Linear-Time Copositivity Detection for Tridiagonal Matrices and Extension to Block-Tridiagonality

Authors:
Immanuel M. Bomze
Affiliations:
-
Venue:
SIAM Journal on Matrix Analysis and Applications
Year:
2000

Citing 0
Cited 3

On Copositive Programming and Standard Quadratic Optimization Problems

Journal of Global Optimization
A Variational Approach to Copositive Matrices

SIAM Review
Think co(mpletely)positive ! Matrix properties, examples and a clustered bibliography on copositive optimization

Journal of Global Optimization

Quantified Score

Hi-index	0.01

Visualization

Abstract

Determining whether a given symmetric matrix is copositive (i.e., whether it generates a quadratic form taking no negative values on the positive orthant) is an NP-hard problem. However, for diagonal matrices this amounts to the trivial check of signs of the diagonal entries. Here, a linear-time algorithm for tridiagonal matrices is presented which similarly checks only for signs of diagonal entries, but (depending on the sign of an off-diagonal entry) sometimes updates the matrix by an ordinary pivot step. Ramifications for the sign-constrained border and generalizations for the block-tridiagonal case are also specified. As a key tool, we establish a monotonicity result for the Löwner ordering of partial Schur complements for general symmetric matrices with a positive definite principal block.