Numerical Computation of the Characteristic Polynomial of a Complex Matrix

Abstract

REHMAN, RIZWANA. Numerical Computation of the Characteristic Polynomial of a Complex Matrix. (Under the direction of Ilse C.F. Ipsen.) In this dissertation we present algorithms, and sensitivity and stability analyses for the numerical computation of characteristic polynomials of complex matrices. In Quantum Physics, for instance, characteristic polynomials are required to calculate thermodynamic properties of systems of fermions. The general consensus seems to be that numerical methods for computing characteristic polynomials are numerically inaccurate and unstable. However, in order to judge the numerical accuracy of a method, one needs to investigate the sensitivity of the coeffcients of the characteristic polynomial to perturbations in the matrix. We derive forward error bounds for the coeffcients of the characteristic polynomial of an n x n complex matrix. These bounds consist of elementary symmetric functions of singular values. Furthermore, we investigate the numerical stability of two methods for the computation of characteristic polynomials. The frst method determines the coeffcients of the characteristic polynomial of a matrix from its eigenvalues. The second method requires a preliminary reduction of a complex matrix A to its Hessenberg form H. The characteristic polynomial of H is obtained from successive computations of characteristic polynomials of leading principal submatrices of H. Our numerical experiments suggest that the second method is more accurate than the determination of the characteristic polynomial from eigenvalues.