Computable backward error bounds for basic algorithms in linear algebra

抄録

Standard error estimates in numerical linear algebra are often of the form γ_k|R||S| where R,S are known matrices and γ_k:=ku/(1-u) with u denoting the relative rounding error unit. Recently we showed that for a number of standard problems γ_k can be replaced by ku for any order of computation and without restriction on the dimension. Such problems include LU- and Cholesky decomposition, triangular system solving by substitution, matrix multiplication and more. The theoretical bound implies a practically computable bound by estimating the error in the floating-point computation of ku|R||S|. Standard techniques, however, imply again a restriction on the dimension. In this note we derive simple computable bounds being valid without restriction on the dimension. As the bounds are mathematically rigorous, they may serve in computer assisted proofs.