mirror of
https://gitlab.com/libeigen/eigen.git
synced 2025-08-12 11:49:02 +08:00
Implement matrix power-matrix product again
This commit is contained in:
Chen-Pang He
parent
87afd99433
commit
446d14f6ad
@ -80,6 +80,10 @@ class NoAlias
|
||||
template<typename Lhs, typename Rhs, int NestingFlags>
|
||||
EIGEN_STRONG_INLINE ExpressionType& operator-=(const CoeffBasedProduct<Lhs,Rhs,NestingFlags>& other)
|
||||
{ return m_expression.derived() -= CoeffBasedProduct<Lhs,Rhs,NestByRefBit>(other.lhs(), other.rhs()); }
|
||||
|
||||
template<typename Derived>
|
||||
EIGEN_STRONG_INLINE ExpressionType& operator=(const MatrixPowerProductBase<Derived>& other)
|
||||
{ other.derived().evalTo(m_expression); return m_expression; }
|
||||
#endif
|
||||
|
||||
ExpressionType& expression() const
|
||||
|
||||
@ -272,6 +272,7 @@ template<typename Derived> class MatrixFunctionReturnValue;
|
||||
template<typename Derived> class MatrixSquareRootReturnValue;
|
||||
template<typename Derived> class MatrixLogarithmReturnValue;
|
||||
template<typename Derived> class MatrixPowerReturnValue;
|
||||
template<typename Derived> class MatrixPowerProductBase;
|
||||
|
||||
namespace internal {
|
||||
template <typename Scalar>
|
||||
|
||||
@ -59,6 +59,7 @@
|
||||
#include "src/MatrixFunctions/MatrixFunction.h"
|
||||
#include "src/MatrixFunctions/MatrixSquareRoot.h"
|
||||
#include "src/MatrixFunctions/MatrixLogarithm.h"
|
||||
#include "src/MatrixFunctions/MatrixPowerBase.h"
|
||||
#include "src/MatrixFunctions/MatrixPower.h"
|
||||
|
||||
|
||||
|
||||
@ -12,209 +12,8 @@
|
||||
|
||||
namespace Eigen {
|
||||
|
||||
namespace internal {
|
||||
|
||||
template<int IsComplex>
|
||||
struct recompose_complex_schur
|
||||
{
|
||||
template<typename ResultType, typename MatrixType>
|
||||
static inline void run(ResultType& res, const MatrixType& T, const MatrixType& U)
|
||||
{ res = U * (T.template triangularView<Upper>() * U.adjoint()); }
|
||||
};
|
||||
|
||||
template<>
|
||||
struct recompose_complex_schur<0>
|
||||
{
|
||||
template<typename ResultType, typename MatrixType>
|
||||
static inline void run(ResultType& res, const MatrixType& T, const MatrixType& U)
|
||||
{ res = (U * (T.template triangularView<Upper>() * U.adjoint())).real(); }
|
||||
};
|
||||
|
||||
template<typename T>
|
||||
inline int binary_powering_cost(T p)
|
||||
{
|
||||
int cost, tmp;
|
||||
frexp(p, &cost);
|
||||
while (std::frexp(p, &tmp), tmp > 0) {
|
||||
p -= std::ldexp(static_cast<T>(0.5), tmp);
|
||||
++cost;
|
||||
}
|
||||
return cost;
|
||||
}
|
||||
|
||||
inline int matrix_power_get_pade_degree(float normIminusT)
|
||||
{
|
||||
const float maxNormForPade[] = { 2.8064004e-1f /* degree = 3 */ , 4.3386528e-1f };
|
||||
int degree = 3;
|
||||
for (; degree <= 4; ++degree)
|
||||
if (normIminusT <= maxNormForPade[degree - 3])
|
||||
break;
|
||||
return degree;
|
||||
}
|
||||
|
||||
inline int matrix_power_get_pade_degree(double normIminusT)
|
||||
{
|
||||
const double maxNormForPade[] = { 1.884160592658218e-2 /* degree = 3 */ , 6.038881904059573e-2, 1.239917516308172e-1,
|
||||
1.999045567181744e-1, 2.789358995219730e-1 };
|
||||
int degree = 3;
|
||||
for (; degree <= 7; ++degree)
|
||||
if (normIminusT <= maxNormForPade[degree - 3])
|
||||
break;
|
||||
return degree;
|
||||
}
|
||||
|
||||
inline int matrix_power_get_pade_degree(long double normIminusT)
|
||||
{
|
||||
#if LDBL_MANT_DIG == 53
|
||||
const int maxPadeDegree = 7;
|
||||
const double maxNormForPade[] = { 1.884160592658218e-2L /* degree = 3 */ , 6.038881904059573e-2L, 1.239917516308172e-1L,
|
||||
1.999045567181744e-1L, 2.789358995219730e-1L };
|
||||
#elif LDBL_MANT_DIG <= 64
|
||||
const int maxPadeDegree = 8;
|
||||
const double maxNormForPade[] = { 6.3854693117491799460e-3L /* degree = 3 */ , 2.6394893435456973676e-2L,
|
||||
6.4216043030404063729e-2L, 1.1701165502926694307e-1L, 1.7904284231268670284e-1L, 2.4471944416607995472e-1L };
|
||||
#elif LDBL_MANT_DIG <= 106
|
||||
const int maxPadeDegree = 10;
|
||||
const double maxNormForPade[] = { 1.0007161601787493236741409687186e-4L /* degree = 3 */ ,
|
||||
1.0007161601787493236741409687186e-3L, 4.7069769360887572939882574746264e-3L, 1.3220386624169159689406653101695e-2L,
|
||||
2.8063482381631737920612944054906e-2L, 4.9625993951953473052385361085058e-2L, 7.7367040706027886224557538328171e-2L,
|
||||
1.1016843812851143391275867258512e-1L };
|
||||
#else
|
||||
const int maxPadeDegree = 10;
|
||||
const double maxNormForPade[] = { 5.524506147036624377378713555116378e-5L /* degree = 3 */ ,
|
||||
6.640600568157479679823602193345995e-4L, 3.227716520106894279249709728084626e-3L,
|
||||
9.619593944683432960546978734646284e-3L, 2.134595382433742403911124458161147e-2L,
|
||||
3.908166513900489428442993794761185e-2L, 6.266780814639442865832535460550138e-2L,
|
||||
9.134603732914548552537150753385375e-2L };
|
||||
#endif
|
||||
int degree = 3;
|
||||
for (; degree <= maxPadeDegree; ++degree)
|
||||
if (normIminusT <= maxNormForPade[degree - 3])
|
||||
break;
|
||||
return degree;
|
||||
}
|
||||
} // namespace internal
|
||||
|
||||
/* (non-doc)
|
||||
* \brief Class for computing triangular matrices to fractional power.
|
||||
*
|
||||
* \tparam MatrixType type of the base.
|
||||
*/
|
||||
template<typename MatrixType, int UpLo = Upper> class MatrixPowerTriangularAtomic
|
||||
{
|
||||
private:
|
||||
typedef typename MatrixType::Scalar Scalar;
|
||||
typedef typename MatrixType::RealScalar RealScalar;
|
||||
typedef Array<Scalar,
|
||||
EIGEN_SIZE_MIN_PREFER_FIXED(MatrixType::RowsAtCompileTime,MatrixType::ColsAtCompileTime),
|
||||
1,ColMajor,
|
||||
EIGEN_SIZE_MIN_PREFER_FIXED(MatrixType::MaxRowsAtCompileTime,MatrixType::MaxColsAtCompileTime)> ArrayType;
|
||||
const MatrixType& m_T;
|
||||
|
||||
void computePade(int degree, const MatrixType& IminusT, MatrixType& res, RealScalar p) const;
|
||||
void compute2x2(MatrixType& res, RealScalar p) const;
|
||||
void computeBig(MatrixType& res, RealScalar p) const;
|
||||
|
||||
public:
|
||||
explicit MatrixPowerTriangularAtomic(const MatrixType& T);
|
||||
void compute(MatrixType& res, RealScalar p) const;
|
||||
};
|
||||
|
||||
template<typename MatrixType, int UpLo>
|
||||
MatrixPowerTriangularAtomic<MatrixType,UpLo>::MatrixPowerTriangularAtomic(const MatrixType& T) :
|
||||
m_T(T)
|
||||
{ eigen_assert(T.rows() == T.cols()); }
|
||||
|
||||
template<typename MatrixType, int UpLo>
|
||||
void MatrixPowerTriangularAtomic<MatrixType,UpLo>::compute(MatrixType& res, RealScalar p) const
|
||||
{
|
||||
switch (m_T.rows()) {
|
||||
case 0:
|
||||
break;
|
||||
case 1:
|
||||
res(0,0) = std::pow(m_T(0,0), p);
|
||||
break;
|
||||
case 2:
|
||||
compute2x2(res, p);
|
||||
break;
|
||||
default:
|
||||
computeBig(res, p);
|
||||
}
|
||||
}
|
||||
|
||||
template<typename MatrixType, int UpLo>
|
||||
void MatrixPowerTriangularAtomic<MatrixType,UpLo>::computePade(int degree, const MatrixType& IminusT, MatrixType& res,
|
||||
RealScalar p) const
|
||||
{
|
||||
int i = degree<<1;
|
||||
res = (p-(i>>1)) / ((i-1)<<1) * IminusT;
|
||||
for (--i; i; --i) {
|
||||
res = (MatrixType::Identity(m_T.rows(), m_T.cols()) + res).template triangularView<UpLo>()
|
||||
.solve((i==1 ? -p : i&1 ? (-p-(i>>1))/(i<<1) : (p-(i>>1))/((i-1)<<1)) * IminusT).eval();
|
||||
}
|
||||
res += MatrixType::Identity(m_T.rows(), m_T.cols());
|
||||
}
|
||||
|
||||
template<typename MatrixType, int UpLo>
|
||||
void MatrixPowerTriangularAtomic<MatrixType,UpLo>::compute2x2(MatrixType& res, RealScalar p) const
|
||||
{
|
||||
using std::abs;
|
||||
using std::pow;
|
||||
|
||||
ArrayType logTdiag = m_T.diagonal().array().log();
|
||||
res(0,0) = pow(m_T(0,0), p);
|
||||
|
||||
for (int i=1; i < m_T.cols(); ++i) {
|
||||
res(i,i) = pow(m_T(i,i), p);
|
||||
if (m_T(i-1,i-1) == m_T(i,i)) {
|
||||
res(i-1,i) = p * pow(m_T(i-1,i), p-1);
|
||||
} else if (2*abs(m_T(i-1,i-1)) < abs(m_T(i,i)) || 2*abs(m_T(i,i)) < abs(m_T(i-1,i-1))) {
|
||||
res(i-1,i) = m_T(i-1,i) * (res(i,i)-res(i-1,i-1)) / (m_T(i,i)-m_T(i-1,i-1));
|
||||
} else {
|
||||
// computation in previous branch is inaccurate if abs(m_T(i,i)) \approx abs(m_T(i-1,i-1))
|
||||
int unwindingNumber = std::ceil(((logTdiag[i]-logTdiag[i-1]).imag() - M_PI) / (2*M_PI));
|
||||
Scalar w = internal::atanh2(m_T(i,i)-m_T(i-1,i-1), m_T(i,i)+m_T(i-1,i-1)) + Scalar(0, M_PI*unwindingNumber);
|
||||
res(i-1,i) = m_T(i-1,i) * RealScalar(2) * std::exp(RealScalar(0.5) * p * (logTdiag[i]+logTdiag[i-1])) *
|
||||
std::sinh(p * w) / (m_T(i,i) - m_T(i-1,i-1));
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
template<typename MatrixType, int UpLo>
|
||||
void MatrixPowerTriangularAtomic<MatrixType,UpLo>::computeBig(MatrixType& res, RealScalar p) const
|
||||
{
|
||||
const int digits = std::numeric_limits<RealScalar>::digits;
|
||||
const RealScalar maxNormForPade = digits <= 24? 4.3386528e-1f: // sigle precision
|
||||
digits <= 53? 2.789358995219730e-1: // double precision
|
||||
digits <= 64? 2.4471944416607995472e-1L: // extended precision
|
||||
digits <= 106? 1.1016843812851143391275867258512e-01: // double-double
|
||||
9.134603732914548552537150753385375e-02; // quadruple precision
|
||||
int degree, degree2, numberOfSquareRoots=0, numberOfExtraSquareRoots=0;
|
||||
MatrixType IminusT, sqrtT, T=m_T;
|
||||
RealScalar normIminusT;
|
||||
|
||||
while (true) {
|
||||
IminusT = MatrixType::Identity(m_T.rows(), m_T.cols()) - T;
|
||||
normIminusT = IminusT.cwiseAbs().colwise().sum().maxCoeff();
|
||||
if (normIminusT < maxNormForPade) {
|
||||
degree = internal::matrix_power_get_pade_degree(normIminusT);
|
||||
degree2 = internal::matrix_power_get_pade_degree(normIminusT/2);
|
||||
if (degree - degree2 <= 1 || numberOfExtraSquareRoots)
|
||||
break;
|
||||
++numberOfExtraSquareRoots;
|
||||
}
|
||||
MatrixSquareRootTriangular<MatrixType>(T).compute(sqrtT);
|
||||
T = sqrtT;
|
||||
++numberOfSquareRoots;
|
||||
}
|
||||
computePade(degree, IminusT, res, p);
|
||||
|
||||
for (; numberOfSquareRoots; --numberOfSquareRoots) {
|
||||
compute2x2(res, std::ldexp(p,-numberOfSquareRoots));
|
||||
res *= res;
|
||||
}
|
||||
compute2x2(res, p);
|
||||
}
|
||||
template<typename MatrixType>
|
||||
class MatrixPowerEvaluator;
|
||||
|
||||
/**
|
||||
* \ingroup MatrixFunctions_Module
|
||||
@ -281,8 +80,8 @@ template<typename MatrixType> class MatrixPower
|
||||
*
|
||||
* \param[in] p exponent, a real scalar.
|
||||
*/
|
||||
const MatrixPowerReturnValue<MatrixPower<MatrixType> > operator()(RealScalar p)
|
||||
{ return MatrixPowerReturnValue<MatrixPower<MatrixType> >(*this, p); }
|
||||
const MatrixPowerEvaluator<MatrixType> operator()(RealScalar p)
|
||||
{ return MatrixPowerEvaluator<MatrixType>(*this, p); }
|
||||
|
||||
/**
|
||||
* \brief Compute the matrix power.
|
||||
@ -451,6 +250,30 @@ void MatrixPower<MatrixType>::computeFracPower(ResultType& res, RealScalar p)
|
||||
}
|
||||
}
|
||||
|
||||
template<typename MatrixType, typename PlainObject>
|
||||
class MatrixPowerMatrixProduct : public MatrixPowerProductBase<MatrixPowerMatrixProduct<MatrixType,PlainObject> >
|
||||
{
|
||||
public:
|
||||
typedef MatrixPowerProductBase<MatrixPowerMatrixProduct<MatrixType,PlainObject> > Base;
|
||||
EIGEN_DENSE_PUBLIC_INTERFACE(MatrixPowerMatrixProduct)
|
||||
|
||||
MatrixPowerMatrixProduct(MatrixPower<MatrixType>& pow, const PlainObject& b, RealScalar p)
|
||||
: m_pow(pow), m_b(b), m_p(p) { }
|
||||
|
||||
template<typename ResultType>
|
||||
inline void evalTo(ResultType& res) const
|
||||
{ m_pow.compute(m_b, res, m_p); }
|
||||
|
||||
Index rows() const { return m_b.rows(); }
|
||||
Index cols() const { return m_b.cols(); }
|
||||
|
||||
private:
|
||||
MatrixPower<MatrixType>& m_pow;
|
||||
const PlainObject& m_b;
|
||||
const RealScalar m_p;
|
||||
MatrixPowerMatrixProduct& operator=(const MatrixPowerMatrixProduct&);
|
||||
};
|
||||
|
||||
/**
|
||||
* \ingroup MatrixFunctions_Module
|
||||
*
|
||||
@ -500,41 +323,68 @@ class MatrixPowerReturnValue : public ReturnByValue<MatrixPowerReturnValue<Deriv
|
||||
};
|
||||
|
||||
template<typename MatrixType>
|
||||
class MatrixPowerReturnValue<MatrixPower<MatrixType> >
|
||||
: public ReturnByValue<MatrixPowerReturnValue<MatrixPower<MatrixType> > >
|
||||
class MatrixPowerEvaluator
|
||||
: public ReturnByValue<MatrixPowerEvaluator<MatrixType> >
|
||||
{
|
||||
public:
|
||||
typedef typename MatrixType::RealScalar RealScalar;
|
||||
typedef typename MatrixType::Index Index;
|
||||
|
||||
MatrixPowerReturnValue(MatrixPower<MatrixType>& ref, RealScalar p)
|
||||
MatrixPowerEvaluator(MatrixPower<MatrixType>& ref, RealScalar p)
|
||||
: m_pow(ref), m_p(p) { }
|
||||
|
||||
template<typename ResultType>
|
||||
inline void evalTo(ResultType& res) const
|
||||
{ m_pow.compute(res, m_p); }
|
||||
|
||||
template<typename Derived>
|
||||
const MatrixPowerMatrixProduct<MatrixType, typename Derived::PlainObject> operator*(const MatrixBase<Derived>& b) const
|
||||
{ return MatrixPowerMatrixProduct<MatrixType, typename Derived::PlainObject>(m_pow, b.derived(), m_p); }
|
||||
|
||||
Index rows() const { return m_pow.rows(); }
|
||||
Index cols() const { return m_pow.cols(); }
|
||||
|
||||
private:
|
||||
MatrixPower<MatrixType>& m_pow;
|
||||
const RealScalar m_p;
|
||||
MatrixPowerReturnValue& operator=(const MatrixPowerReturnValue&);
|
||||
MatrixPowerEvaluator& operator=(const MatrixPowerEvaluator&);
|
||||
};
|
||||
|
||||
namespace internal {
|
||||
template<typename MatrixType, typename PlainObject>
|
||||
struct nested<MatrixPowerMatrixProduct<MatrixType,PlainObject> >
|
||||
{ typedef PlainObject const& type; };
|
||||
|
||||
template<typename Derived>
|
||||
struct traits<MatrixPowerReturnValue<Derived> >
|
||||
{ typedef typename Derived::PlainObject ReturnType; };
|
||||
|
||||
template<typename MatrixType>
|
||||
struct traits<MatrixPowerReturnValue<MatrixPower<MatrixType> > >
|
||||
struct traits<MatrixPowerEvaluator<MatrixType> >
|
||||
{ typedef MatrixType ReturnType; };
|
||||
|
||||
template<typename Derived>
|
||||
struct traits<MatrixPowerProductBase<Derived> >
|
||||
{ typedef typename traits<Derived>::ReturnType ReturnType; };
|
||||
template<typename MatrixType, typename PlainObject>
|
||||
struct traits<MatrixPowerMatrixProduct<MatrixType,PlainObject> >
|
||||
{
|
||||
typedef MatrixXpr XprKind;
|
||||
typedef typename scalar_product_traits<typename MatrixType::Scalar, typename PlainObject::Scalar>::ReturnType Scalar;
|
||||
typedef typename promote_storage_type<typename traits<MatrixType>::StorageKind,
|
||||
typename traits<PlainObject>::StorageKind>::ret StorageKind;
|
||||
typedef typename promote_index_type<typename traits<MatrixType>::Index,
|
||||
typename traits<PlainObject>::Index>::type Index;
|
||||
|
||||
enum {
|
||||
RowsAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED(traits<MatrixType>::RowsAtCompileTime,
|
||||
traits<PlainObject>::RowsAtCompileTime),
|
||||
ColsAtCompileTime = traits<PlainObject>::ColsAtCompileTime,
|
||||
MaxRowsAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED(traits<MatrixType>::MaxRowsAtCompileTime,
|
||||
traits<PlainObject>::MaxRowsAtCompileTime),
|
||||
MaxColsAtCompileTime = traits<PlainObject>::MaxColsAtCompileTime,
|
||||
Flags = (MaxRowsAtCompileTime==1 ? RowMajorBit : 0)
|
||||
| EvalBeforeNestingBit | EvalBeforeAssigningBit | NestByRefBit,
|
||||
CoeffReadCost = 0
|
||||
};
|
||||
};
|
||||
}
|
||||
|
||||
template<typename Derived>
|
||||
@ -544,6 +394,6 @@ const MatrixPowerReturnValue<Derived> MatrixBase<Derived>::pow(RealScalar p) con
|
||||
return MatrixPowerReturnValue<Derived>(derived(), p);
|
||||
}
|
||||
|
||||
} // end namespace Eigen
|
||||
} // namespace Eigen
|
||||
|
||||
#endif // EIGEN_MATRIX_POWER
|
||||
|
||||
247
unsupported/Eigen/src/MatrixFunctions/MatrixPowerBase.h
Normal file
247
unsupported/Eigen/src/MatrixFunctions/MatrixPowerBase.h
Normal file
@ -0,0 +1,247 @@
|
||||
// This file is part of Eigen, a lightweight C++ template library
|
||||
// for linear algebra.
|
||||
//
|
||||
// Copyright (C) 2012 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_POWER_BASE
|
||||
#define EIGEN_MATRIX_POWER_BASE
|
||||
|
||||
namespace Eigen {
|
||||
|
||||
namespace internal {
|
||||
template<int IsComplex>
|
||||
struct recompose_complex_schur
|
||||
{
|
||||
template<typename ResultType, typename MatrixType>
|
||||
static inline void run(ResultType& res, const MatrixType& T, const MatrixType& U)
|
||||
{ res = U * (T.template triangularView<Upper>() * U.adjoint()); }
|
||||
};
|
||||
|
||||
template<>
|
||||
struct recompose_complex_schur<0>
|
||||
{
|
||||
template<typename ResultType, typename MatrixType>
|
||||
static inline void run(ResultType& res, const MatrixType& T, const MatrixType& U)
|
||||
{ res = (U * (T.template triangularView<Upper>() * U.adjoint())).real(); }
|
||||
};
|
||||
|
||||
template<typename Derived>
|
||||
struct traits<MatrixPowerProductBase<Derived> > : traits<Derived>
|
||||
{ };
|
||||
|
||||
template<typename T>
|
||||
inline int binary_powering_cost(T p)
|
||||
{
|
||||
int cost, tmp;
|
||||
frexp(p, &cost);
|
||||
while (std::frexp(p, &tmp), tmp > 0) {
|
||||
p -= std::ldexp(static_cast<T>(0.5), tmp);
|
||||
++cost;
|
||||
}
|
||||
return cost;
|
||||
}
|
||||
|
||||
inline int matrix_power_get_pade_degree(float normIminusT)
|
||||
{
|
||||
const float maxNormForPade[] = { 2.8064004e-1f /* degree = 3 */ , 4.3386528e-1f };
|
||||
int degree = 3;
|
||||
for (; degree <= 4; ++degree)
|
||||
if (normIminusT <= maxNormForPade[degree - 3])
|
||||
break;
|
||||
return degree;
|
||||
}
|
||||
|
||||
inline int matrix_power_get_pade_degree(double normIminusT)
|
||||
{
|
||||
const double maxNormForPade[] = { 1.884160592658218e-2 /* degree = 3 */ , 6.038881904059573e-2, 1.239917516308172e-1,
|
||||
1.999045567181744e-1, 2.789358995219730e-1 };
|
||||
int degree = 3;
|
||||
for (; degree <= 7; ++degree)
|
||||
if (normIminusT <= maxNormForPade[degree - 3])
|
||||
break;
|
||||
return degree;
|
||||
}
|
||||
|
||||
inline int matrix_power_get_pade_degree(long double normIminusT)
|
||||
{
|
||||
#if LDBL_MANT_DIG == 53
|
||||
const int maxPadeDegree = 7;
|
||||
const double maxNormForPade[] = { 1.884160592658218e-2L /* degree = 3 */ , 6.038881904059573e-2L, 1.239917516308172e-1L,
|
||||
1.999045567181744e-1L, 2.789358995219730e-1L };
|
||||
#elif LDBL_MANT_DIG <= 64
|
||||
const int maxPadeDegree = 8;
|
||||
const double maxNormForPade[] = { 6.3854693117491799460e-3L /* degree = 3 */ , 2.6394893435456973676e-2L,
|
||||
6.4216043030404063729e-2L, 1.1701165502926694307e-1L, 1.7904284231268670284e-1L, 2.4471944416607995472e-1L };
|
||||
#elif LDBL_MANT_DIG <= 106
|
||||
const int maxPadeDegree = 10;
|
||||
const double maxNormForPade[] = { 1.0007161601787493236741409687186e-4L /* degree = 3 */ ,
|
||||
1.0007161601787493236741409687186e-3L, 4.7069769360887572939882574746264e-3L, 1.3220386624169159689406653101695e-2L,
|
||||
2.8063482381631737920612944054906e-2L, 4.9625993951953473052385361085058e-2L, 7.7367040706027886224557538328171e-2L,
|
||||
1.1016843812851143391275867258512e-1L };
|
||||
#else
|
||||
const int maxPadeDegree = 10;
|
||||
const double maxNormForPade[] = { 5.524506147036624377378713555116378e-5L /* degree = 3 */ ,
|
||||
6.640600568157479679823602193345995e-4L, 3.227716520106894279249709728084626e-3L,
|
||||
9.619593944683432960546978734646284e-3L, 2.134595382433742403911124458161147e-2L,
|
||||
3.908166513900489428442993794761185e-2L, 6.266780814639442865832535460550138e-2L,
|
||||
9.134603732914548552537150753385375e-2L };
|
||||
#endif
|
||||
int degree = 3;
|
||||
for (; degree <= maxPadeDegree; ++degree)
|
||||
if (normIminusT <= maxNormForPade[degree - 3])
|
||||
break;
|
||||
return degree;
|
||||
}
|
||||
} // namespace internal
|
||||
|
||||
template<typename MatrixType, int UpLo = Upper> class MatrixPowerTriangularAtomic
|
||||
{
|
||||
private:
|
||||
typedef typename MatrixType::Scalar Scalar;
|
||||
typedef typename MatrixType::RealScalar RealScalar;
|
||||
typedef Array<Scalar,
|
||||
EIGEN_SIZE_MIN_PREFER_FIXED(MatrixType::RowsAtCompileTime,MatrixType::ColsAtCompileTime),
|
||||
1,ColMajor,
|
||||
EIGEN_SIZE_MIN_PREFER_FIXED(MatrixType::MaxRowsAtCompileTime,MatrixType::MaxColsAtCompileTime)> ArrayType;
|
||||
const MatrixType& m_T;
|
||||
|
||||
void computePade(int degree, const MatrixType& IminusT, MatrixType& res, RealScalar p) const;
|
||||
void compute2x2(MatrixType& res, RealScalar p) const;
|
||||
void computeBig(MatrixType& res, RealScalar p) const;
|
||||
|
||||
public:
|
||||
explicit MatrixPowerTriangularAtomic(const MatrixType& T);
|
||||
void compute(MatrixType& res, RealScalar p) const;
|
||||
};
|
||||
|
||||
template<typename MatrixType, int UpLo>
|
||||
MatrixPowerTriangularAtomic<MatrixType,UpLo>::MatrixPowerTriangularAtomic(const MatrixType& T) :
|
||||
m_T(T)
|
||||
{ eigen_assert(T.rows() == T.cols()); }
|
||||
|
||||
template<typename MatrixType, int UpLo>
|
||||
void MatrixPowerTriangularAtomic<MatrixType,UpLo>::compute(MatrixType& res, RealScalar p) const
|
||||
{
|
||||
switch (m_T.rows()) {
|
||||
case 0:
|
||||
break;
|
||||
case 1:
|
||||
res(0,0) = std::pow(m_T(0,0), p);
|
||||
break;
|
||||
case 2:
|
||||
compute2x2(res, p);
|
||||
break;
|
||||
default:
|
||||
computeBig(res, p);
|
||||
}
|
||||
}
|
||||
|
||||
template<typename MatrixType, int UpLo>
|
||||
void MatrixPowerTriangularAtomic<MatrixType,UpLo>::computePade(int degree, const MatrixType& IminusT, MatrixType& res,
|
||||
RealScalar p) const
|
||||
{
|
||||
int i = degree<<1;
|
||||
res = (p-(i>>1)) / ((i-1)<<1) * IminusT;
|
||||
for (--i; i; --i) {
|
||||
res = (MatrixType::Identity(m_T.rows(), m_T.cols()) + res).template triangularView<UpLo>()
|
||||
.solve((i==1 ? -p : i&1 ? (-p-(i>>1))/(i<<1) : (p-(i>>1))/((i-1)<<1)) * IminusT).eval();
|
||||
}
|
||||
res += MatrixType::Identity(m_T.rows(), m_T.cols());
|
||||
}
|
||||
|
||||
template<typename MatrixType, int UpLo>
|
||||
void MatrixPowerTriangularAtomic<MatrixType,UpLo>::compute2x2(MatrixType& res, RealScalar p) const
|
||||
{
|
||||
using std::abs;
|
||||
using std::pow;
|
||||
|
||||
ArrayType logTdiag = m_T.diagonal().array().log();
|
||||
res(0,0) = pow(m_T(0,0), p);
|
||||
|
||||
for (int i=1; i < m_T.cols(); ++i) {
|
||||
res(i,i) = pow(m_T(i,i), p);
|
||||
if (m_T(i-1,i-1) == m_T(i,i)) {
|
||||
res(i-1,i) = p * pow(m_T(i-1,i), p-1);
|
||||
} else if (2*abs(m_T(i-1,i-1)) < abs(m_T(i,i)) || 2*abs(m_T(i,i)) < abs(m_T(i-1,i-1))) {
|
||||
res(i-1,i) = m_T(i-1,i) * (res(i,i)-res(i-1,i-1)) / (m_T(i,i)-m_T(i-1,i-1));
|
||||
} else {
|
||||
// computation in previous branch is inaccurate if abs(m_T(i,i)) \approx abs(m_T(i-1,i-1))
|
||||
int unwindingNumber = std::ceil(((logTdiag[i]-logTdiag[i-1]).imag() - M_PI) / (2*M_PI));
|
||||
Scalar w = internal::atanh2(m_T(i,i)-m_T(i-1,i-1), m_T(i,i)+m_T(i-1,i-1)) + Scalar(0, M_PI*unwindingNumber);
|
||||
res(i-1,i) = m_T(i-1,i) * RealScalar(2) * std::exp(RealScalar(0.5) * p * (logTdiag[i]+logTdiag[i-1])) *
|
||||
std::sinh(p * w) / (m_T(i,i) - m_T(i-1,i-1));
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
template<typename MatrixType, int UpLo>
|
||||
void MatrixPowerTriangularAtomic<MatrixType,UpLo>::computeBig(MatrixType& res, RealScalar p) const
|
||||
{
|
||||
const int digits = std::numeric_limits<RealScalar>::digits;
|
||||
const RealScalar maxNormForPade = digits <= 24? 4.3386528e-1f: // sigle precision
|
||||
digits <= 53? 2.789358995219730e-1: // double precision
|
||||
digits <= 64? 2.4471944416607995472e-1L: // extended precision
|
||||
digits <= 106? 1.1016843812851143391275867258512e-01: // double-double
|
||||
9.134603732914548552537150753385375e-02; // quadruple precision
|
||||
int degree, degree2, numberOfSquareRoots=0, numberOfExtraSquareRoots=0;
|
||||
MatrixType IminusT, sqrtT, T=m_T;
|
||||
RealScalar normIminusT;
|
||||
|
||||
while (true) {
|
||||
IminusT = MatrixType::Identity(m_T.rows(), m_T.cols()) - T;
|
||||
normIminusT = IminusT.cwiseAbs().colwise().sum().maxCoeff();
|
||||
if (normIminusT < maxNormForPade) {
|
||||
degree = internal::matrix_power_get_pade_degree(normIminusT);
|
||||
degree2 = internal::matrix_power_get_pade_degree(normIminusT/2);
|
||||
if (degree - degree2 <= 1 || numberOfExtraSquareRoots)
|
||||
break;
|
||||
++numberOfExtraSquareRoots;
|
||||
}
|
||||
MatrixSquareRootTriangular<MatrixType>(T).compute(sqrtT);
|
||||
T = sqrtT;
|
||||
++numberOfSquareRoots;
|
||||
}
|
||||
computePade(degree, IminusT, res, p);
|
||||
|
||||
for (; numberOfSquareRoots; --numberOfSquareRoots) {
|
||||
compute2x2(res, std::ldexp(p,-numberOfSquareRoots));
|
||||
res *= res;
|
||||
}
|
||||
compute2x2(res, p);
|
||||
}
|
||||
|
||||
template<typename Derived>
|
||||
class MatrixPowerProductBase : public MatrixBase<Derived>
|
||||
{
|
||||
public:
|
||||
typedef MatrixBase<Derived> Base;
|
||||
typedef typename Base::PlainObject PlainObject;
|
||||
EIGEN_DENSE_PUBLIC_INTERFACE(MatrixPowerProductBase)
|
||||
|
||||
inline Index rows() const { return derived().rows(); }
|
||||
inline Index cols() const { return derived().cols(); }
|
||||
|
||||
template<typename ResultType>
|
||||
inline void evalTo(ResultType& res) const { derived().evalTo(res); }
|
||||
|
||||
const PlainObject& eval() const
|
||||
{
|
||||
m_result.resize(rows(), cols());
|
||||
derived().evalTo(m_result);
|
||||
return m_result;
|
||||
}
|
||||
|
||||
operator const PlainObject&() const
|
||||
{ return eval(); }
|
||||
|
||||
protected:
|
||||
mutable PlainObject m_result;
|
||||
};
|
||||
|
||||
} // namespace Eigen
|
||||
|
||||
#endif // EIGEN_MATRIX_POWER
|
||||
@ -86,6 +86,28 @@ void testExponentLaws(const MatrixType& m, double tol)
|
||||
}
|
||||
}
|
||||
|
||||
template<typename MatrixType, typename VectorType>
|
||||
void testMatrixVectorProduct(const MatrixType& m, const VectorType& v, double tol)
|
||||
{
|
||||
typedef typename MatrixType::RealScalar RealScalar;
|
||||
MatrixType m1;
|
||||
VectorType v1, v2, v3;
|
||||
RealScalar p;
|
||||
|
||||
for (int i=0; i<g_repeat; ++i) {
|
||||
generateTestMatrix<MatrixType>::run(m1, m.rows());
|
||||
MatrixPower<MatrixType> mpow(m1);
|
||||
|
||||
v1 = VectorType::Random(v.rows(), v.cols());
|
||||
p = internal::random<RealScalar>();
|
||||
|
||||
v2.noalias() = mpow(p) * v1;
|
||||
v3.noalias() = mpow(p).eval() * v1;
|
||||
std::cout << "testMatrixVectorProduct: error powerm = " << relerr(v2, v3) << '\n';
|
||||
VERIFY(v2.isApprox(v3, static_cast<RealScalar>(tol)));
|
||||
}
|
||||
}
|
||||
|
||||
void test_matrix_power()
|
||||
{
|
||||
typedef Matrix<long double,Dynamic,Dynamic> MatrixXe;
|
||||
@ -105,4 +127,14 @@ void test_matrix_power()
|
||||
CALL_SUBTEST_5(testExponentLaws(Matrix3cf(), 1e-4));
|
||||
CALL_SUBTEST_8(testExponentLaws(Matrix4f(), 1e-4));
|
||||
CALL_SUBTEST_6(testExponentLaws(MatrixXf(8,8), 1e-4));
|
||||
|
||||
CALL_SUBTEST_2(testMatrixVectorProduct(Matrix2d(), Vector2d(), 1e-13));
|
||||
CALL_SUBTEST_7(testMatrixVectorProduct(Matrix<double,3,3,RowMajor>(), Vector3d(), 1e-13));
|
||||
CALL_SUBTEST_3(testMatrixVectorProduct(Matrix4cd(), Vector4cd(), 1e-13));
|
||||
CALL_SUBTEST_4(testMatrixVectorProduct(MatrixXd(8,8), MatrixXd(8,2), 1e-13));
|
||||
CALL_SUBTEST_1(testMatrixVectorProduct(Matrix2f(), Vector2f(), 1e-4));
|
||||
CALL_SUBTEST_5(testMatrixVectorProduct(Matrix3cf(), Vector3cf(), 1e-4));
|
||||
CALL_SUBTEST_8(testMatrixVectorProduct(Matrix4f(), Vector4f(), 1e-4));
|
||||
CALL_SUBTEST_6(testMatrixVectorProduct(MatrixXf(8,8), VectorXf(8), 1e-4));
|
||||
CALL_SUBTEST_9(testMatrixVectorProduct(MatrixXe(7,7), MatrixXe(7,9), 1e-13));
|
||||
}
|
||||
|
||||
Loading…
x
Reference in New Issue
Block a user