Abstract
Systems of linear algebraic equations (SLAEs) are considered in this work. If the matrix of a system is nonsingular, a unique solution of the system exists. In the singular case, the system can have no solution or infinitely many solutions. In this case, the notion of a normal solution is introduced. The case of a nonsingular square matrix can be theoretically regarded as good in the sense of solution existence and uniqueness. However, in the theory of computational methods, nonsingular matrices are divided into two categories: ill-conditioned and well-conditioned matrices. A matrix is ill-conditioned if the solution of the system of equations is practically unstable. An important characteristic of the practical solution stability for a system of linear equations is the condition number. Regularization methods are usually applied to obtain a reliable solution. A common strategy is to use Tikhonov’s stabilizer or its modifications or to represent the required solution as the orthogonal sum of two vectors of which one vector is determined in a stable fashion, while seeking the second one requires a stabilization procedure. Methods for numerically solving SLAEs with positive definite symmetric matrices or oscillation-type matrices using regularization are considered in this work, which lead to SLAEs with reduced condition numbers.
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REFERENCES
A. N. Tikhonov and V. Ya. Arsenin, Solutions of Ill-Posed Problems (Nedra, Moscow, 1979; Wiley, Winston, NY, 1977).
O. A. Liskovets, Variational Methods for Solving Unstable Problems (Nauka i Tehnika, Minsk, 1981) [in Russian].
V. K. Ivanov, V. V. Vasin, and V. P. Tanana, Theory of Linear Ill-Posed Problems and Its Applications (Nauka, Moscow, 1978) [in Russian].
S. I. Kabanikhin, Inverse and Ill-Posed Problems (Sib. Nauchn. Izd., Novosibirsk, 2009) [in Russian].
V. V. Voevodin and Yu. A. Kuznetsov, Matrices and Computations (Nauka, Moscow, 1984) [in Russian].
S. G. Mikhlin, Variational Methods in Mathematical Physics (Nauka, Moscow, 1970; Pergamon, Oxford, 1982).
S. G. Mikhlin, The Numerical Performance of Variational Methods (Nauka, Moscow, 1966; Wolters-Noordhoff, Groningen, 1971).
F. R. Gantmakher and M. G. Krein, Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems (GITTL, Moscow, 1950; AMS Chelsea, Providence, RI, 2002).
F. R. Gantmakher, The Theory of Matrices (Nauka, Moscow, 1967; Chelsea, New York, 1989).
N. J. Higham, Functions of Matrices: Theory and Computation (Society for Industrial and Applied Mathematics, Philadelphia, 2008).
W. Gautschi, “The condition of a matrix arising in the numerical inversion of the Laplace transform,” Math. Comput. 23 (105), 109–118 (1969).
I. G. Burova, V. M. Ryabov, M. A. Kalnitskaya, A. V. Malevich, and Yu. K. Demjanovich, “Program for solving a system of linear algebraic equations with a positively defined matrix by the regularization method,” RF Patent No. 2018661356 (2018).
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Translated by N. Berestova
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Lebedeva, A.V., Ryabov, V.M. Numerical Solution of Systems of Linear Algebraic Equations with Ill-Conditioned Matrices. Vestnik St.Petersb. Univ.Math. 52, 388–393 (2019). https://doi.org/10.1134/S1063454119040058
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DOI: https://doi.org/10.1134/S1063454119040058