = m_A.eval();
480 Index rows()
const {
return m_A.rows(); }
481 Index cols()
const {
return m_A.cols(); }
484 typename internal::nested<Derived>::type m_A;
490 template<
typename Derived>
491 struct traits<MatrixLogarithmReturnValue<Derived> >
493 typedef typename Derived::PlainObject ReturnType;
501 template <
typename Derived>
502 const MatrixLogarithmReturnValue<Derived> MatrixBase<Derived>::log()
const
504 eigen_assert(rows() == cols());
505 return MatrixLogarithmReturnValue<Derived>(derived());
510 #endif // EIGEN_MATRIX_LOGARITHM