diff options
Diffstat (limited to 'unsupported/Eigen/MatrixFunctions')
| -rw-r--r-- | unsupported/Eigen/MatrixFunctions | 19 |
1 files changed, 18 insertions, 1 deletions
diff --git a/unsupported/Eigen/MatrixFunctions b/unsupported/Eigen/MatrixFunctions index 0bdd379d7..41dfab390 100644 --- a/unsupported/Eigen/MatrixFunctions +++ b/unsupported/Eigen/MatrixFunctions @@ -239,12 +239,29 @@ 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. -This function computes the matrix power using the Schur-Padé +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] + 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>, |