Effective Nondeterministic Positive Definiteness Test for Unidiagonal Integral Matrices

For standard algorithms verifying positive definiteness of a matrix A ∈ Mn(R) based on Sylvester's criterion, the computationally pessimistic case is this when A is positive definite. We present an algorithm realizing the same task for A ∈ M n (Z), for which the case when A is positive definite...

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Bibliographic Details
Published in2016 18th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC) pp. 65 - 71
Main Author Mroz, Andrzej
Format Conference Proceeding
LanguageEnglish
Published IEEE 01.09.2016
Subjects
Algebra
bigraph
Complexity theory
nondeterministic algorithm
positive definite matrix
Scientific computing
signed graph
Solids
Zinc
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Summary:For standard algorithms verifying positive definiteness of a matrix A ∈ Mn(R) based on Sylvester's criterion, the computationally pessimistic case is this when A is positive definite. We present an algorithm realizing the same task for A ∈ M n (Z), for which the case when A is positive definite is the optimistic one. The algorithm relies on performing certain edge transformations, called inflations, on the signed graph (bigraph) Δ = Δ(A) associated with A. We provide few variants of the algorithm, including Las Vegas type randomized ones (with precisely described maximal number of steps). The algorithms work very well in practice, in many cases with a better speed than the standard tests. On the other hand, our results provide an interesting example of an application of symbolic computing methods originally developed for different purposes, with a big potential for further generalizations in matrix problems.
ISSN:2470-881X
DOI:10.1109/SYNASC.2016.023