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On solving of ill-conditioned linear systems by iterative methods. (Russian. English summary) Zbl 1058.65043

There are many problems of approximation theory and economy (Leontiew input-output models) etc. which lead to the ill-conditioned linear algebraic equations. The matrices of such problems are so ill-conditioned that attempts to solve these systems by direct methods like the Gauss method or the square-root method lead to a quick loss of accuracy even for a sufficiently small order \((\approx 10)\) of the initial matrix. But in practice the matrices can be of order 100–1000 or even more. Here numerical investigations of several known variants of the conjugate gradient method are provided with the Hilbert matrix arising in approximation theory as an example.

MSC:

65F10 Iterative numerical methods for linear systems
65F35 Numerical computation of matrix norms, conditioning, scaling
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