Abstract
We present a new method for the a priori approximation of the orders of magnitude of the entries in the LU factors of a complex or real matrix $A$. This approximation is used to determine the positions of the largest entries in the LU factors of $A$, and these positions are used as the sparsity pattern for an incomplete LU factorization preconditioner. Our method uses max-plus algebra and is based solely on the moduli of the entries of $A$. We also present techniques for predicting which permutation matrices will be chosen by Gaussian elimination with partial pivoting. We exploit the strong connection between the field of Puiseux series and the max-plus semiring to prove properties of the max-plus LU factors. Experiments with a set of test matrices from the University of Florida Sparse Matrix Collection show that our max-plus LU preconditioners outperform traditional level of fill methods and have similar performance to those preconditioners computed with more expensive threshold-based methods.
Highlights
Max-plus algebra is the analogue of linear algebra developed for the binary operations max and plus over the real numbers together with −∞, the latter playing the role of additive identity
Our aim is to show how max-plus algebra can be used to approximate the sizes of the entries in the LU factors of a complex or real matrix A and how these approximations can subsequently be used in the construction of an incomplete LU (ILU) factorization preconditioner for A
We have presented a new method for approximating the order of magnitude of the entries in the LU factors of a matrix A ∈ Cn×n
Summary
Max-plus algebra is the analogue of linear algebra developed for the binary operations max and plus over the real numbers together with −∞, the latter playing the role of additive identity. We use the max-plus LU factors L and U to define the sparsity pattern of the ILU preconditioners by allowing only entries with a value over a certain threshold. This approximation forms the basis of our LU factor approximation. Our definition of the max-plus LU factors of a max-plus matrix was chosen so that that we could use it to approximate the orders of magnitude of the entries in the LU factors of a complex matrix. What do the max-plus LU factors of a max-plus matrix tell us about
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