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Extend MatrixPowerTriangularAtomic for future implementation for triangular matrix power.
This commit is contained in:
Chen-Pang He
parent
ed18d6f2ad
commit
067a5a98c8
@ -74,11 +74,11 @@ class MatrixPower : public MatrixPowerBase<MatrixPower<MatrixType>,MatrixType>
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void compute(const Derived& b, ResultType& res, RealScalar p);
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private:
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EIGEN_MATRIX_POWER_PROTECTED_MEMBERS(MatrixPower)
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typedef Matrix<std::complex<RealScalar>, RowsAtCompileTime, ColsAtCompileTime,
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Options,MaxRowsAtCompileTime,MaxColsAtCompileTime> ComplexMatrix;
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MatrixType m_tmp1, m_tmp2;
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ComplexMatrix m_T, m_U, m_fT;
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bool m_init;
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RealScalar modfAndInit(RealScalar, RealScalar*);
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@ -98,9 +98,8 @@ class MatrixPower : public MatrixPowerBase<MatrixPower<MatrixType>,MatrixType>
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template<typename MatrixType>
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template<typename MatrixExpression>
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MatrixPower<MatrixType>::MatrixPower(const MatrixExpression& A) :
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Base(A),
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m_init(false)
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{ /* empty body */ }
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Base(A, 0)
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{ }
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template<typename MatrixType>
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void MatrixPower<MatrixType>::compute(MatrixType& res, RealScalar p)
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@ -113,7 +112,7 @@ void MatrixPower<MatrixType>::compute(MatrixType& res, RealScalar p)
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break;
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default:
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RealScalar intpart, x = modfAndInit(p, &intpart);
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res = MatrixType::Identity(m_A.rows(), m_A.cols());
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res = m_Id;
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computeIntPower(res, intpart);
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computeFracPower(res, x);
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}
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@ -139,22 +138,19 @@ void MatrixPower<MatrixType>::compute(const Derived& b, ResultType& res, RealSca
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template<typename MatrixType>
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typename MatrixPower<MatrixType>::Base::RealScalar MatrixPower<MatrixType>::modfAndInit(RealScalar x, RealScalar* intpart)
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{
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static RealScalar maxAbsEival, minAbsEival;
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*intpart = std::floor(x);
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RealScalar res = x - *intpart;
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if (!m_init && res) {
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if (!m_conditionNumber && res) {
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const ComplexSchur<MatrixType> schurOfA(m_A);
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m_T = schurOfA.matrixT();
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m_U = schurOfA.matrixU();
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m_init = true;
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const RealArray absTdiag = m_T.diagonal().array().abs();
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maxAbsEival = absTdiag.maxCoeff();
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minAbsEival = absTdiag.minCoeff();
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m_conditionNumber = absTdiag.maxCoeff() / absTdiag.minCoeff();
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}
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if (res > RealScalar(0.5) && res > (1-res) * std::pow(maxAbsEival/minAbsEival, res)) {
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if (res>RealScalar(0.5) && res>(1-res)*std::pow(m_conditionNumber, res)) {
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--res;
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++*intpart;
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}
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@ -195,7 +191,7 @@ template<typename Derived, typename ResultType>
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void MatrixPower<MatrixType>::computeIntPower(const Derived& b, ResultType& res, RealScalar p)
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{
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if (b.cols() >= m_A.cols()) {
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m_tmp2 = MatrixType::Identity(m_A.rows(), m_A.cols());
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m_tmp2 = m_Id;
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computeIntPower(m_tmp2, p);
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res.noalias() = m_tmp2 * b;
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}
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@ -18,7 +18,7 @@ struct recompose_complex_schur
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{
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template<typename ResultType, typename MatrixType>
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static inline void run(ResultType& res, const MatrixType& T, const MatrixType& U)
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{ res = U * (T.template triangularView<Upper>() * U.adjoint()); }
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{ res.noalias() = U * (T.template triangularView<Upper>() * U.adjoint()); }
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};
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template<>
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@ -26,7 +26,21 @@ struct recompose_complex_schur<0>
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{
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template<typename ResultType, typename MatrixType>
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static inline void run(ResultType& res, const MatrixType& T, const MatrixType& U)
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{ res = (U * (T.template triangularView<Upper>() * U.adjoint())).real(); }
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{ res.noalias() = (U * (T.template triangularView<Upper>() * U.adjoint())).real(); }
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};
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template<typename Scalar, int IsComplex=NumTraits<Scalar>::IsComplex>
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struct matrix_power_unwinder
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{
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static inline Scalar run(const Scalar& eival, const Scalar& eival0, int unwindingNumber)
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{ return internal::atanh2(eival-eival0, eival+eival0) + Scalar(0, M_PI*unwindingNumber); }
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};
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template<typename Scalar>
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struct matrix_power_unwinder<Scalar,0>
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{
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static inline Scalar run(Scalar eival, Scalar eival0, int)
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{ return internal::atanh2(eival-eival0, eival+eival0); }
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};
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template<typename T>
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@ -68,21 +82,21 @@ inline int matrix_power_get_pade_degree(double normIminusT)
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inline int matrix_power_get_pade_degree(long double normIminusT)
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{
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#if LDBL_MANT_DIG == 53
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const int maxPadeDegree = 7;
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enum { maxPadeDegree = 7 };
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const double maxNormForPade[] = { 1.884160592658218e-2L /* degree = 3 */ , 6.038881904059573e-2L, 1.239917516308172e-1L,
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1.999045567181744e-1L, 2.789358995219730e-1L };
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#elif LDBL_MANT_DIG <= 64
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const int maxPadeDegree = 8;
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enum { maxPadeDegree = 8 };
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const double maxNormForPade[] = { 6.3854693117491799460e-3L /* degree = 3 */ , 2.6394893435456973676e-2L,
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6.4216043030404063729e-2L, 1.1701165502926694307e-1L, 1.7904284231268670284e-1L, 2.4471944416607995472e-1L };
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#elif LDBL_MANT_DIG <= 106
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const int maxPadeDegree = 10;
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enum { maxPadeDegree = 10 };
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const double maxNormForPade[] = { 1.0007161601787493236741409687186e-4L /* degree = 3 */ ,
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1.0007161601787493236741409687186e-3L, 4.7069769360887572939882574746264e-3L, 1.3220386624169159689406653101695e-2L,
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2.8063482381631737920612944054906e-2L, 4.9625993951953473052385361085058e-2L, 7.7367040706027886224557538328171e-2L,
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1.1016843812851143391275867258512e-1L };
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#else
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const int maxPadeDegree = 10;
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enum { maxPadeDegree = 10 };
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const double maxNormForPade[] = { 5.524506147036624377378713555116378e-5L /* degree = 3 */ ,
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6.640600568157479679823602193345995e-4L, 3.227716520106894279249709728084626e-3L,
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9.619593944683432960546978734646284e-3L, 2.134595382433742403911124458161147e-2L,
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@ -97,6 +111,125 @@ inline int matrix_power_get_pade_degree(long double normIminusT)
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}
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} // namespace internal
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#define MATRIX_POWER_TRIANGULAR_2x2_SPECIALIZATION(Mode) \
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template<typename MatrixType> \
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class MatrixPowerTriangular2x2<MatrixType,Mode> \
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{ \
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private: \
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enum { \
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RowsAtCompileTime = MatrixType::RowsAtCompileTime, \
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MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime \
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}; \
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typedef typename MatrixType::Scalar Scalar; \
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typedef typename MatrixType::RealScalar RealScalar; \
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typedef Array<Scalar,RowsAtCompileTime,1,ColMajor,MaxRowsAtCompileTime> ArrayType; \
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const MatrixType& m_T; \
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public: \
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explicit MatrixPowerTriangular2x2(const MatrixType& T) : m_T(T) { } \
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void compute(MatrixType& res, RealScalar p) const; \
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};
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template<typename MatrixType, unsigned int Mode>
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class MatrixPowerTriangular2x2;
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MATRIX_POWER_TRIANGULAR_2x2_SPECIALIZATION(Upper)
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MATRIX_POWER_TRIANGULAR_2x2_SPECIALIZATION(Lower)
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MATRIX_POWER_TRIANGULAR_2x2_SPECIALIZATION(UnitUpper)
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MATRIX_POWER_TRIANGULAR_2x2_SPECIALIZATION(UnitLower)
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MATRIX_POWER_TRIANGULAR_2x2_SPECIALIZATION(StrictlyUpper)
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MATRIX_POWER_TRIANGULAR_2x2_SPECIALIZATION(StrictlyLower)
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template<typename MatrixType>
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void MatrixPowerTriangular2x2<MatrixType,Upper>::compute(MatrixType& res, RealScalar p) const
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{
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using std::abs;
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using std::pow;
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ArrayType logTdiag = m_T.diagonal().array().log();
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res.coeffRef(0,0) = pow(m_T.coeff(0,0), p);
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for (int i=1; i < m_T.cols(); ++i) {
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res.coeffRef(i,i) = pow(m_T.coeff(i,i), p);
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if (m_T.coeff(i-1,i-1) == m_T.coeff(i,i)) {
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res.coeffRef(i-1,i) = p * pow(m_T.coeff(i-1,i), p-1);
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}
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else if (2*abs(m_T.coeff(i-1,i-1)) < abs(m_T.coeff(i,i)) || 2*abs(m_T.coeff(i,i)) < abs(m_T.coeff(i-1,i-1))) {
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res.coeffRef(i-1,i) = m_T.coeff(i-1,i) * (res.coeff(i,i)-res.coeff(i-1,i-1)) / (m_T.coeff(i,i)-m_T.coeff(i-1,i-1));
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}
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else {
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int unwindingNumber = std::ceil((internal::imag(logTdiag[i]-logTdiag[i-1]) - M_PI) / (2*M_PI));
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Scalar w = internal::matrix_power_unwinder<Scalar>::run(m_T.coeff(i,i), m_T.coeff(i-1,i-1), unwindingNumber);
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res.coeffRef(i-1,i) = m_T.coeff(i-1,i) * RealScalar(2) * std::exp(RealScalar(0.5)*p*(logTdiag[i]+logTdiag[i-1])) *
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std::sinh(p * w) / (m_T.coeff(i,i) - m_T.coeff(i-1,i-1));
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}
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}
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}
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template<typename MatrixType>
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void MatrixPowerTriangular2x2<MatrixType,Lower>::compute(MatrixType& res, RealScalar p) const
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{
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using std::abs;
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using std::pow;
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ArrayType logTdiag = m_T.diagonal().array().log();
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res.coeffRef(0,0) = pow(m_T.coeff(0,0), p);
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for (int i=1; i < m_T.cols(); ++i) {
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res.coeffRef(i,i) = pow(m_T.coeff(i,i), p);
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if (m_T.coeff(i-1,i-1) == m_T.coeff(i,i)) {
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res.coeffRef(i,i-1) = p * pow(m_T.coeff(i,i-1), p-1);
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}
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else if (2*abs(m_T.coeff(i-1,i-1)) < abs(m_T.coeff(i,i)) || 2*abs(m_T.coeff(i,i)) < abs(m_T.coeff(i-1,i-1))) {
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res.coeffRef(i,i-1) = m_T.coeff(i,i-1) * (res.coeff(i,i)-res.coeff(i-1,i-1)) / (m_T.coeff(i,i)-m_T.coeff(i-1,i-1));
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}
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else {
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int unwindingNumber = std::ceil((internal::imag(logTdiag[i]-logTdiag[i-1]) - M_PI) / (2*M_PI));
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Scalar w = internal::matrix_power_unwinder<Scalar>::run(m_T.coeff(i,i), m_T.coeff(i-1,i-1), unwindingNumber);
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res.coeffRef(i,i-1) = m_T.coeff(i,i-1) * RealScalar(2) * std::exp(RealScalar(0.5)*p*(logTdiag[i]+logTdiag[i-1])) *
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std::sinh(p * w) / (m_T.coeff(i,i) - m_T.coeff(i-1,i-1));
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}
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}
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}
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template<typename MatrixType>
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void MatrixPowerTriangular2x2<MatrixType,UnitUpper>::compute(MatrixType& res, RealScalar p) const
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{
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for (int i=1; i < m_T.cols(); ++i)
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res.coeffRef(i-1,i) = p * std::pow(m_T.coeff(i-1,i), p-1);
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}
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template<typename MatrixType>
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void MatrixPowerTriangular2x2<MatrixType,UnitLower>::compute(MatrixType& res, RealScalar p) const
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{
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for (int i=1; i < m_T.cols(); ++i) {
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res.coeffRef(i,i-1) = p * std::pow(m_T.coeff(i,i-1), p-1);
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}
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}
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template<typename MatrixType>
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void MatrixPowerTriangular2x2<MatrixType,StrictlyUpper>::compute(MatrixType& res, RealScalar p) const
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{
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RealScalar diag = !p ? 1 : 0;
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res.coeffRef(0,0) = diag;
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for (int i=1; i < m_T.cols(); ++i) {
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res.coeffRef(i,i) = diag;
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res.coeffRef(i-1,i) = p * std::pow(m_T.coeff(i-1,i), p-1);
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}
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}
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template<typename MatrixType>
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void MatrixPowerTriangular2x2<MatrixType,StrictlyLower>::compute(MatrixType& res, RealScalar p) const
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{
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RealScalar diag = !p ? 1 : 0;
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res.coeffRef(0,0) = diag;
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for (int i=1; i < m_T.cols(); ++i) {
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res.coeffRef(i,i) = diag;
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res.coeffRef(i,i-1) = p * std::pow(m_T.coeff(i,i-1), p-1);
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}
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}
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template<typename MatrixType, unsigned int Mode=Upper>
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class MatrixPowerTriangularAtomic
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{
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@ -113,7 +246,6 @@ class MatrixPowerTriangularAtomic
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const MatrixType m_Id;
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void computePade(int degree, const MatrixType& IminusT, MatrixType& res, RealScalar p) const;
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void compute2x2(MatrixType& res, RealScalar p) const;
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void computeBig(MatrixType& res, RealScalar p) const;
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public:
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@ -125,7 +257,7 @@ template<typename MatrixType, unsigned int Mode>
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MatrixPowerTriangularAtomic<MatrixType,Mode>::MatrixPowerTriangularAtomic(const MatrixType& T) :
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m_T(T),
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m_Id(MatrixType::Identity(T.rows(), T.cols()))
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{ /* empty body */ }
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{ }
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template<typename MatrixType, unsigned int Mode>
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void MatrixPowerTriangularAtomic<MatrixType,Mode>::compute(MatrixType& res, RealScalar p) const
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@ -137,7 +269,7 @@ void MatrixPowerTriangularAtomic<MatrixType,Mode>::compute(MatrixType& res, Real
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res(0,0) = std::pow(m_T(0,0), p);
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break;
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case 2:
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compute2x2(res, p);
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MatrixPowerTriangular2x2<MatrixType,Mode>(m_T).compute(res, p);
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break;
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default:
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computeBig(res, p);
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@ -157,42 +289,16 @@ void MatrixPowerTriangularAtomic<MatrixType,Mode>::computePade(int degree, const
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res += m_Id;
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}
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template<typename MatrixType, unsigned int Mode>
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void MatrixPowerTriangularAtomic<MatrixType,Mode>::compute2x2(MatrixType& res, RealScalar p) const
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{
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using std::abs;
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using std::pow;
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ArrayType logTdiag = m_T.diagonal().array().log();
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res(0,0) = pow(m_T(0,0), p);
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for (int i=1; i < m_T.cols(); ++i) {
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res(i,i) = pow(m_T(i,i), p);
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if (m_T(i-1,i-1) == m_T(i,i)) {
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res(i-1,i) = p * pow(m_T(i-1,i), p-1);
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}
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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))) {
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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));
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}
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else {
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// computation in previous branch is inaccurate if abs(m_T(i,i)) \approx abs(m_T(i-1,i-1))
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int unwindingNumber = std::ceil(((logTdiag[i]-logTdiag[i-1]).imag() - M_PI) / (2*M_PI));
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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);
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res(i-1,i) = m_T(i-1,i) * RealScalar(2) * std::exp(RealScalar(0.5) * p * (logTdiag[i]+logTdiag[i-1])) *
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std::sinh(p * w) / (m_T(i,i) - m_T(i-1,i-1));
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}
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}
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}
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template<typename MatrixType, unsigned int Mode>
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void MatrixPowerTriangularAtomic<MatrixType,Mode>::computeBig(MatrixType& res, RealScalar p) const
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{
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const int digits = std::numeric_limits<RealScalar>::digits;
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enum { digits = std::numeric_limits<RealScalar>::digits };
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const RealScalar maxNormForPade = digits <= 24? 4.3386528e-1f: // sigle precision
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digits <= 53? 2.789358995219730e-1: // double precision
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digits <= 64? 2.4471944416607995472e-1L: // extended precision
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digits <= 106? 1.1016843812851143391275867258512e-01: // double-double
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9.134603732914548552537150753385375e-02; // quadruple precision
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digits <= 106? 1.1016843812851143391275867258512e-1L: // double-double
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9.134603732914548552537150753385375e-2L; // quadruple precision
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const MatrixPowerTriangular2x2<MatrixType,Mode> atomic2x2(m_T);
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MatrixType IminusT, sqrtT, T=m_T;
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RealScalar normIminusT;
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int degree, degree2, numberOfSquareRoots=0, numberOfExtraSquareRoots=0;
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@ -214,14 +320,14 @@ void MatrixPowerTriangularAtomic<MatrixType,Mode>::computeBig(MatrixType& res, R
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computePade(degree, IminusT, res, p);
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for (; numberOfSquareRoots; --numberOfSquareRoots) {
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compute2x2(res, std::ldexp(p,-numberOfSquareRoots));
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atomic2x2.compute(res, std::ldexp(p,-numberOfSquareRoots));
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res *= res;
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}
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compute2x2(res, p);
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atomic2x2.compute(res, p);
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}
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#define EIGEN_MATRIX_POWER_PUBLIC_INTERFACE(Derived) \
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typedef MatrixPowerBase<Derived<MatrixType>, MatrixType> Base; \
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typedef MatrixPowerBase<Derived, MatrixType> Base; \
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using Base::RowsAtCompileTime; \
|
||||
using Base::ColsAtCompileTime; \
|
||||
using Base::Options; \
|
||||
@ -229,8 +335,14 @@ void MatrixPowerTriangularAtomic<MatrixType,Mode>::computeBig(MatrixType& res, R
|
||||
using Base::MaxColsAtCompileTime; \
|
||||
typedef typename Base::Scalar Scalar; \
|
||||
typedef typename Base::RealScalar RealScalar; \
|
||||
typedef typename Base::RealArray RealArray; \
|
||||
using Base::m_A;
|
||||
typedef typename Base::RealArray RealArray;
|
||||
|
||||
#define EIGEN_MATRIX_POWER_PROTECTED_MEMBERS(Derived) \
|
||||
using Base::m_A; \
|
||||
using Base::m_Id; \
|
||||
using Base::m_tmp1; \
|
||||
using Base::m_tmp2; \
|
||||
using Base::m_conditionNumber;
|
||||
|
||||
#define EIGEN_MATRIX_POWER_PRODUCT_PUBLIC_INTERFACE(Derived) \
|
||||
typedef MatrixPowerProductBase<Derived, Lhs, Rhs> Base; \
|
||||
@ -277,34 +389,63 @@ class MatrixPowerBase
|
||||
typedef typename MatrixType::RealScalar RealScalar;
|
||||
typedef typename MatrixType::Index Index;
|
||||
|
||||
explicit MatrixPowerBase(const MatrixType& A)
|
||||
: m_A(A), m_del(false) { }
|
||||
explicit MatrixPowerBase(const MatrixType& A, RealScalar cond);
|
||||
|
||||
template<typename OtherDerived>
|
||||
explicit MatrixPowerBase(const MatrixBase<OtherDerived>& A)
|
||||
: m_A(*new MatrixType(A)), m_del(true) { }
|
||||
explicit MatrixPowerBase(const MatrixBase<OtherDerived>& A, RealScalar cond);
|
||||
|
||||
~MatrixPowerBase()
|
||||
{ if (m_del) delete &m_A; }
|
||||
~MatrixPowerBase();
|
||||
|
||||
void compute(MatrixType& res, RealScalar p)
|
||||
{ static_cast<Derived*>(this)->compute(res,p); }
|
||||
void compute(MatrixType& res, RealScalar p);
|
||||
|
||||
template<typename OtherDerived, typename ResultType>
|
||||
void compute(const OtherDerived& b, ResultType& res, RealScalar p)
|
||||
{ static_cast<Derived*>(this)->compute(b,res,p); }
|
||||
void compute(const OtherDerived& b, ResultType& res, RealScalar p);
|
||||
|
||||
Index rows() const { return m_A.rows(); }
|
||||
Index cols() const { return m_A.cols(); }
|
||||
|
||||
protected:
|
||||
typedef Array<RealScalar,RowsAtCompileTime,1,ColMajor,MaxRowsAtCompileTime> RealArray;
|
||||
|
||||
const MatrixType& m_A;
|
||||
const MatrixType m_Id;
|
||||
MatrixType m_tmp1, m_tmp2;
|
||||
RealScalar m_conditionNumber;
|
||||
|
||||
private:
|
||||
const bool m_del; // whether to delete the pointer at destruction
|
||||
};
|
||||
|
||||
template<typename Derived, typename MatrixType>
|
||||
MatrixPowerBase<Derived,MatrixType>::MatrixPowerBase(const MatrixType& A, RealScalar cond) :
|
||||
m_A(A),
|
||||
m_Id(MatrixType::Identity(A.rows(),A.cols())),
|
||||
m_conditionNumber(cond),
|
||||
m_del(false)
|
||||
{ }
|
||||
|
||||
template<typename Derived, typename MatrixType>
|
||||
template<typename OtherDerived>
|
||||
MatrixPowerBase<Derived,MatrixType>::MatrixPowerBase(const MatrixBase<OtherDerived>& A, RealScalar cond) :
|
||||
m_A(*new MatrixType(A)),
|
||||
m_Id(MatrixType::Identity(A.rows(),A.cols())),
|
||||
m_conditionNumber(cond),
|
||||
m_del(true)
|
||||
{ }
|
||||
|
||||
template<typename Derived, typename MatrixType>
|
||||
MatrixPowerBase<Derived,MatrixType>::~MatrixPowerBase()
|
||||
{ if (m_del) delete &m_A; }
|
||||
|
||||
template<typename Derived, typename MatrixType>
|
||||
void MatrixPowerBase<Derived,MatrixType>::compute(MatrixType& res, RealScalar p)
|
||||
{ static_cast<Derived*>(this)->compute(res,p); }
|
||||
|
||||
template<typename Derived, typename MatrixType>
|
||||
template<typename OtherDerived, typename ResultType>
|
||||
void MatrixPowerBase<Derived,MatrixType>::compute(const OtherDerived& b, ResultType& res, RealScalar p)
|
||||
{ static_cast<Derived*>(this)->compute(b,res,p); }
|
||||
|
||||
template<typename Derived, typename Lhs, typename Rhs>
|
||||
class MatrixPowerProductBase : public MatrixBase<Derived>
|
||||
{
|
||||
|
||||
Loading…
Reference in New Issue
Block a user