Structured backward errors in linearizations

A standard approach to calculate the roots of a univariate polynomial is compute eigenvalues an associated confederate matrix instead, such as, for instance, companion or comrade matrix. The can be computed by Francis's QR algorithm. Unfortunately, even though algorithm provably backward stable, mapping errors back original coefficients still lead huge errors. However, latter statement assumes use non-structure-exploiting In [J. L. Aurentz et al., Fast and stable computation polynomials, SIAM J. Matrix Anal. Appl., 36 (2015), pp. 942–973] it was shown that structure-exploiting matrices leads structured error in proof relied on decomposing into two parts: part related recurrence basis (a monomial case) linked polynomial. this article we prove analysis extended other classes matrices. We first provide alternative stability using algorithms; our new point view shows more explicitly how structured, decoupled gets mapped coefficients. This insight reveals which properties have preserved end up with will show previously formulated fits framework, analyze detail Jacobi polynomials (comrade matrices) Chebyshev (colleague matrices).