On Equivalence and Reducibility of Generating Matrices of RK-Procedures

Este artigo é uma revisão corrigida e melhorada do Relatório Técnico RT-MAP-7701 do Dep. de Matemática Aplicada do Instituto de Matemática e Estatística da Universidade de São Paulo.


In Stetter [1] the following theorem for Runge-Kutta procedures is stated:
"If two m-stage generating matrices are equivalent and irreducible, one is a permutation of the other".
In this paper we extend the theorem for the case of two generating matrices with different number of stages. Two kinds of reducibility are introduced, the definition used in [1] being a particular case of the reducibility of the 1st kind. Then it is proved the more general formulation:
"Let A and B be generating matrices, not equivalent to zero, with mA and mB stages respectively. If A and B are equivalent and irreducible (of both kinds) then mA=mB and one is a permutation of the other". The next theorem shows that the given algebraic conditions are the good ones.
"A generating matrix A, not equivalent to zero, is irreducible (both of the 1st and 2nd kind) if and only if, there does not exist an equivalent generating matrix B with strictly less number of stages".

Veja a integra em Generating Matrices of RK-Procedures