Error Bounds for Dynamic Responses in Forced Vibration (Cabos) ====================================================================== SIAM Journal on Scientific Computing Volume 15-1, January 1994, pp. 1-15 (C) 1994 by Society for Industrial and Applied Mathematics All rights reserved Title: Error Bounds for Dynamic Responses in Forced Vibration Problems Author: Christian Cabos AMS Subject Classifications: 65F30, 65L70, 73K12, 70J35, 65F50, 65F15 Key words: mode superposition, error bounds, Lanczos method, matrix functions, forced vibrations ---- ABSTRACT When using mode superposition in large applications, generally only relatively few approximate eigenmodes are linearly combined. Block Lanczos iteration is an efficient method of determining such modes. In this paper new a~posteriori bounds are developed that estimate the error when approximating the exact result of mode superposition with a linear combination of the output vectors of block Lanczos iteration. Mode superposition can be regarded as a way of computing $g(S)f$, a function $g$ of a selfadjoint matrix $S$ applied to a vector. One formula is developed that estimates the norm of the unknown error vector. A second inequality gives a bound for the error when computing linear functionals $\lsk v,g(S)f\rsk$ of the response. The error bounds require that $f$ and possibly $v$ are contained in the Lanczos starting block and that all Ritz vectors are used to compute the result. No gaps in the spectrum of $S$ need to be known. The bounds can be evaluated at a small cost compared to the eigenpair extraction in large systems. In a forced response calculation for a container ship with $\approx$ 38,000 degrees of freedom the error is overestimated by two to four orders of magnitude. ====================================================================== SIAM 3600 University City Science Center Philadelphia, PA 19104-2688, USA Phone: 215-382-9800, 800-447-7426 (USA only) Fax: 215-386-7999 E-mail: journals@siam.org