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On the numerical solution of ill‐conditioned linear systems by regularization and iteration
Author(s) -
Spigler Renato
Publication year - 2021
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.2335
Subject(s) - tikhonov regularization , mathematics , linear system , regularization (linguistics) , diagonal matrix , diagonal , positive definite matrix , condition number , iterative method , matrix (chemical analysis) , system of linear equations , numerical analysis , spectrum (functional analysis) , mathematical analysis , mathematical optimization , eigenvalues and eigenvectors , inverse problem , computer science , geometry , physics , materials science , quantum mechanics , artificial intelligence , composite material
Summary We propose to reduce the (spectral) condition number of a given linear system by adding a suitable diagonal matrix to the system matrix, in particular by shifting its spectrum. Iterative procedures are then adopted to recover the solution of the original system. The case of real symmetric positive definite matrices is considered in particular, and several numerical examples are given. This approach has some close relations with Riley's method and with Tikhonov regularization. Moreover, we identify approximately the aforementioned procedure with a true action of preconditioning.