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

Bunch, James R.; Demmel, James W.; Van Loan, Charles F.

SIAM J. Matrix Anal. Appl. 10, No. 4, 494-499 (1989).

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

MSC:

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

Keywords:

strong stability; error analysis; symmetric perturbations; symmetric matrices; Gaussian elimination with pivoting; symmetric systems; positive definite systems

Citations:

Zbl 0242.65032