On systems of linear equations with nonnegative coefficients. Log-convexity of the Perron root and the l1-norm of the positive solution with applications

Holger Boche, Sławomir Stańczak

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

We consider a system of linear equations with positive coefficients, where the entries of the nonnegative irreducible coefficient matrix depend on a parameter vector. We say that the parameter vector is feasible if there exists a positive solution to this system. A set of all feasible parameter vectors is called the feasibility set. If all the positive entries are log-convex functions, the paper shows that the associated Perron root is log-convex on the parameter set and the l1-norm of the solution is log-convex on the feasibility set. These results imply that the feasibility set is a convex set regardless whether the l1-norm of the solution is bounded by some positive and real number or not. Finally, we show important applications of these results to wireless communication networks and prove some other interesting results for this special case.

Original languageEnglish
Pages (from-to)397-414
Number of pages18
JournalApplicable Algebra in Engineering, Communications and Computing
Volume14
Issue number6
DOIs
StatePublished - Mar 2004
Externally publishedYes

Keywords

  • Log-convexity
  • Parametric solutions
  • Perron-Frobenius theory
  • Spectral radius (Perron root)

Fingerprint

Dive into the research topics of 'On systems of linear equations with nonnegative coefficients. Log-convexity of the Perron root and the l1-norm of the positive solution with applications'. Together they form a unique fingerprint.

Cite this