Geodesic
On incomplete factorization methods with generalized compensation
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 1 (1998) no. 4, pp. 321-336.

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The iterative incomplete factorization methods are described on the base of definition of preconditioning $B$ matrix from generalized compensation principle $B_{y_k}=A_{y_k}$, $k=1,\dots,m$, where $A$ is the matrix of original system of linear algebraic equations and $\{y_k\}$ is the set of so called probe vectors. The correctness of such algorithms and conditions of positive definiteness of preconditioning matrices are investigated for solution to the Stieltjes type block-tridiagonal systems. The estimates of condition number of matrix product $B^{-1}A$, that define the iterative convergence rate, are derived in the terms of the properties of original matrices. 
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V. P. Il'in; K. Yu. Laevskii. On incomplete factorization methods with generalized compensation. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 1 (1998) no. 4, pp. 321-336. http://geodesic.mathdoc.fr/item/SJVM_1998_1_4_a3/

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