Various conditions are developed that guarantee existence of analytic roots of a given analytic matrix function with invertible values defined on a simply connected domain.

Two main approaches are used, nowadays, to compute the roots of a zero-dimensional polynomial system. The first one involves Gröbner basis computation, and applies to any zero-dimensional system. But, it is performed with exact arithmetic and, usually, large numbers appear during the computation. The other approach is based on resultant formulations and can be performed with floating point arit...

Various conditions are developed that guarantee existence of analytic roots of a given analytic matrix function with invertible values defined on a simply connected domain.

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In classical linear algebra, the eigenvalues of a matrix are sometimes defined as the roots of the characteristic polynomial. An algorithm to compute the roots of a polynomial by computing the eigenvalues of the corresponding companion matrix turns the tables on the usual definition. We derive a firstorder error analysis of this algorithm that sheds light on both the underlying geometry of the ...

Matrix methods are increasingly popular for polynomial root-finding. The idea is to approximate the roots as the eigenvalues of the companion or generalized companion matrix associated with an input polynomial. The algorithms also solve secular equation. QR algorithm is the most customary method for eigen-solving, but we explore the inverse Rayleigh quotient iteration instead, which turns out t...