# Handout20 - COMPUTER SCIENCE 349A Handout Number 20...

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1 COMPUTER SCIENCE 349A Handout Number 20 STABILITY AND CONDITION : SYSTEMS OF LINEAR EQUATIONS STABILITY OF ALGORITHMS FOR SOLVING b Ax = Given a nonsingular matrix A , a vector b and some algorithm for computing the solution of b Ax = , let x ˆ denote the computed solution using this algorithm. The computation is said to be stable if there exist small perturbations E and e of A and b , respectively, such that x ˆ is close to the exact solution y of the perturbed linear system e b y E A + = + ) (. That is, the computed solution x ˆ is very close to the exact solution of some small perturbation of the given problem. KNOWN RESULTS Gaussian elimination without pivoting may be unstable. For example, if there exists a pivot which is very small relative to other entries in the coefficient matrix, then the computed solution may be very inaccurate and the computation may be unstable. In practice, Gaussian elimination with partial pivoting is almost always stable.

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## This note was uploaded on 02/08/2010 for the course CSC 349A taught by Professor Olesky during the Spring '08 term at University of Victoria.

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Handout20 - COMPUTER SCIENCE 349A Handout Number 20...

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