Pattern-multiplicative average of nonnegative matrices revisited:  Eigenvalue approximation is the best of versatile optimization tools

Logofet, D.O.

Автор: Logofet D.O.
Журнал: Mathematics
Том: 2023
Номер: 11
Год издания: 2023
Издательство: MDPI
Местоположение издательства: Basel, Switzerland
Первая страница: 1
Последняя страница: 10
Номер статьи: 3237
DOI: 10.3390/math11143237
Аннотация: Given several nonnegative matrices with a single pattern of allocation among their zero/nonzero elements, the average matrix should have the same pattern, too. This is the first tenet of the pattern-multiplicative average (PMA)concept, while the second one suggests the multiplicative(or geometric) nature of averaging. The original concept of PMA was motivated by the practice ofmatrix population models as a tool to assess the population viability from long-term monitoringdata. The task has reduced to searching for an approximate solution to an overdetermined systemof polynomial equations for unknown elements of the average matrix (G), and hence to a nonlinearconstrained minimization problem for the matrix norm. Former practical solutions faced certaintechnical problems, which required sophisticated algorithms but returned acceptable estimates.Now, we formulate (for the first time in ecological modeling and nonnegative matrix theory) the PMA problem as an eigenvalue approximation one and reduce it to a standard problem of linear programing (LP). The basic equation of averaging also determines the exact value of λ1(G), the dominant eigenvalue of matrix G, and the corresponding eigenvector. These are bound by the well-known linear equations, which enable an LP formulation of the former nonlinear problem. The LP approach is realized for 13 fixed-pattern matrices gained in a case study of Androsace albana, an alpineshort-lived perennial, monitored on permanent plots over 14 years. A standard software routinereveals the unique exact solution, rather than an approximate one, to the PMA problem, whichturns the LP approach into ‘’the best of versatile optimization tools”. The exact solution turns out to be peculiar in reaching zero bounds for certain nonnegative entries of G, which deserves modified problem formulation separating the lower bounds from zero.
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Pattern-multiplicative average of nonnegative matrices revisited: Eigenvalue approximation is the best of versatile optimization toolsстатья

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