Computing Characteristic Polynomials from Eigenvalues

Abstract. This paper concerns the computation of the coefficients ck of the characteristic polynomial of a real or complex matrix A. We analyze the forward error in the coefficients ck when they are computed from the eigenvalues of A, as is done by MATLAB’s poly function. In particular, we derive absolute and relative perturbation bounds for elementary symmetric functions, which we use in turn to derive perturbation bounds for the coefficients ck with regard to absolute and relative changes in the eigenvalues λj of A. We present the so-called Summation Algorithm for computing the coefficients ck from the eigenvalues λj , which is essentially the algorithm used by poly. We derive roundoff error bounds and running error bounds for the Summation Algorithm. The roundoff error bounds imply that the Summation Algorithm is forward stable. The running error bounds can be used to estimate the accuracy of the computed coefficients “on the fly,” and they tend to be less pessimistic tha...

• Type: Journal
• Year: 2011
• Where: SIAMMAX
• Authors: Rizwana Rehman, Ilse C. F. Ipsen
• Keywords: Elementary Symmetric Functions, Roundoff Error, SIAMMAX 2011, Summation Algorithm