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Least squares residuals and minimal residual methods. (English) Zbl 1012.65037

Minimal residual methods for solving linear systems can be formulated and implemented using different orthogonalization processes. Using general theoretical results about the least squares residual, this paper shows that the choice of the basis is fundamental for getting a numerically stable implementation. It is explained that using the best orthogonalization technique in building the basis does not compensate for the possible loss of accuracy in a given method which is related to the choice of the basis.

MSC:

65F20 Numerical solutions to overdetermined systems, pseudoinverses
65F25 Orthogonalization in numerical linear algebra
65F10 Iterative numerical methods for linear systems

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