diff options
Diffstat (limited to '')
| -rw-r--r-- | src/EigenUnsupported/MatrixFunctions | 504 |
1 files changed, 0 insertions, 504 deletions
diff --git a/src/EigenUnsupported/MatrixFunctions b/src/EigenUnsupported/MatrixFunctions deleted file mode 100644 index 20c23d1..0000000 --- a/src/EigenUnsupported/MatrixFunctions +++ /dev/null @@ -1,504 +0,0 @@ -// This file is part of Eigen, a lightweight C++ template library -// for linear algebra. -// -// Copyright (C) 2009 Jitse Niesen <jitse@maths.leeds.ac.uk> -// 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_FUNCTIONS -#define EIGEN_MATRIX_FUNCTIONS - -#include <cfloat> -#include <list> - -#include "../../Eigen/Core" -#include "../../Eigen/LU" -#include "../../Eigen/Eigenvalues" - -/** - * \defgroup MatrixFunctions_Module Matrix functions module - * \brief This module aims to provide various methods for the computation of - * matrix functions. - * - * To use this module, add - * \code - * #include <unsupported/Eigen/MatrixFunctions> - * \endcode - * at the start of your source file. - * - * This module defines the following MatrixBase methods. - * - \ref matrixbase_cos "MatrixBase::cos()", for computing the matrix cosine - * - \ref matrixbase_cosh "MatrixBase::cosh()", for computing the matrix hyperbolic cosine - * - \ref matrixbase_exp "MatrixBase::exp()", for computing the matrix exponential - * - \ref matrixbase_log "MatrixBase::log()", for computing the matrix logarithm - * - \ref matrixbase_pow "MatrixBase::pow()", for computing the matrix power - * - \ref matrixbase_matrixfunction "MatrixBase::matrixFunction()", for computing general matrix functions - * - \ref matrixbase_sin "MatrixBase::sin()", for computing the matrix sine - * - \ref matrixbase_sinh "MatrixBase::sinh()", for computing the matrix hyperbolic sine - * - \ref matrixbase_sqrt "MatrixBase::sqrt()", for computing the matrix square root - * - * These methods are the main entry points to this module. - * - * %Matrix functions are defined as follows. Suppose that \f$ f \f$ - * is an entire function (that is, a function on the complex plane - * that is everywhere complex differentiable). Then its Taylor - * series - * \f[ f(0) + f'(0) x + \frac{f''(0)}{2} x^2 + \frac{f'''(0)}{3!} x^3 + \cdots \f] - * converges to \f$ f(x) \f$. In this case, we can define the matrix - * function by the same series: - * \f[ f(M) = f(0) + f'(0) M + \frac{f''(0)}{2} M^2 + \frac{f'''(0)}{3!} M^3 + \cdots \f] - * - */ - -#include "../../Eigen/src/Core/util/DisableStupidWarnings.h" - -#include "src/MatrixFunctions/MatrixExponential.h" -#include "src/MatrixFunctions/MatrixFunction.h" -#include "src/MatrixFunctions/MatrixSquareRoot.h" -#include "src/MatrixFunctions/MatrixLogarithm.h" -#include "src/MatrixFunctions/MatrixPower.h" - -#include "../../Eigen/src/Core/util/ReenableStupidWarnings.h" - - -/** -\page matrixbaseextra_page -\ingroup MatrixFunctions_Module - -\section matrixbaseextra MatrixBase methods defined in the MatrixFunctions module - -The remainder of the page documents the following MatrixBase methods -which are defined in the MatrixFunctions module. - - - -\subsection matrixbase_cos MatrixBase::cos() - -Compute the matrix cosine. - -\code -const MatrixFunctionReturnValue<Derived> MatrixBase<Derived>::cos() const -\endcode - -\param[in] M a square matrix. -\returns expression representing \f$ \cos(M) \f$. - -This function computes the matrix cosine. Use ArrayBase::cos() for computing the entry-wise cosine. - -The implementation calls \ref matrixbase_matrixfunction "matrixFunction()" with StdStemFunctions::cos(). - -\sa \ref matrixbase_sin "sin()" for an example. - - - -\subsection matrixbase_cosh MatrixBase::cosh() - -Compute the matrix hyberbolic cosine. - -\code -const MatrixFunctionReturnValue<Derived> MatrixBase<Derived>::cosh() const -\endcode - -\param[in] M a square matrix. -\returns expression representing \f$ \cosh(M) \f$ - -This function calls \ref matrixbase_matrixfunction "matrixFunction()" with StdStemFunctions::cosh(). - -\sa \ref matrixbase_sinh "sinh()" for an example. - - - -\subsection matrixbase_exp MatrixBase::exp() - -Compute the matrix exponential. - -\code -const MatrixExponentialReturnValue<Derived> MatrixBase<Derived>::exp() const -\endcode - -\param[in] M matrix whose exponential is to be computed. -\returns expression representing the matrix exponential of \p M. - -The matrix exponential of \f$ M \f$ is defined by -\f[ \exp(M) = \sum_{k=0}^\infty \frac{M^k}{k!}. \f] -The matrix exponential can be used to solve linear ordinary -differential equations: the solution of \f$ y' = My \f$ with the -initial condition \f$ y(0) = y_0 \f$ is given by -\f$ y(t) = \exp(M) y_0 \f$. - -The matrix exponential is different from applying the exp function to all the entries in the matrix. -Use ArrayBase::exp() if you want to do the latter. - -The cost of the computation is approximately \f$ 20 n^3 \f$ for -matrices of size \f$ n \f$. The number 20 depends weakly on the -norm of the matrix. - -The matrix exponential is computed using the scaling-and-squaring -method combined with Padé approximation. The matrix is first -rescaled, then the exponential of the reduced matrix is computed -approximant, and then the rescaling is undone by repeated -squaring. The degree of the Padé approximant is chosen such -that the approximation error is less than the round-off -error. However, errors may accumulate during the squaring phase. - -Details of the algorithm can be found in: Nicholas J. Higham, "The -scaling and squaring method for the matrix exponential revisited," -<em>SIAM J. %Matrix Anal. Applic.</em>, <b>26</b>:1179–1193, -2005. - -Example: The following program checks that -\f[ \exp \left[ \begin{array}{ccc} - 0 & \frac14\pi & 0 \\ - -\frac14\pi & 0 & 0 \\ - 0 & 0 & 0 - \end{array} \right] = \left[ \begin{array}{ccc} - \frac12\sqrt2 & -\frac12\sqrt2 & 0 \\ - \frac12\sqrt2 & \frac12\sqrt2 & 0 \\ - 0 & 0 & 1 - \end{array} \right]. \f] -This corresponds to a rotation of \f$ \frac14\pi \f$ radians around -the z-axis. - -\include MatrixExponential.cpp -Output: \verbinclude MatrixExponential.out - -\note \p M has to be a matrix of \c float, \c double, `long double` -\c complex<float>, \c complex<double>, or `complex<long double>` . - - -\subsection matrixbase_log MatrixBase::log() - -Compute the matrix logarithm. - -\code -const MatrixLogarithmReturnValue<Derived> MatrixBase<Derived>::log() const -\endcode - -\param[in] M invertible matrix whose logarithm is to be computed. -\returns expression representing the matrix logarithm root of \p M. - -The matrix logarithm of \f$ M \f$ is a matrix \f$ X \f$ such that -\f$ \exp(X) = M \f$ where exp denotes the matrix exponential. As for -the scalar logarithm, the equation \f$ \exp(X) = M \f$ may have -multiple solutions; this function returns a matrix whose eigenvalues -have imaginary part in the interval \f$ (-\pi,\pi] \f$. - -The matrix logarithm is different from applying the log function to all the entries in the matrix. -Use ArrayBase::log() if you want to do the latter. - -In the real case, the matrix \f$ M \f$ should be invertible and -it should have no eigenvalues which are real and negative (pairs of -complex conjugate eigenvalues are allowed). In the complex case, it -only needs to be invertible. - -This function computes the matrix logarithm using the Schur-Parlett -algorithm as implemented by MatrixBase::matrixFunction(). The -logarithm of an atomic block is computed by MatrixLogarithmAtomic, -which uses direct computation for 1-by-1 and 2-by-2 blocks and an -inverse scaling-and-squaring algorithm for bigger blocks, with the -square roots computed by MatrixBase::sqrt(). - -Details of the algorithm can be found in Section 11.6.2 of: -Nicholas J. Higham, -<em>Functions of Matrices: Theory and Computation</em>, -SIAM 2008. ISBN 978-0-898716-46-7. - -Example: The following program checks that -\f[ \log \left[ \begin{array}{ccc} - \frac12\sqrt2 & -\frac12\sqrt2 & 0 \\ - \frac12\sqrt2 & \frac12\sqrt2 & 0 \\ - 0 & 0 & 1 - \end{array} \right] = \left[ \begin{array}{ccc} - 0 & \frac14\pi & 0 \\ - -\frac14\pi & 0 & 0 \\ - 0 & 0 & 0 - \end{array} \right]. \f] -This corresponds to a rotation of \f$ \frac14\pi \f$ radians around -the z-axis. This is the inverse of the example used in the -documentation of \ref matrixbase_exp "exp()". - -\include MatrixLogarithm.cpp -Output: \verbinclude MatrixLogarithm.out - -\note \p M has to be a matrix of \c float, \c double, `long -double`, \c complex<float>, \c complex<double>, or `complex<long double>`. - -\sa MatrixBase::exp(), MatrixBase::matrixFunction(), - class MatrixLogarithmAtomic, MatrixBase::sqrt(). - - -\subsection matrixbase_pow MatrixBase::pow() - -Compute the matrix raised to arbitrary real power. - -\code -const MatrixPowerReturnValue<Derived> MatrixBase<Derived>::pow(RealScalar p) const -\endcode - -\param[in] M base of the matrix power, should be a square matrix. -\param[in] p exponent of the matrix power. - -The matrix power \f$ M^p \f$ is defined as \f$ \exp(p \log(M)) \f$, -where exp denotes the matrix exponential, and log denotes the matrix -logarithm. This is different from raising all the entries in the matrix -to the p-th power. Use ArrayBase::pow() if you want to do the latter. - -If \p p is complex, the scalar type of \p M should be the type of \p -p . \f$ M^p \f$ simply evaluates into \f$ \exp(p \log(M)) \f$. -Therefore, the matrix \f$ M \f$ should meet the conditions to be an -argument of matrix logarithm. - -If \p p is real, it is casted into the real scalar type of \p M. Then -this function computes the matrix power using the Schur-Padé -algorithm as implemented by class MatrixPower. The exponent is split -into integral part and fractional part, where the fractional part is -in the interval \f$ (-1, 1) \f$. The main diagonal and the first -super-diagonal is directly computed. - -If \p M is singular with a semisimple zero eigenvalue and \p p is -positive, the Schur factor \f$ T \f$ is reordered with Givens -rotations, i.e. - -\f[ T = \left[ \begin{array}{cc} - T_1 & T_2 \\ - 0 & 0 - \end{array} \right] \f] - -where \f$ T_1 \f$ is invertible. Then \f$ T^p \f$ is given by - -\f[ T^p = \left[ \begin{array}{cc} - T_1^p & T_1^{-1} T_1^p T_2 \\ - 0 & 0 - \end{array}. \right] \f] - -\warning Fractional power of a matrix with a non-semisimple zero -eigenvalue is not well-defined. We introduce an assertion failure -against inaccurate result, e.g. \code -#include <unsupported/Eigen/MatrixFunctions> -#include <iostream> - -int main() -{ - Eigen::Matrix4d A; - A << 0, 0, 2, 3, - 0, 0, 4, 5, - 0, 0, 6, 7, - 0, 0, 8, 9; - std::cout << A.pow(0.37) << std::endl; - - // The 1 makes eigenvalue 0 non-semisimple. - A.coeffRef(0, 1) = 1; - - // This fails if EIGEN_NO_DEBUG is undefined. - std::cout << A.pow(0.37) << std::endl; - - return 0; -} -\endcode - -Details of the algorithm can be found in: Nicholas J. Higham and -Lijing Lin, "A Schur-Padé algorithm for fractional powers of a -matrix," <em>SIAM J. %Matrix Anal. Applic.</em>, -<b>32(3)</b>:1056–1078, 2011. - -Example: The following program checks that -\f[ \left[ \begin{array}{ccc} - \cos1 & -\sin1 & 0 \\ - \sin1 & \cos1 & 0 \\ - 0 & 0 & 1 - \end{array} \right]^{\frac14\pi} = \left[ \begin{array}{ccc} - \frac12\sqrt2 & -\frac12\sqrt2 & 0 \\ - \frac12\sqrt2 & \frac12\sqrt2 & 0 \\ - 0 & 0 & 1 - \end{array} \right]. \f] -This corresponds to \f$ \frac14\pi \f$ rotations of 1 radian around -the z-axis. - -\include MatrixPower.cpp -Output: \verbinclude MatrixPower.out - -MatrixBase::pow() is user-friendly. However, there are some -circumstances under which you should use class MatrixPower directly. -MatrixPower can save the result of Schur decomposition, so it's -better for computing various powers for the same matrix. - -Example: -\include MatrixPower_optimal.cpp -Output: \verbinclude MatrixPower_optimal.out - -\note \p M has to be a matrix of \c float, \c double, `long -double`, \c complex<float>, \c complex<double>, or -\c complex<long double> . - -\sa MatrixBase::exp(), MatrixBase::log(), class MatrixPower. - - -\subsection matrixbase_matrixfunction MatrixBase::matrixFunction() - -Compute a matrix function. - -\code -const MatrixFunctionReturnValue<Derived> MatrixBase<Derived>::matrixFunction(typename internal::stem_function<typename internal::traits<Derived>::Scalar>::type f) const -\endcode - -\param[in] M argument of matrix function, should be a square matrix. -\param[in] f an entire function; \c f(x,n) should compute the n-th -derivative of f at x. -\returns expression representing \p f applied to \p M. - -Suppose that \p M is a matrix whose entries have type \c Scalar. -Then, the second argument, \p f, should be a function with prototype -\code -ComplexScalar f(ComplexScalar, int) -\endcode -where \c ComplexScalar = \c std::complex<Scalar> if \c Scalar is -real (e.g., \c float or \c double) and \c ComplexScalar = -\c Scalar if \c Scalar is complex. The return value of \c f(x,n) -should be \f$ f^{(n)}(x) \f$, the n-th derivative of f at x. - -This routine uses the algorithm described in: -Philip Davies and Nicholas J. Higham, -"A Schur-Parlett algorithm for computing matrix functions", -<em>SIAM J. %Matrix Anal. Applic.</em>, <b>25</b>:464–485, 2003. - -The actual work is done by the MatrixFunction class. - -Example: The following program checks that -\f[ \exp \left[ \begin{array}{ccc} - 0 & \frac14\pi & 0 \\ - -\frac14\pi & 0 & 0 \\ - 0 & 0 & 0 - \end{array} \right] = \left[ \begin{array}{ccc} - \frac12\sqrt2 & -\frac12\sqrt2 & 0 \\ - \frac12\sqrt2 & \frac12\sqrt2 & 0 \\ - 0 & 0 & 1 - \end{array} \right]. \f] -This corresponds to a rotation of \f$ \frac14\pi \f$ radians around -the z-axis. This is the same example as used in the documentation -of \ref matrixbase_exp "exp()". - -\include MatrixFunction.cpp -Output: \verbinclude MatrixFunction.out - -Note that the function \c expfn is defined for complex numbers -\c x, even though the matrix \c A is over the reals. Instead of -\c expfn, we could also have used StdStemFunctions::exp: -\code -A.matrixFunction(StdStemFunctions<std::complex<double> >::exp, &B); -\endcode - - - -\subsection matrixbase_sin MatrixBase::sin() - -Compute the matrix sine. - -\code -const MatrixFunctionReturnValue<Derived> MatrixBase<Derived>::sin() const -\endcode - -\param[in] M a square matrix. -\returns expression representing \f$ \sin(M) \f$. - -This function computes the matrix sine. Use ArrayBase::sin() for computing the entry-wise sine. - -The implementation calls \ref matrixbase_matrixfunction "matrixFunction()" with StdStemFunctions::sin(). - -Example: \include MatrixSine.cpp -Output: \verbinclude MatrixSine.out - - - -\subsection matrixbase_sinh MatrixBase::sinh() - -Compute the matrix hyperbolic sine. - -\code -MatrixFunctionReturnValue<Derived> MatrixBase<Derived>::sinh() const -\endcode - -\param[in] M a square matrix. -\returns expression representing \f$ \sinh(M) \f$ - -This function calls \ref matrixbase_matrixfunction "matrixFunction()" with StdStemFunctions::sinh(). - -Example: \include MatrixSinh.cpp -Output: \verbinclude MatrixSinh.out - - -\subsection matrixbase_sqrt MatrixBase::sqrt() - -Compute the matrix square root. - -\code -const MatrixSquareRootReturnValue<Derived> MatrixBase<Derived>::sqrt() const -\endcode - -\param[in] M invertible matrix whose square root is to be computed. -\returns expression representing the matrix square root of \p M. - -The matrix square root of \f$ M \f$ is the matrix \f$ M^{1/2} \f$ -whose square is the original matrix; so if \f$ S = M^{1/2} \f$ then -\f$ S^2 = M \f$. This is different from taking the square root of all -the entries in the matrix; use ArrayBase::sqrt() if you want to do the -latter. - -In the <b>real case</b>, the matrix \f$ M \f$ should be invertible and -it should have no eigenvalues which are real and negative (pairs of -complex conjugate eigenvalues are allowed). In that case, the matrix -has a square root which is also real, and this is the square root -computed by this function. - -The matrix square root is computed by first reducing the matrix to -quasi-triangular form with the real Schur decomposition. The square -root of the quasi-triangular matrix can then be computed directly. The -cost is approximately \f$ 25 n^3 \f$ real flops for the real Schur -decomposition and \f$ 3\frac13 n^3 \f$ real flops for the remainder -(though the computation time in practice is likely more than this -indicates). - -Details of the algorithm can be found in: Nicholas J. Highan, -"Computing real square roots of a real matrix", <em>Linear Algebra -Appl.</em>, 88/89:405–430, 1987. - -If the matrix is <b>positive-definite symmetric</b>, then the square -root is also positive-definite symmetric. In this case, it is best to -use SelfAdjointEigenSolver::operatorSqrt() to compute it. - -In the <b>complex case</b>, the matrix \f$ M \f$ should be invertible; -this is a restriction of the algorithm. The square root computed by -this algorithm is the one whose eigenvalues have an argument in the -interval \f$ (-\frac12\pi, \frac12\pi] \f$. This is the usual branch -cut. - -The computation is the same as in the real case, except that the -complex Schur decomposition is used to reduce the matrix to a -triangular matrix. The theoretical cost is the same. Details are in: -Åke Björck and Sven Hammarling, "A Schur method for the -square root of a matrix", <em>Linear Algebra Appl.</em>, -52/53:127–140, 1983. - -Example: The following program checks that the square root of -\f[ \left[ \begin{array}{cc} - \cos(\frac13\pi) & -\sin(\frac13\pi) \\ - \sin(\frac13\pi) & \cos(\frac13\pi) - \end{array} \right], \f] -corresponding to a rotation over 60 degrees, is a rotation over 30 degrees: -\f[ \left[ \begin{array}{cc} - \cos(\frac16\pi) & -\sin(\frac16\pi) \\ - \sin(\frac16\pi) & \cos(\frac16\pi) - \end{array} \right]. \f] - -\include MatrixSquareRoot.cpp -Output: \verbinclude MatrixSquareRoot.out - -\sa class RealSchur, class ComplexSchur, class MatrixSquareRoot, - SelfAdjointEigenSolver::operatorSqrt(). - -*/ - -#endif // EIGEN_MATRIX_FUNCTIONS - |