The Strong Stability of Algorithms for Solving Symmetric Linear Systems

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Abstract
An algorithm for solving linear equations is stable on the class of nonsingular symmetric matrices or on the class of symmetric positive definite matrices if the computed solution solves a system that is near the original problem. Here it is shown that any stable algorithm is also strongly stable on the same matrix class if the computed solution solves a nearby problem that is also symmetric or symmetric positive definite. An algorithm for solving linear equations is stable on the class of nonsingular symmetric matrices or on the class of symmetric positive definite matrices if the computed solution solves a system that is near the original problem. Here it is shown that any stable algorithm is also strongly stable on the same matrix class if the computed solution solves a nearby problem that is also symmetric or symmetric positive definite.

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