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T h e A p p r o x i m a t e S o l u t i o n of Matrix Problems* A. S. HOUSEHOLDER Oak Ridge National Laboratory, Oak Ridge, Tennessee 1. Statement of the Problem; Notational Conventions These notes have to do ing approximate solutions t e m of linear equations, an appraisal of the error the'system with methods of obtaining and methods of appraisof matrix problems. If the problem is to solve a sysor to invert a matrix, one might suppose t h a t can be made directly b y substitution. B u t consider A x = h, (1.1) and let h = h(A) be a proper value of A, and u a proper vector belonging to A u = ku. Then A ( x A- u) = h A- hu. Hence if X is small, any component of error along u can be completely obscured in the rounding process, so t h a t an " a p p r o x i m a t e " solution x* =" ~c A- u, even if crude, m a y satisfy the system exactly to within machine errors. The
Journal of the ACM (JACM) – Association for Computing Machinery
Published: Jul 1, 1958
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