TY - JOUR

T1 - Numerical Solution of Systems of Linear Algebraic Equations with Ill-Conditioned Matrices

AU - Lebedeva, A. V.

AU - Ryabov, V. M.

N1 - Lebedeva, A.V. & Ryabov, V.M. Vestnik St.Petersb. Univ.Math. (2019) 52: 388. https://proxy.library.spbu.ru:2060/10.1134/S1063454119040058

PY - 2019

Y1 - 2019

N2 - 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.

AB - 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.

KW - condition number

KW - ill-conditioned problems

KW - ill-posed problems

KW - regularization method

KW - system of linear algebraic equations

UR - http://www.scopus.com/inward/record.url?scp=85077021768&partnerID=8YFLogxK

UR - https://link.springer.com/article/10.1134/S1063454119040058

U2 - 10.1134/S1063454119040058

DO - 10.1134/S1063454119040058

M3 - Article

AN - SCOPUS:85077021768

VL - 52

SP - 388

EP - 393

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 4

ER -