## Block triangular preconditioners for M-matrices and Markov chains

Michele Benzi and Bora Uçar

### Abstract

We consider preconditioned Krylov subspace methods for solving large sparse linear systems under the assumption that the coefficient matrix is a (possibly singular) $M$-matrix. The matrices are partitioned into $2\times 2$ block form using graph partitioning. Approximations to the Schur complement are used to produce various preconditioners of block triangular and block diagonal type. A few properties of the preconditioners are established, and extensive numerical experiments are used to illustrate the performance of the various preconditioners on singular linear systems arising from Markov modeling.

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### Key words

$M$-matrices, preconditioning, discrete Markov chains, iterative methods, graph partitioning

### AMS subject classifications

05C50, 60J10, 60J22, 65F10, 65F35, 65F50

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