The strong stability of algorithms for solving symmetric linear systems. (English) Zbl 0687.65021

An algorithm for solving linear equations of the form \(Ax=b\) is “strongly stable” on a class C of matrices if for every \(A\in C\) and for arbitrary vector b the computed solution \(\hat x\) solves a system \(\hat A\hat x=\hat b\) with \(\hat A\in C,\) which is near the original one (in the sense that \(\| \hat A-A\| /\| A\|\) and \(\| \hat b- b\| /\| b\|\) are “small”). If we do not require the condition \(\hat A\in C,\) then we simply say that the algorithm is “stable”. The concept of strong stability is important for instance in the error analysis in view of a new perturbation theorem about symmetric perturbations of symmetric matrices.
In previous works the strong stability of certain algorithms for suitable classes of matrices was established. Here a general theorem is proved on this subject. It states that if a method is stable for the class of nonsingular symmetric matrices or the class of symmetric positive definite matrices, then it is strongly stable for the same class. As new application of this result it is shown that Gaussian elimination with pivoting and J. O. Aasen’s method [BIT 11, 233-242 (1971; Zbl 0242.65032)] are strongly stable when applied to symmetric systems. Other applications in the class of positive definite systems are briefly discussed.
Reviewer: F.Flandoli


65F05 Direct numerical methods for linear systems and matrix inversion
65F30 Other matrix algorithms (MSC2010)
65G50 Roundoff error


Zbl 0242.65032
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