Matrix analysis
Accuracy and Stability of Numerical Algorithms
Accuracy and Stability of Numerical Algorithms
Perturbation bounds for polynomials
Numerische Mathematik
Perturbation Bounds for Determinants and Characteristic Polynomials
SIAM Journal on Matrix Analysis and Applications
Numerical computation of the characteristic polynomial of a complex matrix
Numerical computation of the characteristic polynomial of a complex matrix
Accurate computing elementary symmetric functions
ACM Communications in Computer Algebra
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This paper concerns the computation of the coefficients $c_k$ of the characteristic polynomial of a real or complex matrix $A$. We analyze the forward error in the coefficients $c_k$ 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 $c_k$ with regard to absolute and relative changes in the eigenvalues $\lambda_j$ of $A$. We present the so-called Summation Algorithm for computing the coefficients $c_k$ from the eigenvalues $\lambda_j$, 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 than the roundoff error bounds. Numerical experiments illustrate that our bounds give useful estimates for the accuracy of the coefficients $c_k$. In particular, the bounds confirm that poly computes the coefficients $c_k$ to high relative accuracy if the eigenvalues are positive and given to high relative accuracy.