On square roots of M-matrices

We consider the problem of solving large sparse linear systems where the coefficient matrix is possibly singular but the equations are consistent. Block two-stage methods in which the inner iterations are performed using alternating methods are studied. These methods are ideal for parallel processing and provide a very general setting to study parallel block methods including overlapping. Convergence properties of these methods are established when the matrix in question is either M-matrix or symmetric matrix. Different parallel versions of these methods and implementation strategies, with and without overlapping blocks, are explored. The reported experiments show the behavior and effectiveness of the designed parallel algorithms by exploiting the benefits of shared memory inside the nodes of current SMP supercomputers.

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“…, satisfy condition (b) of Theorem 2. Therefore, using the result in [18,Theorem 2], the proof is complete. (7), are semiconvergent.…”

Section: Convergencementioning

confidence: 89%

Parallel alternating iterative algorithms with and without overlapping on multicore architectures

Migallón

Penadés

2016

Advances in Engineering Software

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“…Since Q f is a symmetric positive semidefinite M matrix, G is a positive definite M matrix and G s = √ G is also a positive definite M matrix [25], and all of its entries are non-zero [23]. Using Lemma 5.5 of [23], G + G + 4 α √ G is an M matrix with negative off-diagonal entries.…”

Section: Discussionmentioning

confidence: 99%

Design Of Real-Time Implementable Distributed Suboptimal Control: An LQR Perspective

Jaleel

Shamma

2018

IEEE Trans. Control Netw. Syst.

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“…We wish to show that lim k−→∞ T k θ exists, i.e., it is convergent 1 . This limit if it exists is a projection onto the null space of the singular matrix A.…”

Section: Convergence Of Additive Schwarz For Markov Chainsmentioning

confidence: 99%

“…can be replaced with T having positive diagonal entries [1]. Equivalent conditions for (ii) can be found in [22].…”

Section: Convergence Of Additive Schwarz For Markov Chainsmentioning

confidence: 99%

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Additive Schwarz Iterations for Markov Chains

Bru¹,

Pedroche²,

Szyld³

2005

SIAM J. Matrix Anal. & Appl.

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Abstract.A convergence analysis is presented for additive Schwarz iterations when applied to consistent singular systems of equations of the form Ax = b. The theory applies to singular M -matrices with one-dimensional null space and is applicable in particular to systems representing ergodic Markov chains, and to certain discretizations of partial differential equations. Additive Schwarz can be seen as a generalization of block Jacobi, where the set of indices defining the diagonal blocks have nonempty intersection; this is called the overlap. The presence of overlap is known to accelerate the convergence of the methods in the nonsingular case. By providing convergence results, as well as some characteristics of the induced splitting, we hope to encourage the use of this additional computational tool for the solution of Markov chains and other singular systems. We present several numerical examples showing that additive Schwarz performs better than block Jacobi. For completeness, a few numerical experiments with block Gauss-Seidel and multiplicative Schwarz are also included.

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On square roots of M-matrices

Cited by 70 publications

References 1 publication

Parallel alternating iterative algorithms with and without overlapping on multicore architectures

Parallel alternating iterative algorithms with and without overlapping on multicore architectures

Design Of Real-Time Implementable Distributed Suboptimal Control: An LQR Perspective

Additive Schwarz Iterations for Markov Chains

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