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Jacobi-Davidson methods for polynomial two-parameter eigenvalue problems
Hochstenbach, Michiel E. ; Muhič, Andrej, matematik ; Plestenjak, Bor
Za polinomske dvoparametrične probleme lastnih vrednosti (PMEP) lahko uporabimo metode Jacobi-Davidsonovega tipa. Tovrstne probleme lahko lineariziramo v singularne dvoparametrične probleme lastnih ...vrednosti z matrikami velikosti ▫$k(k+1)n/2$▫, kjer je ▫$k$▫ stopnja polinoma in ▫$n$▫ velikost matričnih koeficientov v PMEP. Če je ▫$k^2n$▫ relativno majhen, lahko problem rešimo numerično tako, da izračunamo skupni regularni del pridruženega para singularnih matričnih šopov. Za velik ▫$k^2n$▫ izračun vseh lastnih vrednosti ni možen in moramo uporabiti iterativne metode. Če je ▫$k$▫ velik, predlagamo, da se probem najprej linearizira in nato uporabi Jacobi-Davidsonovo metodo na dobljenem singularnem dvoparametričnem problemu lastnih vrednosti. Dobljeno metodo lahko npr. uporabimo za računanje ničel sistema dveh skalarnih polinomov dveh spremenljivk v bližini dane tarče. Na drugi strani, če je ▫$k$▫ majhen, lahko Jacobi-Davidsonovo metodo uporabimo direktno na originalnih matrikah. Originalne matrike projiciramo na podprostore majhne dimenzije in projicirane polinomske dvoparametrične probleme lastnih vrednosti rešimo z linearizacijo.
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