Lines Matching full:matrix
24 * \defgroup MatrixFunctions_Module Matrix functions module
26 * matrix functions.
35 * - \ref matrixbase_cos "MatrixBase::cos()", for computing the matrix cosine
36 * - \ref matrixbase_cosh "MatrixBase::cosh()", for computing the matrix hyperbolic cosine
37 * - \ref matrixbase_exp "MatrixBase::exp()", for computing the matrix exponential
38 * - \ref matrixbase_log "MatrixBase::log()", for computing the matrix logarithm
39 * - \ref matrixbase_pow "MatrixBase::pow()", for computing the matrix power
40 …* - \ref matrixbase_matrixfunction "MatrixBase::matrixFunction()", for computing general matrix f…
41 * - \ref matrixbase_sin "MatrixBase::sin()", for computing the matrix sine
42 * - \ref matrixbase_sinh "MatrixBase::sinh()", for computing the matrix hyperbolic sine
43 * - \ref matrixbase_sqrt "MatrixBase::sqrt()", for computing the matrix square root
47 * %Matrix functions are defined as follows. Suppose that \f$ f \f$
52 * converges to \f$ f(x) \f$. In this case, we can define the matrix
78 Compute the matrix cosine.
84 \param[in] M a square matrix.
95 Compute the matrix hyberbolic cosine.
101 \param[in] M a square matrix.
112 Compute the matrix exponential.
118 \param[in] M matrix whose exponential is to be computed.
119 \returns expression representing the matrix exponential of \p M.
121 The matrix exponential of \f$ M \f$ is defined by
123 The matrix exponential can be used to solve linear ordinary
130 norm of the matrix.
132 The matrix exponential is computed using the scaling-and-squaring
133 method combined with Padé approximation. The matrix is first
134 rescaled, then the exponential of the reduced matrix is computed
141 scaling and squaring method for the matrix exponential revisited,"
142 <em>SIAM J. %Matrix Anal. Applic.</em>, <b>26</b>:1179–1193,
161 \note \p M has to be a matrix of \c float, \c double, \c long double
167 Compute the matrix logarithm.
173 \param[in] M invertible matrix whose logarithm is to be computed.
174 \returns expression representing the matrix logarithm root of \p M.
176 The matrix logarithm of \f$ M \f$ is a matrix \f$ X \f$ such that
177 \f$ \exp(X) = M \f$ where exp denotes the matrix exponential. As for
179 multiple solutions; this function returns a matrix whose eigenvalues
182 In the real case, the matrix \f$ M \f$ should be invertible and
187 This function computes the matrix logarithm using the Schur-Parlett
216 \note \p M has to be a matrix of \c float, \c double, <tt>long
226 Compute the matrix raised to arbitrary real power.
232 \param[in] M base of the matrix power, should be a square matrix.
233 \param[in] p exponent of the matrix power, should be real.
235 The matrix power \f$ M^p \f$ is defined as \f$ \exp(p \log(M)) \f$,
236 where exp denotes the matrix exponential, and log denotes the matrix
239 The matrix \f$ M \f$ should meet the conditions to be an argument of
240 matrix logarithm. If \p p is not of the real scalar type of \p M, it
243 This function computes the matrix power using the Schur-Padé
251 matrix," <em>SIAM J. %Matrix Anal. Applic.</em>,
273 better for computing various powers for the same matrix.
279 \note \p M has to be a matrix of \c float, \c double, <tt>long
288 Compute a matrix function.
294 \param[in] M argument of matrix function, should be a square matrix.
299 Suppose that \p M is a matrix whose entries have type \c Scalar.
311 "A Schur-Parlett algorithm for computing matrix functions",
312 <em>SIAM J. %Matrix Anal. Applic.</em>, <b>25</b>:464–485, 2003.
334 \c x, even though the matrix \c A is over the reals. Instead of
344 Compute the matrix sine.
350 \param[in] M a square matrix.
362 Compute the matrix hyperbolic sine.
368 \param[in] M a square matrix.
379 Compute the matrix square root.
385 \param[in] M invertible matrix whose square root is to be computed.
386 \returns expression representing the matrix square root of \p M.
388 The matrix square root of \f$ M \f$ is the matrix \f$ M^{1/2} \f$
389 whose square is the original matrix; so if \f$ S = M^{1/2} \f$ then
392 In the <b>real case</b>, the matrix \f$ M \f$ should be invertible and
394 complex conjugate eigenvalues are allowed). In that case, the matrix
398 The matrix square root is computed by first reducing the matrix to
400 root of the quasi-triangular matrix can then be computed directly. The
407 "Computing real square roots of a real matrix", <em>Linear Algebra
410 If the matrix is <b>positive-definite symmetric</b>, then the square
414 In the <b>complex case</b>, the matrix \f$ M \f$ should be invertible;
421 complex Schur decomposition is used to reduce the matrix to a
422 triangular matrix. The theoretical cost is the same. Details are in:
424 square root of a matrix", <em>Linear Algebra Appl.</em>,