Ela a Factorization of the Inverse of the Shifted Companion Matrix
نویسنده
چکیده
A common method for computing the zeros of an n-th degree polynomial is to compute the eigenvalues of its companion matrix C. A method for factoring the shifted companion matrix C − ρI is presented. The factorization is a product of 2n − 1 essentially 2 × 2 matrices and requires O(n) storage. Once the factorization is computed it immediately yields factorizations of both (C − ρI) and (C − ρI). The cost of multiplying a vector by (C − ρI) is shown to be O(n), and therefore, any shift and invert eigenvalue method can be implemented efficiently. This factorization is generalized to arbitrary forms of the companion matrix.
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