MatrixLogarithm.h
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// 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 Chen-Pang He <jdh8@ms63.hinet.net>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
#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:
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&);
};
/** \brief Compute logarithm of triangular matrix with clustered eigenvalues. */
template <typename MatrixType>
MatrixType MatrixLogarithmAtomic<MatrixType>::compute(const MatrixType& A)
{
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;
}
/** \brief Compute logarithm of 2x2 triangular matrix. */
template <typename MatrixType>
void MatrixLogarithmAtomic<MatrixType>::compute2x2(const MatrixType& A, MatrixType& result)
{
using std::abs;
using std::ceil;
using std::imag;
using std::log;
Scalar logA00 = log(A(0,0));
Scalar logA11 = log(A(1,1));
result(0,0) = logA00;
result(1,0) = Scalar(0);
result(1,1) = logA11;
if (A(0,0) == A(1,1)) {
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;
}
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)
{
const float maxNormForPade[] = { 2.5111573934555054e-1 /* degree = 3 */ , 4.0535837411880493e-1,
5.3149729967117310e-1 };
int degree = 3;
for (; degree <= maxPadeDegree; ++degree)
if (normTminusI <= maxNormForPade[degree - minPadeDegree])
break;
return degree;
}
/* \brief Get suitable degree for Pade approximation. (specialized for RealScalar = double) */
template <typename MatrixType>
int MatrixLogarithmAtomic<MatrixType>::getPadeDegree(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;
for (; degree <= maxPadeDegree; ++degree)
if (normTminusI <= maxNormForPade[degree - minPadeDegree])
break;
return degree;
}
/* \brief Get suitable degree for Pade approximation. (specialized for RealScalar = long double) */
template <typename MatrixType>
int MatrixLogarithmAtomic<MatrixType>::getPadeDegree(long double normTminusI)
{
#if LDBL_MANT_DIG == 53 // double precision
const long double maxNormForPade[] = { 1.6206284795015624e-2L /* degree = 3 */ , 5.3873532631381171e-2L,
1.1352802267628681e-1L, 1.8662860613541288e-1L, 2.642960831111435e-1L };
#elif LDBL_MANT_DIG <= 64 // extended precision
const long double maxNormForPade[] = { 5.48256690357782863103e-3L /* degree = 3 */, 2.34559162387971167321e-2L,
5.84603923897347449857e-2L, 1.08486423756725170223e-1L, 1.68385767881294446649e-1L,
2.32777776523703892094e-1L };
#elif LDBL_MANT_DIG <= 106 // double-double
const long double maxNormForPade[] = { 8.58970550342939562202529664318890e-5L /* degree = 3 */,
9.34074328446359654039446552677759e-4L, 4.26117194647672175773064114582860e-3L,
1.21546224740281848743149666560464e-2L, 2.61100544998339436713088248557444e-2L,
4.66170074627052749243018566390567e-2L, 7.32585144444135027565872014932387e-2L,
1.05026503471351080481093652651105e-1L };
#else // quadruple precision
const long double maxNormForPade[] = { 4.7419931187193005048501568167858103e-5L /* degree = 3 */,
5.8853168473544560470387769480192666e-4L, 2.9216120366601315391789493628113520e-3L,
8.8415758124319434347116734705174308e-3L, 1.9850836029449446668518049562565291e-2L,
3.6688019729653446926585242192447447e-2L, 5.9290962294020186998954055264528393e-2L,
8.6998436081634343903250580992127677e-2L, 1.1880960220216759245467951592883642e-1L };
#endif
int degree = 3;
for (; degree <= maxPadeDegree; ++degree)
if (normTminusI <= maxNormForPade[degree - minPadeDegree])
break;
return degree;
}
/* \brief Compute Pade approximation to matrix logarithm */
template <typename MatrixType>
void MatrixLogarithmAtomic<MatrixType>::computePade(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
}
}
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);
}
template <typename MatrixType>
void MatrixLogarithmAtomic<MatrixType>::computePade4(MatrixType& result, const MatrixType& T)
{
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);
}
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);
}
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);
}
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);
}
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);
}
template <typename MatrixType>
void MatrixLogarithmAtomic<MatrixType>::computePade9(MatrixType& result, const MatrixType& T)
{
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);
}
template <typename MatrixType>
void MatrixLogarithmAtomic<MatrixType>::computePade10(MatrixType& result, const MatrixType& T)
{
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);
}
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);
}
/** \ingroup MatrixFunctions_Module
*
* \brief Proxy for the matrix logarithm of some matrix (expression).
*
* \tparam Derived Type of the argument to the matrix function.
*
* This class holds the argument to the matrix function until it is
* assigned or evaluated for some other reason (so the argument
* should not be changed in the meantime). It is the return type of
* MatrixBase::log() and most of the time this is the only way it
* is used.
*/
template<typename Derived> class MatrixLogarithmReturnValue
: public ReturnByValue<MatrixLogarithmReturnValue<Derived> >
{
public:
typedef typename Derived::Scalar Scalar;
typedef typename Derived::Index Index;
/** \brief Constructor.
*
* \param[in] A %Matrix (expression) forming the argument of the matrix logarithm.
*/
MatrixLogarithmReturnValue(const Derived& A) : m_A(A) { }
/** \brief Compute the matrix logarithm.
*
* \param[out] result Logarithm of \p A, where \A is as specified in the constructor.
*/
template <typename ResultType>
inline void evalTo(ResultType& result) const
{
typedef typename Derived::PlainObject PlainObject;
typedef internal::traits<PlainObject> 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;
AtomicType atomic;
const PlainObject Aevaluated = m_A.eval();
MatrixFunction<PlainObject, AtomicType> mf(Aevaluated, atomic);
mf.compute(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&);
};
namespace internal {
template<typename Derived>
struct traits<MatrixLogarithmReturnValue<Derived> >
{
typedef typename Derived::PlainObject ReturnType;
};
}
/********** MatrixBase method **********/
template <typename Derived>
const MatrixLogarithmReturnValue<Derived> MatrixBase<Derived>::log() const
{
eigen_assert(rows() == cols());
return MatrixLogarithmReturnValue<Derived>(derived());
}
} // end namespace Eigen
#endif // EIGEN_MATRIX_LOGARITHM