Computational error bounds for multiple or nearly multiple eigenvalues

AbstractIn this paper bounds for clusters of eigenvalues of non-selfadjoint matrices are investigated. We describe a method for the computation of rigorous error bounds for multiple or nearly multiple eigenvalues, and for a basis of the corresponding invariant subspaces. The input matrix may be real or complex, dense or sparse. The method is based on a quadratically convergent Newton-like method; it includes the case of defective eigenvalues, uncertain input matrices and the generalized eigenvalue problem. Computational results show that verified bounds are still computed even if other eigenvalues or clusters are nearby the eigenvalues under consideration

VARIATIONAL CHARACTERIZATIONS OF THE SIGN-REAL AND THE SIGN-COMPLEX SPECTRAL RADIUS ∗

Key words. Generalized spectral radius, sign-real spectral radius, sign-complex spectral radius, Perron-Frobenius theory. AMS subject classifications. 15A48, 15A18 Abstract. The sign-real and the sign-complex spectral radius, also called the generalized spectral radius, proved to be an interesting generalization of the classical Perron-Frobenius theory (for nonnegative matrices) to general real and to general complex matrices, respectively. Especially the generalization of the well-known Collatz-Wielandt max-min characterization shows one of the many one-to-one correspondences to classical Perron-Frobenius theory. In this paper the corresponding inf-max characterization as well as variational characterizations of the generalized (real and complex) spectral radius are presented. Again those are almost identical to the corresponding results in classical Perron-Frobenius theory. 1. Introduction. Denote R+: = {x ≥ 0: x ∈ R}, andletK ∈{R+, R, C}. The generalized spectral radius is defined [6] by (1.1) ρ K (A):=max{|λ | : ∃ 0 = x ∈ K n, ∃ λ ∈ K, |Ax | = |λx|} for A ∈ Mn(K)

On relative errors of floating-point operations: optimal bounds and applications

International audienceRounding error analyses of numerical algorithms are most often carried out via repeated applications of the so-called standard models of floating-point arithmetic. Given a round-to-nearest function fl and barring underflow and overflow, such models bound the relative errors E 1 (t) = |t − fl(t)|/|t| and E 2 (t) = |t − fl(t)|/|fl(t)| by the unit roundoff u. This paper investigates the possibility and the usefulness of refining these bounds, both in the case of an arbitrary real t and in the case where t is the exact result of an arithmetic operation on some floating-point numbers. We show that E 1 (t) and E 2 (t) are optimally bounded by u/(1 + u) and u, respectively, when t is real or, under mild assumptions on the base and the precision, when t = x ± y or t = xy with x, y two floating-point numbers. We prove that while this remains true for division in base β > 2, smaller, attainable bounds can be derived for both division in base β = 2 and square root. This set of optimal bounds is then applied to the rounding error analysis of various numerical algorithms: in all cases, we obtain significantly shorter proofs of the best-known error bounds for such algorithms, and/or improvements on these bounds themselves

Improved error bounds for inner products in floating-point arithmetic

International audienceGiven two floating-point vectors $x,y$ of dimension $n$ and assuming rounding to nearest, we show that if no underflow or overflow occurs, any evaluation order for inner product returns a floating-point number $\hat r$ such that $|{\hat r}-x^Ty| \le nu|x|^T|y|$ with $u$ the unit roundoff. This result, which holds for any radix and with no restriction on $n$, can be seen as a generalization of a similar bound given in~\cite{Rump12} for recursive summation in radix $2$, namely $|{\hat r}- x^Te| \le (n-1)u|x|^Te$ with $e=[1,1,\ldots,1]^T$. As a direct consequence, the error bound for the floating-point approximation $\hat C$ of classical matrix multiplication with inner dimension $n$ simplifies to $|\hat{C}-AB|\le nu|A||B|$

05391 Abstracts Collection -- Algebraic and Numerical Algorithms and Computer-assisted Proofs

From 25.09.05 to 30.09.05, the Dagstuhl Seminar 05391 ``Algebraic and Numerical Algorithms and Computer-assisted Proofs\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. Links to extended abstracts or full papers are provided, if available

05391 Executive Summary -- Numerical and Algebraic Algorithms and Computer-assisted Proofs

The common goal of self-validating methods and computer algebra methods is to solve mathematical problems with complete rigor and with the aid of computers. The seminar focused on several aspects of such methods for computer-assisted proofs