Abstract
Cardinal's matrix version of the Sebastiao e Silva polynomial root-finder rapidly approximates the roots as the eigenvalues of the associated Frobenius matrix. We preserve rapid convergence to the roots but amend the algorithm to allow input polynomials with multiple roots and root clusters. As in Cardinal's algorithm, we repeatedly square the Frobenius matrix in nearly linear arithmetic time per squaring, which yields dramatic speedup versus the recent effective polynomial root-finder based on the application of the inverse power method to the Frobenius matrix.
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