Error Bounds for Dynamic Responses in Forced Vibration (Cabos)
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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
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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.
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