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2006.172: Bounding the error in Gaussian elimination for tridiagonal systems

2006.172: Nicholas J. Higham (1990) Bounding the error in Gaussian elimination for tridiagonal systems. SIAM Journal On Matrix Analysis And Applications, 11 (4). pp. 521-530. ISSN 1095-7162

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If $\hat x$ is the computed solution to a tridiagonal system $Ax = b$ obtained by Gaussian elimination, what is the “best” bound available for the error $x - \hat x$ and how can it be computed efficiently? This question is answered using backward error analysis, perturbation theory, and properties of the $LU$ factorization of $A$. For three practically important classes of tridiagonal matrix, those that are symmetric positive definite, totally nonnegative, or $M$-matrices, it is shown that $(A + E)\hat x = b$ where the backward error matrix $E$ is small componentwise relative to $A$. For these classes of matrices the appropriate forward error bound involves Skeel’s condition number cond $(A,x)$, which, it is shown, can be computed exactly in $O(n)$ operations. For diagonally dominant tridiagonal $A$ the same type of backward error result holds, and the author obtains a useful upper bound for cond $(A,x)$ that can be computed in $O(n)$ operations. Error bounds and their computation for general tridiagonal matrices are discussed also.

Item Type:Article
Uncontrolled Keywords:tridiagonal matrix, forward error analysis, backward error analysis, condition number, comparison matrix, M -matrix, totally nonnegative, positive definite,, diagonally dominant, LAPACK
Subjects:MSC 2000 > 15 Linear and multilinear algebra; matrix theory
MSC 2000 > 65 Numerical analysis
MIMS number:2006.172
Deposited By:Miss Louise Stait
Deposited On:03 July 2006

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