# Bounding the error in Gaussian elimination for tridiagonal systems

Higham, Nicholas J. (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

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.