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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Vydáno v:Proceedings (International Symposium on Symbolic and Numeric Algorithms for Scientific Computing) s. 65 - 71
Hlavní autor: Mroz, Andrzej
Médium: Konferenční příspěvek
Jazyk:angličtina
Vydáno: IEEE 01.09.2016
Témata:
Algebra
bigraph
Complexity theory
nondeterministic algorithm
positive definite matrix
Scientific computing
signed graph
Solids
Zinc
ISSN:2470-881X
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Shrnutí: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