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Summary: Stability and sensitivity of tridiagonal LU factorization
without pivoting
M. ISABEL BUENO and FROILŽAN M. DOPICO
Department of Mathematics, Universidad Carlos III de Madrid,
Avda. de la Universidad, 30. 28911 LeganŽes, Spain. emails: mbueno@math.uc3m.es,
dopico@math.uc3m.es
Abstract.
In this paper the accuracy of LU factorization of tridiagonal matrices without piv-
oting is considered. Two types of componentwise condition numbers for the L and U
factors of tridiadonal matrices are presented and compared. One type is a condition
number with respect to small relative perturbations of each entry of the matrix. The
other type is a condition number with respect to small componentwise perturbations
of the kind appearing in the backward error analysis of the usual algorithm for the LU
factorization. We show that both condition numbers are of similar magnitude. This
means that the algorithm is componentwise forward stable, i.e., the forward errors are
of similar magnitude to those produced by a componentwise backward stable method.
Moreover the presented condition numbers can be computed in O(n) flops, which allows
to estimate with low cost the forward errors.
AMS subject classification: 65F35, 65F50, 15A12, 15A23, 65G50.
Key words: tridiagonal matrices, LU factorization, condition numbers, error analy-
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