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Write doc for positive power of a matrix with a semisimple zero eigenvalue.
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@ -239,12 +239,29 @@ p . \f$ M^p \f$ simply evaluates into \f$ \exp(p \log(M)) \f$.
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Therefore, the matrix \f$ M \f$ should meet the conditions to be an
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Therefore, the matrix \f$ M \f$ should meet the conditions to be an
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argument of matrix logarithm.
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argument of matrix logarithm.
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This function computes the matrix power using the Schur-Padé
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If \p p is real, it is casted into the real scalar type of \p M. Then
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this function computes the matrix power using the Schur-Padé
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algorithm as implemented by class MatrixPower. The exponent is split
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algorithm as implemented by class MatrixPower. The exponent is split
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into integral part and fractional part, where the fractional part is
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into integral part and fractional part, where the fractional part is
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in the interval \f$ (-1, 1) \f$. The main diagonal and the first
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in the interval \f$ (-1, 1) \f$. The main diagonal and the first
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super-diagonal is directly computed.
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super-diagonal is directly computed.
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If \p M is singular with a semisimple zero eigenvalue and \p p is
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positive, the Schur factor \f$ T \f$ is reordered with Givens
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rotations, i.e.
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\f[ T = \left[ \begin{array}{cc}
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T_1 & T_2 \\
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0 & 0
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\end{array} \right] \f]
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where \f$ T_1 \f$ is invertible. Then \f$ T^p \f$ is given by
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\f[ T^p = \left[ \begin{array}{cc}
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T_1^p & T_1^{-1} T_1^p T_2 \\
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0 & 0
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\end{array}. \right] \f]
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Details of the algorithm can be found in: Nicholas J. Higham and
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Details of the algorithm can be found in: Nicholas J. Higham and
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Lijing Lin, "A Schur-Padé algorithm for fractional powers of a
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Lijing Lin, "A Schur-Padé algorithm for fractional powers of a
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matrix," <em>SIAM J. %Matrix Anal. Applic.</em>,
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matrix," <em>SIAM J. %Matrix Anal. Applic.</em>,
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