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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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unsupported/Eigen/MatrixFunctions
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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
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>,