Chistyakov P.
Tikhonov's regularization method for solving SLAE
Tikhonov's regularization method for solving SLAEs. The paper discusses
the Tikhonov regularization method for solving systems of linear
algebraic equations with perturbations in the system matrix and on the
right side. The system is assumed to have the most general form, with a
rectangular matrix over the field of complex numbers. In the general
case, such a system may be undecidable in the classical sense. It is
substantiated that the regularization parameter can be chosen to depend
only on the error level in the system matrix (independent of the error
level of the right-hand side). A parameter selection rule is proposed that
improves previously known estimates. In a sense, this new rule for
choosing a regularization parameter is optimal. The proof of the main
theorem is based on the mathematical apparatus of the Moore-Penrose
pseudoinverse matrix and singular matrix decomposition. The paper proves
the convergence of Tikhonov-regularized solutions to a normal
pseudo-solution of the SLAE. An estimate of the error of the regularized
solution is obtained. This estimate cannot be improved.
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