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-rw-r--r--eigen/unsupported/Eigen/src/MatrixFunctions/MatrixLogarithm.h507
1 files changed, 197 insertions, 310 deletions
diff --git a/eigen/unsupported/Eigen/src/MatrixFunctions/MatrixLogarithm.h b/eigen/unsupported/Eigen/src/MatrixFunctions/MatrixLogarithm.h
index c744fc0..1acfbed 100644
--- a/eigen/unsupported/Eigen/src/MatrixFunctions/MatrixLogarithm.h
+++ b/eigen/unsupported/Eigen/src/MatrixFunctions/MatrixLogarithm.h
@@ -1,7 +1,7 @@
// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
-// Copyright (C) 2011 Jitse Niesen <jitse@maths.leeds.ac.uk>
+// Copyright (C) 2011, 2013 Jitse Niesen <jitse@maths.leeds.ac.uk>
// Copyright (C) 2011 Chen-Pang He <jdh8@ms63.hinet.net>
//
// This Source Code Form is subject to the terms of the Mozilla
@@ -11,91 +11,33 @@
#ifndef EIGEN_MATRIX_LOGARITHM
#define EIGEN_MATRIX_LOGARITHM
-#ifndef M_PI
-#define M_PI 3.141592653589793238462643383279503L
-#endif
-
namespace Eigen {
-/** \ingroup MatrixFunctions_Module
- * \class MatrixLogarithmAtomic
- * \brief Helper class for computing matrix logarithm of atomic matrices.
- *
- * \internal
- * Here, an atomic matrix is a triangular matrix whose diagonal
- * entries are close to each other.
- *
- * \sa class MatrixFunctionAtomic, MatrixBase::log()
- */
-template <typename MatrixType>
-class MatrixLogarithmAtomic
-{
-public:
-
- typedef typename MatrixType::Scalar Scalar;
- // typedef typename MatrixType::Index Index;
- typedef typename NumTraits<Scalar>::Real RealScalar;
- // typedef typename internal::stem_function<Scalar>::type StemFunction;
- // typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, 1> VectorType;
-
- /** \brief Constructor. */
- MatrixLogarithmAtomic() { }
-
- /** \brief Compute matrix logarithm of atomic matrix
- * \param[in] A argument of matrix logarithm, should be upper triangular and atomic
- * \returns The logarithm of \p A.
- */
- MatrixType compute(const MatrixType& A);
-
-private:
+namespace internal {
- void compute2x2(const MatrixType& A, MatrixType& result);
- void computeBig(const MatrixType& A, MatrixType& result);
- int getPadeDegree(float normTminusI);
- int getPadeDegree(double normTminusI);
- int getPadeDegree(long double normTminusI);
- void computePade(MatrixType& result, const MatrixType& T, int degree);
- void computePade3(MatrixType& result, const MatrixType& T);
- void computePade4(MatrixType& result, const MatrixType& T);
- void computePade5(MatrixType& result, const MatrixType& T);
- void computePade6(MatrixType& result, const MatrixType& T);
- void computePade7(MatrixType& result, const MatrixType& T);
- void computePade8(MatrixType& result, const MatrixType& T);
- void computePade9(MatrixType& result, const MatrixType& T);
- void computePade10(MatrixType& result, const MatrixType& T);
- void computePade11(MatrixType& result, const MatrixType& T);
-
- static const int minPadeDegree = 3;
- static const int maxPadeDegree = std::numeric_limits<RealScalar>::digits<= 24? 5: // single precision
- std::numeric_limits<RealScalar>::digits<= 53? 7: // double precision
- std::numeric_limits<RealScalar>::digits<= 64? 8: // extended precision
- std::numeric_limits<RealScalar>::digits<=106? 10: // double-double
- 11; // quadruple precision
-
- // Prevent copying
- MatrixLogarithmAtomic(const MatrixLogarithmAtomic&);
- MatrixLogarithmAtomic& operator=(const MatrixLogarithmAtomic&);
+template <typename Scalar>
+struct matrix_log_min_pade_degree
+{
+ static const int value = 3;
};
-/** \brief Compute logarithm of triangular matrix with clustered eigenvalues. */
-template <typename MatrixType>
-MatrixType MatrixLogarithmAtomic<MatrixType>::compute(const MatrixType& A)
+template <typename Scalar>
+struct matrix_log_max_pade_degree
{
- using std::log;
- MatrixType result(A.rows(), A.rows());
- if (A.rows() == 1)
- result(0,0) = log(A(0,0));
- else if (A.rows() == 2)
- compute2x2(A, result);
- else
- computeBig(A, result);
- return result;
-}
+ typedef typename NumTraits<Scalar>::Real RealScalar;
+ static const int value = std::numeric_limits<RealScalar>::digits<= 24? 5: // single precision
+ std::numeric_limits<RealScalar>::digits<= 53? 7: // double precision
+ std::numeric_limits<RealScalar>::digits<= 64? 8: // extended precision
+ std::numeric_limits<RealScalar>::digits<=106? 10: // double-double
+ 11; // quadruple precision
+};
/** \brief Compute logarithm of 2x2 triangular matrix. */
template <typename MatrixType>
-void MatrixLogarithmAtomic<MatrixType>::compute2x2(const MatrixType& A, MatrixType& result)
+void matrix_log_compute_2x2(const MatrixType& A, MatrixType& result)
{
+ typedef typename MatrixType::Scalar Scalar;
+ typedef typename MatrixType::RealScalar RealScalar;
using std::abs;
using std::ceil;
using std::imag;
@@ -108,59 +50,31 @@ void MatrixLogarithmAtomic<MatrixType>::compute2x2(const MatrixType& A, MatrixTy
result(1,0) = Scalar(0);
result(1,1) = logA11;
- if (A(0,0) == A(1,1)) {
+ Scalar y = A(1,1) - A(0,0);
+ if (y==Scalar(0))
+ {
result(0,1) = A(0,1) / A(0,0);
- } else if ((abs(A(0,0)) < 0.5*abs(A(1,1))) || (abs(A(0,0)) > 2*abs(A(1,1)))) {
- result(0,1) = A(0,1) * (logA11 - logA00) / (A(1,1) - A(0,0));
- } else {
- // computation in previous branch is inaccurate if A(1,1) \approx A(0,0)
- int unwindingNumber = static_cast<int>(ceil((imag(logA11 - logA00) - M_PI) / (2*M_PI)));
- Scalar y = A(1,1) - A(0,0), x = A(1,1) + A(0,0);
- result(0,1) = A(0,1) * (Scalar(2) * numext::atanh2(y,x) + Scalar(0,2*M_PI*unwindingNumber)) / y;
}
-}
-
-/** \brief Compute logarithm of triangular matrices with size > 2.
- * \details This uses a inverse scale-and-square algorithm. */
-template <typename MatrixType>
-void MatrixLogarithmAtomic<MatrixType>::computeBig(const MatrixType& A, MatrixType& result)
-{
- using std::pow;
- int numberOfSquareRoots = 0;
- int numberOfExtraSquareRoots = 0;
- int degree;
- MatrixType T = A, sqrtT;
- const RealScalar maxNormForPade = maxPadeDegree<= 5? 5.3149729967117310e-1: // single precision
- maxPadeDegree<= 7? 2.6429608311114350e-1: // double precision
- maxPadeDegree<= 8? 2.32777776523703892094e-1L: // extended precision
- maxPadeDegree<=10? 1.05026503471351080481093652651105e-1L: // double-double
- 1.1880960220216759245467951592883642e-1L; // quadruple precision
-
- while (true) {
- RealScalar normTminusI = (T - MatrixType::Identity(T.rows(), T.rows())).cwiseAbs().colwise().sum().maxCoeff();
- if (normTminusI < maxNormForPade) {
- degree = getPadeDegree(normTminusI);
- int degree2 = getPadeDegree(normTminusI / RealScalar(2));
- if ((degree - degree2 <= 1) || (numberOfExtraSquareRoots == 1))
- break;
- ++numberOfExtraSquareRoots;
- }
- MatrixSquareRootTriangular<MatrixType>(T).compute(sqrtT);
- T = sqrtT.template triangularView<Upper>();
- ++numberOfSquareRoots;
+ else if ((abs(A(0,0)) < RealScalar(0.5)*abs(A(1,1))) || (abs(A(0,0)) > 2*abs(A(1,1))))
+ {
+ result(0,1) = A(0,1) * (logA11 - logA00) / y;
+ }
+ else
+ {
+ // computation in previous branch is inaccurate if A(1,1) \approx A(0,0)
+ int unwindingNumber = static_cast<int>(ceil((imag(logA11 - logA00) - RealScalar(EIGEN_PI)) / RealScalar(2*EIGEN_PI)));
+ result(0,1) = A(0,1) * (numext::log1p(y/A(0,0)) + Scalar(0,2*EIGEN_PI*unwindingNumber)) / y;
}
-
- computePade(result, T, degree);
- result *= pow(RealScalar(2), numberOfSquareRoots);
}
/* \brief Get suitable degree for Pade approximation. (specialized for RealScalar = float) */
-template <typename MatrixType>
-int MatrixLogarithmAtomic<MatrixType>::getPadeDegree(float normTminusI)
+inline int matrix_log_get_pade_degree(float normTminusI)
{
const float maxNormForPade[] = { 2.5111573934555054e-1 /* degree = 3 */ , 4.0535837411880493e-1,
5.3149729967117310e-1 };
- int degree = 3;
+ const int minPadeDegree = matrix_log_min_pade_degree<float>::value;
+ const int maxPadeDegree = matrix_log_max_pade_degree<float>::value;
+ int degree = minPadeDegree;
for (; degree <= maxPadeDegree; ++degree)
if (normTminusI <= maxNormForPade[degree - minPadeDegree])
break;
@@ -168,12 +82,13 @@ int MatrixLogarithmAtomic<MatrixType>::getPadeDegree(float normTminusI)
}
/* \brief Get suitable degree for Pade approximation. (specialized for RealScalar = double) */
-template <typename MatrixType>
-int MatrixLogarithmAtomic<MatrixType>::getPadeDegree(double normTminusI)
+inline int matrix_log_get_pade_degree(double normTminusI)
{
const double maxNormForPade[] = { 1.6206284795015624e-2 /* degree = 3 */ , 5.3873532631381171e-2,
1.1352802267628681e-1, 1.8662860613541288e-1, 2.642960831111435e-1 };
- int degree = 3;
+ const int minPadeDegree = matrix_log_min_pade_degree<double>::value;
+ const int maxPadeDegree = matrix_log_max_pade_degree<double>::value;
+ int degree = minPadeDegree;
for (; degree <= maxPadeDegree; ++degree)
if (normTminusI <= maxNormForPade[degree - minPadeDegree])
break;
@@ -181,8 +96,7 @@ int MatrixLogarithmAtomic<MatrixType>::getPadeDegree(double normTminusI)
}
/* \brief Get suitable degree for Pade approximation. (specialized for RealScalar = long double) */
-template <typename MatrixType>
-int MatrixLogarithmAtomic<MatrixType>::getPadeDegree(long double normTminusI)
+inline int matrix_log_get_pade_degree(long double normTminusI)
{
#if LDBL_MANT_DIG == 53 // double precision
const long double maxNormForPade[] = { 1.6206284795015624e-2L /* degree = 3 */ , 5.3873532631381171e-2L,
@@ -204,7 +118,9 @@ int MatrixLogarithmAtomic<MatrixType>::getPadeDegree(long double normTminusI)
3.6688019729653446926585242192447447e-2L, 5.9290962294020186998954055264528393e-2L,
8.6998436081634343903250580992127677e-2L, 1.1880960220216759245467951592883642e-1L };
#endif
- int degree = 3;
+ const int minPadeDegree = matrix_log_min_pade_degree<long double>::value;
+ const int maxPadeDegree = matrix_log_max_pade_degree<long double>::value;
+ int degree = minPadeDegree;
for (; degree <= maxPadeDegree; ++degree)
if (normTminusI <= maxNormForPade[degree - minPadeDegree])
break;
@@ -213,197 +129,168 @@ int MatrixLogarithmAtomic<MatrixType>::getPadeDegree(long double normTminusI)
/* \brief Compute Pade approximation to matrix logarithm */
template <typename MatrixType>
-void MatrixLogarithmAtomic<MatrixType>::computePade(MatrixType& result, const MatrixType& T, int degree)
+void matrix_log_compute_pade(MatrixType& result, const MatrixType& T, int degree)
{
- switch (degree) {
- case 3: computePade3(result, T); break;
- case 4: computePade4(result, T); break;
- case 5: computePade5(result, T); break;
- case 6: computePade6(result, T); break;
- case 7: computePade7(result, T); break;
- case 8: computePade8(result, T); break;
- case 9: computePade9(result, T); break;
- case 10: computePade10(result, T); break;
- case 11: computePade11(result, T); break;
- default: assert(false); // should never happen
- }
-}
+ typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
+ const int minPadeDegree = 3;
+ const int maxPadeDegree = 11;
+ assert(degree >= minPadeDegree && degree <= maxPadeDegree);
+
+ const RealScalar nodes[][maxPadeDegree] = {
+ { 0.1127016653792583114820734600217600L, 0.5000000000000000000000000000000000L, // degree 3
+ 0.8872983346207416885179265399782400L },
+ { 0.0694318442029737123880267555535953L, 0.3300094782075718675986671204483777L, // degree 4
+ 0.6699905217924281324013328795516223L, 0.9305681557970262876119732444464048L },
+ { 0.0469100770306680036011865608503035L, 0.2307653449471584544818427896498956L, // degree 5
+ 0.5000000000000000000000000000000000L, 0.7692346550528415455181572103501044L,
+ 0.9530899229693319963988134391496965L },
+ { 0.0337652428984239860938492227530027L, 0.1693953067668677431693002024900473L, // degree 6
+ 0.3806904069584015456847491391596440L, 0.6193095930415984543152508608403560L,
+ 0.8306046932331322568306997975099527L, 0.9662347571015760139061507772469973L },
+ { 0.0254460438286207377369051579760744L, 0.1292344072003027800680676133596058L, // degree 7
+ 0.2970774243113014165466967939615193L, 0.5000000000000000000000000000000000L,
+ 0.7029225756886985834533032060384807L, 0.8707655927996972199319323866403942L,
+ 0.9745539561713792622630948420239256L },
+ { 0.0198550717512318841582195657152635L, 0.1016667612931866302042230317620848L, // degree 8
+ 0.2372337950418355070911304754053768L, 0.4082826787521750975302619288199080L,
+ 0.5917173212478249024697380711800920L, 0.7627662049581644929088695245946232L,
+ 0.8983332387068133697957769682379152L, 0.9801449282487681158417804342847365L },
+ { 0.0159198802461869550822118985481636L, 0.0819844463366821028502851059651326L, // degree 9
+ 0.1933142836497048013456489803292629L, 0.3378732882980955354807309926783317L,
+ 0.5000000000000000000000000000000000L, 0.6621267117019044645192690073216683L,
+ 0.8066857163502951986543510196707371L, 0.9180155536633178971497148940348674L,
+ 0.9840801197538130449177881014518364L },
+ { 0.0130467357414141399610179939577740L, 0.0674683166555077446339516557882535L, // degree 10
+ 0.1602952158504877968828363174425632L, 0.2833023029353764046003670284171079L,
+ 0.4255628305091843945575869994351400L, 0.5744371694908156054424130005648600L,
+ 0.7166976970646235953996329715828921L, 0.8397047841495122031171636825574368L,
+ 0.9325316833444922553660483442117465L, 0.9869532642585858600389820060422260L },
+ { 0.0108856709269715035980309994385713L, 0.0564687001159523504624211153480364L, // degree 11
+ 0.1349239972129753379532918739844233L, 0.2404519353965940920371371652706952L,
+ 0.3652284220238275138342340072995692L, 0.5000000000000000000000000000000000L,
+ 0.6347715779761724861657659927004308L, 0.7595480646034059079628628347293048L,
+ 0.8650760027870246620467081260155767L, 0.9435312998840476495375788846519636L,
+ 0.9891143290730284964019690005614287L } };
+
+ const RealScalar weights[][maxPadeDegree] = {
+ { 0.2777777777777777777777777777777778L, 0.4444444444444444444444444444444444L, // degree 3
+ 0.2777777777777777777777777777777778L },
+ { 0.1739274225687269286865319746109997L, 0.3260725774312730713134680253890003L, // degree 4
+ 0.3260725774312730713134680253890003L, 0.1739274225687269286865319746109997L },
+ { 0.1184634425280945437571320203599587L, 0.2393143352496832340206457574178191L, // degree 5
+ 0.2844444444444444444444444444444444L, 0.2393143352496832340206457574178191L,
+ 0.1184634425280945437571320203599587L },
+ { 0.0856622461895851725201480710863665L, 0.1803807865240693037849167569188581L, // degree 6
+ 0.2339569672863455236949351719947755L, 0.2339569672863455236949351719947755L,
+ 0.1803807865240693037849167569188581L, 0.0856622461895851725201480710863665L },
+ { 0.0647424830844348466353057163395410L, 0.1398526957446383339507338857118898L, // degree 7
+ 0.1909150252525594724751848877444876L, 0.2089795918367346938775510204081633L,
+ 0.1909150252525594724751848877444876L, 0.1398526957446383339507338857118898L,
+ 0.0647424830844348466353057163395410L },
+ { 0.0506142681451881295762656771549811L, 0.1111905172266872352721779972131204L, // degree 8
+ 0.1568533229389436436689811009933007L, 0.1813418916891809914825752246385978L,
+ 0.1813418916891809914825752246385978L, 0.1568533229389436436689811009933007L,
+ 0.1111905172266872352721779972131204L, 0.0506142681451881295762656771549811L },
+ { 0.0406371941807872059859460790552618L, 0.0903240803474287020292360156214564L, // degree 9
+ 0.1303053482014677311593714347093164L, 0.1561735385200014200343152032922218L,
+ 0.1651196775006298815822625346434870L, 0.1561735385200014200343152032922218L,
+ 0.1303053482014677311593714347093164L, 0.0903240803474287020292360156214564L,
+ 0.0406371941807872059859460790552618L },
+ { 0.0333356721543440687967844049466659L, 0.0747256745752902965728881698288487L, // degree 10
+ 0.1095431812579910219977674671140816L, 0.1346333596549981775456134607847347L,
+ 0.1477621123573764350869464973256692L, 0.1477621123573764350869464973256692L,
+ 0.1346333596549981775456134607847347L, 0.1095431812579910219977674671140816L,
+ 0.0747256745752902965728881698288487L, 0.0333356721543440687967844049466659L },
+ { 0.0278342835580868332413768602212743L, 0.0627901847324523123173471496119701L, // degree 11
+ 0.0931451054638671257130488207158280L, 0.1165968822959952399592618524215876L,
+ 0.1314022722551233310903444349452546L, 0.1364625433889503153572417641681711L,
+ 0.1314022722551233310903444349452546L, 0.1165968822959952399592618524215876L,
+ 0.0931451054638671257130488207158280L, 0.0627901847324523123173471496119701L,
+ 0.0278342835580868332413768602212743L } };
-template <typename MatrixType>
-void MatrixLogarithmAtomic<MatrixType>::computePade3(MatrixType& result, const MatrixType& T)
-{
- const int degree = 3;
- const RealScalar nodes[] = { 0.1127016653792583114820734600217600L, 0.5000000000000000000000000000000000L,
- 0.8872983346207416885179265399782400L };
- const RealScalar weights[] = { 0.2777777777777777777777777777777778L, 0.4444444444444444444444444444444444L,
- 0.2777777777777777777777777777777778L };
- eigen_assert(degree <= maxPadeDegree);
MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
result.setZero(T.rows(), T.rows());
- for (int k = 0; k < degree; ++k)
- result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
- .template triangularView<Upper>().solve(TminusI);
-}
+ for (int k = 0; k < degree; ++k) {
+ RealScalar weight = weights[degree-minPadeDegree][k];
+ RealScalar node = nodes[degree-minPadeDegree][k];
+ result += weight * (MatrixType::Identity(T.rows(), T.rows()) + node * TminusI)
+ .template triangularView<Upper>().solve(TminusI);
+ }
+}
+/** \brief Compute logarithm of triangular matrices with size > 2.
+ * \details This uses a inverse scale-and-square algorithm. */
template <typename MatrixType>
-void MatrixLogarithmAtomic<MatrixType>::computePade4(MatrixType& result, const MatrixType& T)
+void matrix_log_compute_big(const MatrixType& A, MatrixType& result)
{
- const int degree = 4;
- const RealScalar nodes[] = { 0.0694318442029737123880267555535953L, 0.3300094782075718675986671204483777L,
- 0.6699905217924281324013328795516223L, 0.9305681557970262876119732444464048L };
- const RealScalar weights[] = { 0.1739274225687269286865319746109997L, 0.3260725774312730713134680253890003L,
- 0.3260725774312730713134680253890003L, 0.1739274225687269286865319746109997L };
- eigen_assert(degree <= maxPadeDegree);
- MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
- result.setZero(T.rows(), T.rows());
- for (int k = 0; k < degree; ++k)
- result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
- .template triangularView<Upper>().solve(TminusI);
-}
+ typedef typename MatrixType::Scalar Scalar;
+ typedef typename NumTraits<Scalar>::Real RealScalar;
+ using std::pow;
-template <typename MatrixType>
-void MatrixLogarithmAtomic<MatrixType>::computePade5(MatrixType& result, const MatrixType& T)
-{
- const int degree = 5;
- const RealScalar nodes[] = { 0.0469100770306680036011865608503035L, 0.2307653449471584544818427896498956L,
- 0.5000000000000000000000000000000000L, 0.7692346550528415455181572103501044L,
- 0.9530899229693319963988134391496965L };
- const RealScalar weights[] = { 0.1184634425280945437571320203599587L, 0.2393143352496832340206457574178191L,
- 0.2844444444444444444444444444444444L, 0.2393143352496832340206457574178191L,
- 0.1184634425280945437571320203599587L };
- eigen_assert(degree <= maxPadeDegree);
- MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
- result.setZero(T.rows(), T.rows());
- for (int k = 0; k < degree; ++k)
- result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
- .template triangularView<Upper>().solve(TminusI);
-}
+ int numberOfSquareRoots = 0;
+ int numberOfExtraSquareRoots = 0;
+ int degree;
+ MatrixType T = A, sqrtT;
-template <typename MatrixType>
-void MatrixLogarithmAtomic<MatrixType>::computePade6(MatrixType& result, const MatrixType& T)
-{
- const int degree = 6;
- const RealScalar nodes[] = { 0.0337652428984239860938492227530027L, 0.1693953067668677431693002024900473L,
- 0.3806904069584015456847491391596440L, 0.6193095930415984543152508608403560L,
- 0.8306046932331322568306997975099527L, 0.9662347571015760139061507772469973L };
- const RealScalar weights[] = { 0.0856622461895851725201480710863665L, 0.1803807865240693037849167569188581L,
- 0.2339569672863455236949351719947755L, 0.2339569672863455236949351719947755L,
- 0.1803807865240693037849167569188581L, 0.0856622461895851725201480710863665L };
- eigen_assert(degree <= maxPadeDegree);
- MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
- result.setZero(T.rows(), T.rows());
- for (int k = 0; k < degree; ++k)
- result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
- .template triangularView<Upper>().solve(TminusI);
-}
+ int maxPadeDegree = matrix_log_max_pade_degree<Scalar>::value;
+ const RealScalar maxNormForPade = maxPadeDegree<= 5? 5.3149729967117310e-1L: // single precision
+ maxPadeDegree<= 7? 2.6429608311114350e-1L: // double precision
+ maxPadeDegree<= 8? 2.32777776523703892094e-1L: // extended precision
+ maxPadeDegree<=10? 1.05026503471351080481093652651105e-1L: // double-double
+ 1.1880960220216759245467951592883642e-1L; // quadruple precision
-template <typename MatrixType>
-void MatrixLogarithmAtomic<MatrixType>::computePade7(MatrixType& result, const MatrixType& T)
-{
- const int degree = 7;
- const RealScalar nodes[] = { 0.0254460438286207377369051579760744L, 0.1292344072003027800680676133596058L,
- 0.2970774243113014165466967939615193L, 0.5000000000000000000000000000000000L,
- 0.7029225756886985834533032060384807L, 0.8707655927996972199319323866403942L,
- 0.9745539561713792622630948420239256L };
- const RealScalar weights[] = { 0.0647424830844348466353057163395410L, 0.1398526957446383339507338857118898L,
- 0.1909150252525594724751848877444876L, 0.2089795918367346938775510204081633L,
- 0.1909150252525594724751848877444876L, 0.1398526957446383339507338857118898L,
- 0.0647424830844348466353057163395410L };
- eigen_assert(degree <= maxPadeDegree);
- MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
- result.setZero(T.rows(), T.rows());
- for (int k = 0; k < degree; ++k)
- result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
- .template triangularView<Upper>().solve(TminusI);
-}
+ while (true) {
+ RealScalar normTminusI = (T - MatrixType::Identity(T.rows(), T.rows())).cwiseAbs().colwise().sum().maxCoeff();
+ if (normTminusI < maxNormForPade) {
+ degree = matrix_log_get_pade_degree(normTminusI);
+ int degree2 = matrix_log_get_pade_degree(normTminusI / RealScalar(2));
+ if ((degree - degree2 <= 1) || (numberOfExtraSquareRoots == 1))
+ break;
+ ++numberOfExtraSquareRoots;
+ }
+ matrix_sqrt_triangular(T, sqrtT);
+ T = sqrtT.template triangularView<Upper>();
+ ++numberOfSquareRoots;
+ }
-template <typename MatrixType>
-void MatrixLogarithmAtomic<MatrixType>::computePade8(MatrixType& result, const MatrixType& T)
-{
- const int degree = 8;
- const RealScalar nodes[] = { 0.0198550717512318841582195657152635L, 0.1016667612931866302042230317620848L,
- 0.2372337950418355070911304754053768L, 0.4082826787521750975302619288199080L,
- 0.5917173212478249024697380711800920L, 0.7627662049581644929088695245946232L,
- 0.8983332387068133697957769682379152L, 0.9801449282487681158417804342847365L };
- const RealScalar weights[] = { 0.0506142681451881295762656771549811L, 0.1111905172266872352721779972131204L,
- 0.1568533229389436436689811009933007L, 0.1813418916891809914825752246385978L,
- 0.1813418916891809914825752246385978L, 0.1568533229389436436689811009933007L,
- 0.1111905172266872352721779972131204L, 0.0506142681451881295762656771549811L };
- eigen_assert(degree <= maxPadeDegree);
- MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
- result.setZero(T.rows(), T.rows());
- for (int k = 0; k < degree; ++k)
- result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
- .template triangularView<Upper>().solve(TminusI);
+ matrix_log_compute_pade(result, T, degree);
+ result *= pow(RealScalar(2), numberOfSquareRoots);
}
+/** \ingroup MatrixFunctions_Module
+ * \class MatrixLogarithmAtomic
+ * \brief Helper class for computing matrix logarithm of atomic matrices.
+ *
+ * Here, an atomic matrix is a triangular matrix whose diagonal entries are close to each other.
+ *
+ * \sa class MatrixFunctionAtomic, MatrixBase::log()
+ */
template <typename MatrixType>
-void MatrixLogarithmAtomic<MatrixType>::computePade9(MatrixType& result, const MatrixType& T)
+class MatrixLogarithmAtomic
{
- const int degree = 9;
- const RealScalar nodes[] = { 0.0159198802461869550822118985481636L, 0.0819844463366821028502851059651326L,
- 0.1933142836497048013456489803292629L, 0.3378732882980955354807309926783317L,
- 0.5000000000000000000000000000000000L, 0.6621267117019044645192690073216683L,
- 0.8066857163502951986543510196707371L, 0.9180155536633178971497148940348674L,
- 0.9840801197538130449177881014518364L };
- const RealScalar weights[] = { 0.0406371941807872059859460790552618L, 0.0903240803474287020292360156214564L,
- 0.1303053482014677311593714347093164L, 0.1561735385200014200343152032922218L,
- 0.1651196775006298815822625346434870L, 0.1561735385200014200343152032922218L,
- 0.1303053482014677311593714347093164L, 0.0903240803474287020292360156214564L,
- 0.0406371941807872059859460790552618L };
- eigen_assert(degree <= maxPadeDegree);
- MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
- result.setZero(T.rows(), T.rows());
- for (int k = 0; k < degree; ++k)
- result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
- .template triangularView<Upper>().solve(TminusI);
-}
+public:
+ /** \brief Compute matrix logarithm of atomic matrix
+ * \param[in] A argument of matrix logarithm, should be upper triangular and atomic
+ * \returns The logarithm of \p A.
+ */
+ MatrixType compute(const MatrixType& A);
+};
template <typename MatrixType>
-void MatrixLogarithmAtomic<MatrixType>::computePade10(MatrixType& result, const MatrixType& T)
+MatrixType MatrixLogarithmAtomic<MatrixType>::compute(const MatrixType& A)
{
- const int degree = 10;
- const RealScalar nodes[] = { 0.0130467357414141399610179939577740L, 0.0674683166555077446339516557882535L,
- 0.1602952158504877968828363174425632L, 0.2833023029353764046003670284171079L,
- 0.4255628305091843945575869994351400L, 0.5744371694908156054424130005648600L,
- 0.7166976970646235953996329715828921L, 0.8397047841495122031171636825574368L,
- 0.9325316833444922553660483442117465L, 0.9869532642585858600389820060422260L };
- const RealScalar weights[] = { 0.0333356721543440687967844049466659L, 0.0747256745752902965728881698288487L,
- 0.1095431812579910219977674671140816L, 0.1346333596549981775456134607847347L,
- 0.1477621123573764350869464973256692L, 0.1477621123573764350869464973256692L,
- 0.1346333596549981775456134607847347L, 0.1095431812579910219977674671140816L,
- 0.0747256745752902965728881698288487L, 0.0333356721543440687967844049466659L };
- eigen_assert(degree <= maxPadeDegree);
- MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
- result.setZero(T.rows(), T.rows());
- for (int k = 0; k < degree; ++k)
- result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
- .template triangularView<Upper>().solve(TminusI);
+ using std::log;
+ MatrixType result(A.rows(), A.rows());
+ if (A.rows() == 1)
+ result(0,0) = log(A(0,0));
+ else if (A.rows() == 2)
+ matrix_log_compute_2x2(A, result);
+ else
+ matrix_log_compute_big(A, result);
+ return result;
}
-template <typename MatrixType>
-void MatrixLogarithmAtomic<MatrixType>::computePade11(MatrixType& result, const MatrixType& T)
-{
- const int degree = 11;
- const RealScalar nodes[] = { 0.0108856709269715035980309994385713L, 0.0564687001159523504624211153480364L,
- 0.1349239972129753379532918739844233L, 0.2404519353965940920371371652706952L,
- 0.3652284220238275138342340072995692L, 0.5000000000000000000000000000000000L,
- 0.6347715779761724861657659927004308L, 0.7595480646034059079628628347293048L,
- 0.8650760027870246620467081260155767L, 0.9435312998840476495375788846519636L,
- 0.9891143290730284964019690005614287L };
- const RealScalar weights[] = { 0.0278342835580868332413768602212743L, 0.0627901847324523123173471496119701L,
- 0.0931451054638671257130488207158280L, 0.1165968822959952399592618524215876L,
- 0.1314022722551233310903444349452546L, 0.1364625433889503153572417641681711L,
- 0.1314022722551233310903444349452546L, 0.1165968822959952399592618524215876L,
- 0.0931451054638671257130488207158280L, 0.0627901847324523123173471496119701L,
- 0.0278342835580868332413768602212743L };
- eigen_assert(degree <= maxPadeDegree);
- MatrixType TminusI = T - MatrixType::Identity(T.rows(), T.rows());
- result.setZero(T.rows(), T.rows());
- for (int k = 0; k < degree; ++k)
- result += weights[k] * (MatrixType::Identity(T.rows(), T.rows()) + nodes[k] * TminusI)
- .template triangularView<Upper>().solve(TminusI);
-}
+} // end of namespace internal
/** \ingroup MatrixFunctions_Module
*
@@ -421,15 +308,19 @@ template<typename Derived> class MatrixLogarithmReturnValue
: public ReturnByValue<MatrixLogarithmReturnValue<Derived> >
{
public:
-
typedef typename Derived::Scalar Scalar;
typedef typename Derived::Index Index;
+protected:
+ typedef typename internal::ref_selector<Derived>::type DerivedNested;
+
+public:
+
/** \brief Constructor.
*
* \param[in] A %Matrix (expression) forming the argument of the matrix logarithm.
*/
- MatrixLogarithmReturnValue(const Derived& A) : m_A(A) { }
+ explicit MatrixLogarithmReturnValue(const Derived& A) : m_A(A) { }
/** \brief Compute the matrix logarithm.
*
@@ -438,28 +329,24 @@ public:
template <typename ResultType>
inline void evalTo(ResultType& result) const
{
- typedef typename Derived::PlainObject PlainObject;
- typedef internal::traits<PlainObject> Traits;
+ typedef typename internal::nested_eval<Derived, 10>::type DerivedEvalType;
+ typedef typename internal::remove_all<DerivedEvalType>::type DerivedEvalTypeClean;
+ typedef internal::traits<DerivedEvalTypeClean> Traits;
static const int RowsAtCompileTime = Traits::RowsAtCompileTime;
static const int ColsAtCompileTime = Traits::ColsAtCompileTime;
- static const int Options = PlainObject::Options;
typedef std::complex<typename NumTraits<Scalar>::Real> ComplexScalar;
- typedef Matrix<ComplexScalar, Dynamic, Dynamic, Options, RowsAtCompileTime, ColsAtCompileTime> DynMatrixType;
- typedef MatrixLogarithmAtomic<DynMatrixType> AtomicType;
+ typedef Matrix<ComplexScalar, Dynamic, Dynamic, 0, RowsAtCompileTime, ColsAtCompileTime> DynMatrixType;
+ typedef internal::MatrixLogarithmAtomic<DynMatrixType> AtomicType;
AtomicType atomic;
- const PlainObject Aevaluated = m_A.eval();
- MatrixFunction<PlainObject, AtomicType> mf(Aevaluated, atomic);
- mf.compute(result);
+ internal::matrix_function_compute<DerivedEvalTypeClean>::run(m_A, atomic, result);
}
Index rows() const { return m_A.rows(); }
Index cols() const { return m_A.cols(); }
private:
- typename internal::nested<Derived>::type m_A;
-
- MatrixLogarithmReturnValue& operator=(const MatrixLogarithmReturnValue&);
+ const DerivedNested m_A;
};
namespace internal {
generated by cgit v1.2.3 (git 2.39.1) at 2025-04-16 22:34:58 +0000