Saberi Najafi, H.; Edalatpanah, S. A. A new family of \((I+S)\)-type preconditioner with some applications. (English) Zbl 1331.65058 Comput. Appl. Math. 34, No. 3, 917-931 (2015). To construct a preconditioner \(P\) for a matrix \(A=I+U+L\) (\(U\) and \(L\) are the strict upper- and lower triangular parts, respectively), one has to find an approximation for \(A^{-1}\). These are often of the form \(P=I+S\). Approximations for \(A^{-1}\) can be obtained by solving \(X^{-1}-A=0\) using Newton’s method or a higher-order method. The matrix \(I+S\) can be used as a starting value. Newton’s method gives for example \(X_{k+1}=X_k(2I-AX_n)\), thus \(P=I+K=(I+S)[(I-S)+(L+U)(I+S)]\). It is proved and illustrated by numerical examples, and under certain conditions, a speed-up is obtained with such an iterated preconditioner. Reviewer: Adhemar Bultheel (Leuven) Cited in 5 Documents MSC: 65F08 Preconditioners for iterative methods 65F10 Iterative numerical methods for linear systems Keywords:preconditioned iterative methods; \(H\)-matrices; spectral radius; accelerated overrelaxation method; convection-diffusion equation; comparison theorems; Newton’s method; numerical example PDFBibTeX XMLCite \textit{H. Saberi Najafi} and \textit{S. A. Edalatpanah}, Comput. Appl. Math. 34, No. 3, 917--931 (2015; Zbl 1331.65058) Full Text: DOI References: [1] Axelsson O (1985) A survey of preconditioned iterative methods for linear systems of algebraic equations. BIT 25:166-187 · Zbl 0566.65017 [2] Bai ZZ, Golub GH, Ng M (2003) Hermitian and skew-Hermitian splitting methods for non Hermitian positive definite linear systems. 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