Condition numbers for inversion of Fiedler companion matrices

Eigenvalue condition numbers and Pseudospectra of Fiedler matrices

The aim of the present paper is to analyze the behavior of Fiedler companion matrices in the polynomial root-finding problem from the point of view of conditioning of eigenvalues. More precisely, we compare: (a) the condition number of a given root λ of a monic polynomial p(z) with the condition number of λ as an eigenvalue of any Fiedler matrix of p(z), (b) the condition number of λ as an eige...

Backward stability of polynomial root-finding using Fiedler companion matrices

Computing roots of scalar polynomials as the eigenvalues of Frobenius companion matrices using backward stable eigenvalue algorithms is a classical approach. The introduction of new families of companion matrices allows for the use of other matrices in the root-finding problem. In this paper, we analyze the backward stability of polynomial root-finding algorithms via Fiedler companion matrices....

Condition Numbers of Matrices

denote its Euclidean operator norm (often called the 2-norm). If  is nonsingular, then its condition number () is defined by () = kk°°−1°° = 1() () where 1 ≥ 1 ≥    ≥  ≥ 0 are the singular values of . The s constitute lengths of the semi-axes of the hyperellipsoid  = { : kk = 1} in -dimensional space; thus  measures elongation of  at its extreme [1]. The role that  ...

Condition numbers of random matrices

A bound for condition numbers of matrices