FSAI Preconditioning for Singular M-Matrices
Research Paper#Numerical Linear Algebra, Preconditioning, M-matrices🔬 Research|Analyzed: Jan 4, 2026 00:10•
Published: Dec 25, 2025 17:29
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•ArXivAnalysis
This paper investigates the application of the Factorized Sparse Approximate Inverse (FSAI) preconditioner to singular irreducible M-matrices, which are common in Markov chain modeling and graph Laplacian problems. The authors identify restrictions on the nonzero pattern necessary for stable FSAI construction and demonstrate that the resulting preconditioner preserves key properties of the original system, such as non-negativity and the M-matrix structure. This is significant because it provides a method for efficiently solving linear systems arising from these types of matrices, which are often large and sparse, by improving the convergence rate of iterative solvers.
Key Takeaways
- •Applies FSAI preconditioning to singular irreducible M-matrices.
- •Identifies restrictions on the nonzero pattern for stable FSAI construction.
- •Proves the preconditioner preserves key properties like non-negativity and M-matrix structure.
- •Provides a method for efficiently solving linear systems arising from Markov chain modeling and graph Laplacian problems.
Reference / Citation
View Original"The lower triangular matrix $L_G$ and the upper triangular matrix $U_G$, generated by FSAI, are non-singular and non-negative. The diagonal entries of $L_GAU_G$ are positive and $L_GAU_G$, the preconditioned matrix, is a singular M-matrix."