Stability of Gram-Schmidt orthogonalization and the way of its increase

Abstract

The orthogonal methods used when solving the system of the linear equations, are more stable. However the experience of use of the programs realizing these methods, has shown, that Gram-Schmidt and Lanczos orthogonalization methods can demonstrate the results of unacceptable accuracy. While transformational methods (methods of rotations and reflections) give reliable calculations of high accuracy when solving the same problems. For the specified reasons in methods of simplification of the form of matrixes (including in QR-decomposition) users began to prefer transformations of Hausholder reflection and Jacobi rotation (Hivens). In this article under the example of QR-decomposition the nature of instability of Gram-Schmidt orthogonalization is revealed. This decomposition is chosen to simplify the statement of essence of the phenomenon what does not break the commonality of research. To diminish the influence of the revealed lack it is offered to use procedure of bidimentional orthonormalizational basis construction. The executed tests have shown the efficiency of application of the specified procedure.