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Diffstat (limited to 'unsupported/Eigen/MatrixFunctions')
-rw-r--r-- | unsupported/Eigen/MatrixFunctions | 213 |
1 files changed, 209 insertions, 4 deletions
diff --git a/unsupported/Eigen/MatrixFunctions b/unsupported/Eigen/MatrixFunctions index 292357fda..e0bc4732c 100644 --- a/unsupported/Eigen/MatrixFunctions +++ b/unsupported/Eigen/MatrixFunctions @@ -40,6 +40,22 @@ namespace Eigen { * \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_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 + * + * 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 @@ -49,16 +65,205 @@ namespace Eigen { * 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] * - * \code - * #include <unsupported/Eigen/MatrixFunctions> - * \endcode */ #include "src/MatrixFunctions/MatrixExponential.h" #include "src/MatrixFunctions/MatrixFunction.h" -} +/** +\page matrixbaseextra MatrixBase methods defined in the MatrixFunctions module +\ingroup MatrixFunctions_Module + +The remainder of the page documents the following MatrixBase methods +which are defined in the MatrixFunctions module. + + + +\section 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$. + +This function calls \ref matrixbase_matrixfunction "matrixFunction()" with StdStemFunctions::cos(). + +\sa \ref matrixbase_sin "sin()" for an example. + + + +\section matrixbase_cosh MatrixBase::cosh() + +Compute the matrix hyberbolic cosine. + +\code +const MatrixFunctionReturnValue<Derived> MatrixBase<Derived>::cosh() const +\endcode + +\param[in] M a square matrix. +\returns expression representing \f$ \cosh(M) \f$ + +This function calls \ref matrixbase_matrixfunction "matrixFunction()" with StdStemFunctions::cosh(). + +\sa \ref matrixbase_sinh "sinh()" for an example. + + + +\section 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$. + +The cost of the computation is approximately \f$ 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, +\c complex<float> or \c complex<double> . + + + +\section matrixbase_matrixfunction MatrixBase::matrixFunction() + +Compute a matrix function. + +\code +const MatrixFunctionReturnValue<Derived> MatrixBase<Derived>::matrixFunction(typename ei_stem_function<typename ei_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 + + + +\section 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$. + +This function calls \ref matrixbase_matrixfunction "matrixFunction()" with StdStemFunctions::sin(). + +Example: \include MatrixSine.cpp +Output: \verbinclude MatrixSine.out + + + +\section matrixbase_sinh const MatrixBase::sinh() + +Compute the matrix hyperbolic sine. + +\code +MatrixFunctionReturnValue<Derived> MatrixBase<Derived>::sinh() const +\endcode + +\param[in] M a square matrix. +\returns expression representing \f$ \sinh(M) \f$ + +This function calls \ref matrixbase_matrixfunction "matrixFunction()" with StdStemFunctions::sinh(). + +Example: \include MatrixSinh.cpp +Output: \verbinclude MatrixSinh.out + +*/ + +} + #endif // EIGEN_MATRIX_FUNCTIONS |