Abstract
We describe a recurrence method for computing primary $p$th roots of a matrix $A$ with a cost, in terms of elementary arithmetic operations and memory, which is logarithmic with respect to $p$. When $A$ is real and the primary root is real as well, the algorithm is based on the real Schur form of $A$ and uses real arithmetic. The numerical experiments confirm the good behavior of the new algorithm in finite arithmetic. The case of arbitrary fractional powers of $A$ is also considered.
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