diff options
Diffstat (limited to 'unsupported/Eigen/src/MatrixFunctions/MatrixPower.h')
| -rw-r--r-- | unsupported/Eigen/src/MatrixFunctions/MatrixPower.h | 397 |
1 files changed, 299 insertions, 98 deletions
diff --git a/unsupported/Eigen/src/MatrixFunctions/MatrixPower.h b/unsupported/Eigen/src/MatrixFunctions/MatrixPower.h index 78a307e96..ebc433d89 100644 --- a/unsupported/Eigen/src/MatrixFunctions/MatrixPower.h +++ b/unsupported/Eigen/src/MatrixFunctions/MatrixPower.h @@ -14,16 +14,48 @@ namespace Eigen { template<typename MatrixType> class MatrixPower; +/** + * \ingroup MatrixFunctions_Module + * + * \brief Proxy for the matrix power of some matrix. + * + * \tparam MatrixType type of the base, a matrix. + * + * This class holds the arguments to the matrix power 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 + * MatrixPower::operator() and related functions and most of the + * time this is the only way it is used. + */ +/* TODO This class is only used by MatrixPower, so it should be nested + * into MatrixPower, like MatrixPower::ReturnValue. However, my + * compiler complained about unused template parameter in the + * following declaration in namespace internal. + * + * template<typename MatrixType> + * struct traits<MatrixPower<MatrixType>::ReturnValue>; + */ template<typename MatrixType> -class MatrixPowerRetval : public ReturnByValue< MatrixPowerRetval<MatrixType> > +class MatrixPowerParenthesesReturnValue : public ReturnByValue< MatrixPowerParenthesesReturnValue<MatrixType> > { public: typedef typename MatrixType::RealScalar RealScalar; typedef typename MatrixType::Index Index; - MatrixPowerRetval(MatrixPower<MatrixType>& pow, RealScalar p) : m_pow(pow), m_p(p) + /** + * \brief Constructor. + * + * \param[in] pow %MatrixPower storing the base. + * \param[in] p scalar, the exponent of the matrix power. + */ + MatrixPowerParenthesesReturnValue(MatrixPower<MatrixType>& pow, RealScalar p) : m_pow(pow), m_p(p) { } + /** + * \brief Compute the matrix power. + * + * \param[out] result + */ template<typename ResultType> inline void evalTo(ResultType& res) const { m_pow.compute(res, m_p); } @@ -34,11 +66,25 @@ class MatrixPowerRetval : public ReturnByValue< MatrixPowerRetval<MatrixType> > private: MatrixPower<MatrixType>& m_pow; const RealScalar m_p; - MatrixPowerRetval& operator=(const MatrixPowerRetval&); }; +/** + * \ingroup MatrixFunctions_Module + * + * \brief Class for computing matrix powers. + * + * \tparam MatrixType type of the base, expected to be an instantiation + * of the Matrix class template. + * + * This class is capable of computing triangular real/complex matrices + * raised to a power in the interval \f$ (-1, 1) \f$. + * + * \note Currently this class is only used by MatrixPower. One may + * insist that this be nested into MatrixPower. This class is here to + * faciliate future development of triangular matrix functions. + */ template<typename MatrixType> -class MatrixPowerAtomic +class MatrixPowerAtomic : internal::noncopyable { private: enum { @@ -49,14 +95,14 @@ class MatrixPowerAtomic typedef typename MatrixType::RealScalar RealScalar; typedef std::complex<RealScalar> ComplexScalar; typedef typename MatrixType::Index Index; - typedef Array<Scalar, RowsAtCompileTime, 1, ColMajor, MaxRowsAtCompileTime> ArrayType; + typedef Block<MatrixType,Dynamic,Dynamic> ResultType; const MatrixType& m_A; RealScalar m_p; - void computePade(int degree, const MatrixType& IminusT, MatrixType& res) const; - void compute2x2(MatrixType& res, RealScalar p) const; - void computeBig(MatrixType& res) const; + void computePade(int degree, const MatrixType& IminusT, ResultType& res) const; + void compute2x2(ResultType& res, RealScalar p) const; + void computeBig(ResultType& res) const; static int getPadeDegree(float normIminusT); static int getPadeDegree(double normIminusT); static int getPadeDegree(long double normIminusT); @@ -64,24 +110,45 @@ class MatrixPowerAtomic static RealScalar computeSuperDiag(RealScalar, RealScalar, RealScalar p); public: + /** + * \brief Constructor. + * + * \param[in] T the base of the matrix power. + * \param[in] p the exponent of the matrix power, should be in + * \f$ (-1, 1) \f$. + * + * The class stores a reference to T, so it should not be changed + * (or destroyed) before evaluation. Only the upper triangular + * part of T is read. + */ MatrixPowerAtomic(const MatrixType& T, RealScalar p); - void compute(MatrixType& res) const; + + /** + * \brief Compute the matrix power. + * + * \param[out] res \f$ A^p \f$ where A and p are specified in the + * constructor. + */ + void compute(ResultType& res) const; }; template<typename MatrixType> MatrixPowerAtomic<MatrixType>::MatrixPowerAtomic(const MatrixType& T, RealScalar p) : m_A(T), m_p(p) -{ eigen_assert(T.rows() == T.cols()); } +{ + eigen_assert(T.rows() == T.cols()); + eigen_assert(p > -1 && p < 1); +} template<typename MatrixType> -void MatrixPowerAtomic<MatrixType>::compute(MatrixType& res) const +void MatrixPowerAtomic<MatrixType>::compute(ResultType& res) const { - res.resizeLike(m_A); + using std::pow; switch (m_A.rows()) { case 0: break; case 1: - res(0,0) = std::pow(m_A(0,0), m_p); + res(0,0) = pow(m_A(0,0), m_p); break; case 2: compute2x2(res, m_p); @@ -92,24 +159,24 @@ void MatrixPowerAtomic<MatrixType>::compute(MatrixType& res) const } template<typename MatrixType> -void MatrixPowerAtomic<MatrixType>::computePade(int degree, const MatrixType& IminusT, MatrixType& res) const +void MatrixPowerAtomic<MatrixType>::computePade(int degree, const MatrixType& IminusT, ResultType& res) const { - int i = degree<<1; - res = (m_p-degree) / ((i-1)<<1) * IminusT; + int i = 2*degree; + res = (m_p-degree) / (2*i-2) * IminusT; + for (--i; i; --i) { res = (MatrixType::Identity(IminusT.rows(), IminusT.cols()) + res).template triangularView<Upper>() - .solve((i==1 ? -m_p : i&1 ? (-m_p-(i>>1))/(i<<1) : (m_p-(i>>1))/((i-1)<<1)) * IminusT).eval(); + .solve((i==1 ? -m_p : i&1 ? (-m_p-i/2)/(2*i) : (m_p-i/2)/(2*i-2)) * IminusT).eval(); } res += MatrixType::Identity(IminusT.rows(), IminusT.cols()); } // This function assumes that res has the correct size (see bug 614) template<typename MatrixType> -void MatrixPowerAtomic<MatrixType>::compute2x2(MatrixType& res, RealScalar p) const +void MatrixPowerAtomic<MatrixType>::compute2x2(ResultType& res, RealScalar p) const { using std::abs; using std::pow; - res.coeffRef(0,0) = pow(m_A.coeff(0,0), p); for (Index i=1; i < m_A.cols(); ++i) { @@ -125,32 +192,20 @@ void MatrixPowerAtomic<MatrixType>::compute2x2(MatrixType& res, RealScalar p) co } template<typename MatrixType> -void MatrixPowerAtomic<MatrixType>::computeBig(MatrixType& res) const +void MatrixPowerAtomic<MatrixType>::computeBig(ResultType& res) const { + using std::ldexp; 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-1L: // double-double - 9.134603732914548552537150753385375e-2L; // quadruple precision + const RealScalar maxNormForPade = digits <= 24? 4.3386528e-1L // single precision + : digits <= 53? 2.789358995219730e-1L // double precision + : digits <= 64? 2.4471944416607995472e-1L // extended precision + : digits <= 106? 1.1016843812851143391275867258512e-1L // double-double + : 9.134603732914548552537150753385375e-2L; // quadruple precision MatrixType IminusT, sqrtT, T = m_A.template triangularView<Upper>(); RealScalar normIminusT; int degree, degree2, numberOfSquareRoots = 0; bool hasExtraSquareRoot = false; - /* FIXME - * For singular T, norm(I - T) >= 1 but maxNormForPade < 1, leads to infinite - * loop. We should move 0 eigenvalues to bottom right corner. We need not - * worry about tiny values (e.g. 1e-300) because they will reach 1 if - * repetitively sqrt'ed. - * - * If the 0 eigenvalues are semisimple, they can form a 0 matrix at the - * bottom right corner. - * - * [ T A ]^p [ T^p (T^-1 T^p A) ] - * [ ] = [ ] - * [ 0 0 ] [ 0 0 ] - */ for (Index i=0; i < m_A.cols(); ++i) eigen_assert(m_A(i,i) != RealScalar(0)); @@ -164,14 +219,14 @@ void MatrixPowerAtomic<MatrixType>::computeBig(MatrixType& res) const break; hasExtraSquareRoot = true; } - MatrixSquareRootTriangular<MatrixType>(T).compute(sqrtT); + matrix_sqrt_triangular(T, sqrtT); T = sqrtT.template triangularView<Upper>(); ++numberOfSquareRoots; } computePade(degree, IminusT, res); for (; numberOfSquareRoots; --numberOfSquareRoots) { - compute2x2(res, std::ldexp(m_p, -numberOfSquareRoots)); + compute2x2(res, ldexp(m_p, -numberOfSquareRoots)); res = res.template triangularView<Upper>() * res; } compute2x2(res, m_p); @@ -209,7 +264,7 @@ inline int MatrixPowerAtomic<MatrixType>::getPadeDegree(long double normIminusT) 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, + const long 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; @@ -236,19 +291,28 @@ template<typename MatrixType> inline typename MatrixPowerAtomic<MatrixType>::ComplexScalar MatrixPowerAtomic<MatrixType>::computeSuperDiag(const ComplexScalar& curr, const ComplexScalar& prev, RealScalar p) { - ComplexScalar logCurr = std::log(curr); - ComplexScalar logPrev = std::log(prev); - int unwindingNumber = std::ceil((numext::imag(logCurr - logPrev) - M_PI) / (2*M_PI)); - ComplexScalar w = numext::atanh2(curr - prev, curr + prev) + ComplexScalar(0, M_PI*unwindingNumber); - return RealScalar(2) * std::exp(RealScalar(0.5) * p * (logCurr + logPrev)) * std::sinh(p * w) / (curr - prev); + using std::ceil; + using std::exp; + using std::log; + using std::sinh; + + ComplexScalar logCurr = log(curr); + ComplexScalar logPrev = log(prev); + int unwindingNumber = ceil((numext::imag(logCurr - logPrev) - RealScalar(EIGEN_PI)) / RealScalar(2*EIGEN_PI)); + ComplexScalar w = numext::log1p((curr-prev)/prev)/RealScalar(2) + ComplexScalar(0, EIGEN_PI*unwindingNumber); + return RealScalar(2) * exp(RealScalar(0.5) * p * (logCurr + logPrev)) * sinh(p * w) / (curr - prev); } template<typename MatrixType> inline typename MatrixPowerAtomic<MatrixType>::RealScalar MatrixPowerAtomic<MatrixType>::computeSuperDiag(RealScalar curr, RealScalar prev, RealScalar p) { - RealScalar w = numext::atanh2(curr - prev, curr + prev); - return 2 * std::exp(p * (std::log(curr) + std::log(prev)) / 2) * std::sinh(p * w) / (curr - prev); + using std::exp; + using std::log; + using std::sinh; + + RealScalar w = numext::log1p((curr-prev)/prev)/RealScalar(2); + return 2 * exp(p * (log(curr) + log(prev)) / 2) * sinh(p * w) / (curr - prev); } /** @@ -271,15 +335,9 @@ MatrixPowerAtomic<MatrixType>::computeSuperDiag(RealScalar curr, RealScalar prev * Output: \verbinclude MatrixPower_optimal.out */ template<typename MatrixType> -class MatrixPower +class MatrixPower : internal::noncopyable { private: - enum { - RowsAtCompileTime = MatrixType::RowsAtCompileTime, - ColsAtCompileTime = MatrixType::ColsAtCompileTime, - MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime, - MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime - }; typedef typename MatrixType::Scalar Scalar; typedef typename MatrixType::RealScalar RealScalar; typedef typename MatrixType::Index Index; @@ -293,7 +351,11 @@ class MatrixPower * The class stores a reference to A, so it should not be changed * (or destroyed) before evaluation. */ - explicit MatrixPower(const MatrixType& A) : m_A(A), m_conditionNumber(0) + explicit MatrixPower(const MatrixType& A) : + m_A(A), + m_conditionNumber(0), + m_rank(A.cols()), + m_nulls(0) { eigen_assert(A.rows() == A.cols()); } /** @@ -303,8 +365,8 @@ class MatrixPower * \return The expression \f$ A^p \f$, where A is specified in the * constructor. */ - const MatrixPowerRetval<MatrixType> operator()(RealScalar p) - { return MatrixPowerRetval<MatrixType>(*this, p); } + const MatrixPowerParenthesesReturnValue<MatrixType> operator()(RealScalar p) + { return MatrixPowerParenthesesReturnValue<MatrixType>(*this, p); } /** * \brief Compute the matrix power. @@ -321,21 +383,54 @@ class MatrixPower private: typedef std::complex<RealScalar> ComplexScalar; - typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, MatrixType::Options, - MaxRowsAtCompileTime, MaxColsAtCompileTime> ComplexMatrix; + typedef Matrix<ComplexScalar, Dynamic, Dynamic, 0, + MatrixType::RowsAtCompileTime, MatrixType::ColsAtCompileTime> ComplexMatrix; + /** \brief Reference to the base of matrix power. */ typename MatrixType::Nested m_A; + + /** \brief Temporary storage. */ MatrixType m_tmp; - ComplexMatrix m_T, m_U, m_fT; + + /** \brief Store the result of Schur decomposition. */ + ComplexMatrix m_T, m_U; + + /** \brief Store fractional power of m_T. */ + ComplexMatrix m_fT; + + /** + * \brief Condition number of m_A. + * + * It is initialized as 0 to avoid performing unnecessary Schur + * decomposition, which is the bottleneck. + */ RealScalar m_conditionNumber; - RealScalar modfAndInit(RealScalar, RealScalar*); + /** \brief Rank of m_A. */ + Index m_rank; + + /** \brief Rank deficiency of m_A. */ + Index m_nulls; + + /** + * \brief Split p into integral part and fractional part. + * + * \param[in] p The exponent. + * \param[out] p The fractional part ranging in \f$ (-1, 1) \f$. + * \param[out] intpart The integral part. + * + * Only if the fractional part is nonzero, it calls initialize(). + */ + void split(RealScalar& p, RealScalar& intpart); + + /** \brief Perform Schur decomposition for fractional power. */ + void initialize(); template<typename ResultType> - void computeIntPower(ResultType&, RealScalar); + void computeIntPower(ResultType& res, RealScalar p); template<typename ResultType> - void computeFracPower(ResultType&, RealScalar); + void computeFracPower(ResultType& res, RealScalar p); template<int Rows, int Cols, int Options, int MaxRows, int MaxCols> static void revertSchur( @@ -354,59 +449,102 @@ template<typename MatrixType> template<typename ResultType> void MatrixPower<MatrixType>::compute(ResultType& res, RealScalar p) { + using std::pow; switch (cols()) { case 0: break; case 1: - res(0,0) = std::pow(m_A.coeff(0,0), p); + res(0,0) = pow(m_A.coeff(0,0), p); break; default: - RealScalar intpart, x = modfAndInit(p, &intpart); + RealScalar intpart; + split(p, intpart); + + res = MatrixType::Identity(rows(), cols()); computeIntPower(res, intpart); - computeFracPower(res, x); + if (p) computeFracPower(res, p); } } template<typename MatrixType> -typename MatrixPower<MatrixType>::RealScalar -MatrixPower<MatrixType>::modfAndInit(RealScalar x, RealScalar* intpart) +void MatrixPower<MatrixType>::split(RealScalar& p, RealScalar& intpart) { - typedef Array<RealScalar, RowsAtCompileTime, 1, ColMajor, MaxRowsAtCompileTime> RealArray; + using std::floor; + using std::pow; - *intpart = std::floor(x); - RealScalar res = x - *intpart; + intpart = floor(p); + p -= intpart; - if (!m_conditionNumber && res) { - const ComplexSchur<MatrixType> schurOfA(m_A); - m_T = schurOfA.matrixT(); - m_U = schurOfA.matrixU(); - - const RealArray absTdiag = m_T.diagonal().array().abs(); - m_conditionNumber = absTdiag.maxCoeff() / absTdiag.minCoeff(); + // Perform Schur decomposition if it is not yet performed and the power is + // not an integer. + if (!m_conditionNumber && p) + initialize(); + + // Choose the more stable of intpart = floor(p) and intpart = ceil(p). + if (p > RealScalar(0.5) && p > (1-p) * pow(m_conditionNumber, p)) { + --p; + ++intpart; + } +} + +template<typename MatrixType> +void MatrixPower<MatrixType>::initialize() +{ + const ComplexSchur<MatrixType> schurOfA(m_A); + JacobiRotation<ComplexScalar> rot; + ComplexScalar eigenvalue; + + m_fT.resizeLike(m_A); + m_T = schurOfA.matrixT(); + m_U = schurOfA.matrixU(); + m_conditionNumber = m_T.diagonal().array().abs().maxCoeff() / m_T.diagonal().array().abs().minCoeff(); + + // Move zero eigenvalues to the bottom right corner. + for (Index i = cols()-1; i>=0; --i) { + if (m_rank <= 2) + return; + if (m_T.coeff(i,i) == RealScalar(0)) { + for (Index j=i+1; j < m_rank; ++j) { + eigenvalue = m_T.coeff(j,j); + rot.makeGivens(m_T.coeff(j-1,j), eigenvalue); + m_T.applyOnTheRight(j-1, j, rot); + m_T.applyOnTheLeft(j-1, j, rot.adjoint()); + m_T.coeffRef(j-1,j-1) = eigenvalue; + m_T.coeffRef(j,j) = RealScalar(0); + m_U.applyOnTheRight(j-1, j, rot); + } + --m_rank; + } } - if (res>RealScalar(0.5) && res>(1-res)*std::pow(m_conditionNumber, res)) { - --res; - ++*intpart; + m_nulls = rows() - m_rank; + if (m_nulls) { + eigen_assert(m_T.bottomRightCorner(m_nulls, m_nulls).isZero() + && "Base of matrix power should be invertible or with a semisimple zero eigenvalue."); + m_fT.bottomRows(m_nulls).fill(RealScalar(0)); } - return res; } template<typename MatrixType> template<typename ResultType> void MatrixPower<MatrixType>::computeIntPower(ResultType& res, RealScalar p) { - RealScalar pp = std::abs(p); + using std::abs; + using std::fmod; + RealScalar pp = abs(p); - if (p<0) m_tmp = m_A.inverse(); - else m_tmp = m_A; + if (p<0) + m_tmp = m_A.inverse(); + else + m_tmp = m_A; - res = MatrixType::Identity(rows(), cols()); - while (pp >= 1) { - if (std::fmod(pp, 2) >= 1) + while (true) { + if (fmod(pp, 2) >= 1) res = m_tmp * res; - m_tmp *= m_tmp; pp /= 2; + if (pp < 1) + break; + m_tmp *= m_tmp; } } @@ -414,12 +552,17 @@ template<typename MatrixType> template<typename ResultType> void MatrixPower<MatrixType>::computeFracPower(ResultType& res, RealScalar p) { - if (p) { - eigen_assert(m_conditionNumber); - MatrixPowerAtomic<ComplexMatrix>(m_T, p).compute(m_fT); - revertSchur(m_tmp, m_fT, m_U); - res = m_tmp * res; + Block<ComplexMatrix,Dynamic,Dynamic> blockTp(m_fT, 0, 0, m_rank, m_rank); + eigen_assert(m_conditionNumber); + eigen_assert(m_rank + m_nulls == rows()); + + MatrixPowerAtomic<ComplexMatrix>(m_T.topLeftCorner(m_rank, m_rank), p).compute(blockTp); + if (m_nulls) { + m_fT.topRightCorner(m_rank, m_nulls) = m_T.topLeftCorner(m_rank, m_rank).template triangularView<Upper>() + .solve(blockTp * m_T.topRightCorner(m_rank, m_nulls)); } + revertSchur(m_tmp, m_fT, m_U); + res = m_tmp * res; } template<typename MatrixType> @@ -463,7 +606,7 @@ class MatrixPowerReturnValue : public ReturnByValue< MatrixPowerReturnValue<Deri * \brief Constructor. * * \param[in] A %Matrix (expression), the base of the matrix power. - * \param[in] p scalar, the exponent of the matrix power. + * \param[in] p real scalar, the exponent of the matrix power. */ MatrixPowerReturnValue(const Derived& A, RealScalar p) : m_A(A), m_p(p) { } @@ -484,25 +627,83 @@ class MatrixPowerReturnValue : public ReturnByValue< MatrixPowerReturnValue<Deri private: const Derived& m_A; const RealScalar m_p; - MatrixPowerReturnValue& operator=(const MatrixPowerReturnValue&); +}; + +/** + * \ingroup MatrixFunctions_Module + * + * \brief Proxy for the matrix power of some matrix (expression). + * + * \tparam Derived type of the base, a matrix (expression). + * + * This class holds the arguments to the matrix power 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::pow() and related functions and most of the + * time this is the only way it is used. + */ +template<typename Derived> +class MatrixComplexPowerReturnValue : public ReturnByValue< MatrixComplexPowerReturnValue<Derived> > +{ + public: + typedef typename Derived::PlainObject PlainObject; + typedef typename std::complex<typename Derived::RealScalar> ComplexScalar; + typedef typename Derived::Index Index; + + /** + * \brief Constructor. + * + * \param[in] A %Matrix (expression), the base of the matrix power. + * \param[in] p complex scalar, the exponent of the matrix power. + */ + MatrixComplexPowerReturnValue(const Derived& A, const ComplexScalar& p) : m_A(A), m_p(p) + { } + + /** + * \brief Compute the matrix power. + * + * Because \p p is complex, \f$ A^p \f$ is simply evaluated as \f$ + * \exp(p \log(A)) \f$. + * + * \param[out] result \f$ A^p \f$ where \p A and \p p are as in the + * constructor. + */ + template<typename ResultType> + inline void evalTo(ResultType& res) const + { res = (m_p * m_A.log()).exp(); } + + Index rows() const { return m_A.rows(); } + Index cols() const { return m_A.cols(); } + + private: + const Derived& m_A; + const ComplexScalar m_p; }; namespace internal { template<typename MatrixPowerType> -struct traits< MatrixPowerRetval<MatrixPowerType> > +struct traits< MatrixPowerParenthesesReturnValue<MatrixPowerType> > { typedef typename MatrixPowerType::PlainObject ReturnType; }; template<typename Derived> struct traits< MatrixPowerReturnValue<Derived> > { typedef typename Derived::PlainObject ReturnType; }; +template<typename Derived> +struct traits< MatrixComplexPowerReturnValue<Derived> > +{ typedef typename Derived::PlainObject ReturnType; }; + } template<typename Derived> const MatrixPowerReturnValue<Derived> MatrixBase<Derived>::pow(const RealScalar& p) const { return MatrixPowerReturnValue<Derived>(derived(), p); } +template<typename Derived> +const MatrixComplexPowerReturnValue<Derived> MatrixBase<Derived>::pow(const std::complex<RealScalar>& p) const +{ return MatrixComplexPowerReturnValue<Derived>(derived(), p); } + } // namespace Eigen #endif // EIGEN_MATRIX_POWER |