TY - JOUR
T1 - On systems of linear equations with nonnegative coefficients. Log-convexity of the Perron root and the l1-norm of the positive solution with applications
AU - Boche, Holger
AU - Stańczak, Sławomir
PY - 2004/3
Y1 - 2004/3
N2 - 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.
AB - 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.
KW - Log-convexity
KW - Parametric solutions
KW - Perron-Frobenius theory
KW - Spectral radius (Perron root)
UR - http://www.scopus.com/inward/record.url?scp=1842482413&partnerID=8YFLogxK
U2 - 10.1007/s00200-003-0142-4
DO - 10.1007/s00200-003-0142-4
M3 - Article
AN - SCOPUS:1842482413
SN - 0938-1279
VL - 14
SP - 397
EP - 414
JO - Applicable Algebra in Engineering, Communications and Computing
JF - Applicable Algebra in Engineering, Communications and Computing
IS - 6
ER -