Write doc for positive power of a matrix with a semisimple zero eigenvalue.

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Chen-Pang He 2013-07-10 02:44:38 +08:00
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@ -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 Therefore, the matrix \f$ M \f$ should meet the conditions to be an
argument of matrix logarithm. 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 algorithm as implemented by class MatrixPower. The exponent is split
into integral part and fractional part, where the fractional part is into integral part and fractional part, where the fractional part is
in the interval \f$ (-1, 1) \f$. The main diagonal and the first in the interval \f$ (-1, 1) \f$. The main diagonal and the first
super-diagonal is directly computed. super-diagonal is directly computed.
If \p M is singular with a semisimple zero eigenvalue and \p p is
positive, the Schur factor \f$ 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 Details of the algorithm can be found in: Nicholas J. Higham and
Lijing Lin, "A Schur-Padé algorithm for fractional powers of a Lijing Lin, "A Schur-Padé algorithm for fractional powers of a
matrix," <em>SIAM J. %Matrix Anal. Applic.</em>, matrix," <em>SIAM J. %Matrix Anal. Applic.</em>,