Polynomial roots from companion matrix eigenvalues

Journal Article · Sat Apr 01 04:00:00 EST 1995 · Mathematics of Computation

DOI:https://doi.org/10.1090/S0025-5718-1995-1262279-2· OSTI ID:181773

Edelman, A; Murakami, H

In classical linear algebra, the eigenvalues of a matrix are sometimes defined as the roots of the characteristic polynomial. An algorithm to compute the roots of a polynomial by computing the eigenvalues of the corresponding companion matrix turns the tables on the usual definition. The authors derive a first-order error analysis of this algorithm that sheds light on both the underlying geometry of the problem as well as the numerical error of the algorithm. The authors` error analysis expands on work by Van Dooren and Dewilde in that it states that the algorithm is backward normwise stable in a sense that must be defined carefully. Regarding the stronger concept of a small componentwise backward error, this analysis predicts a small such error on a test suite of eight random polynomials suggested by Toh and Trefethen. However, the authors construct examples for which a small componentwise relative backward error is neither predicted nor obtained in practice. They extend their results to polynomial matrices, where the result is essentially the same, but the geometry becomes more complicated.

• Sponsoring Organization: USDOE
• OSTI ID: 181773
• Journal Information: Mathematics of Computation, Journal Name: Mathematics of Computation Journal Issue: 210 Vol. 64; ISSN 0025-5718; ISSN MCMPAF
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS
ALGEBRA
ALGORITHMS
EIGENVALUES
ERRORS
MATRICES
NUMERICAL SOLUTION
POLYNOMIALS