ABSTRACT
Carrying out convergence analyses in the preceding chapters, we invariably faced the need to find the necessary and sufficient condition for convergence of the sequence generated by a majorant generator (a difference equation) of the form
u+ := F (u, v) , v+ := G(u, v) , u ∈ R , v ∈ Rm , (7.1) resulting from the use of Kantorovich’s majorization technique (Propositions 2.11, 3.14, 4.5, 5.5). These conditions have been found as an inequality of the type u0 ≤ f∞(v0), with the function f∞ being the limit of the sequence fn defined recursively. At the same time, it has been shown that this limit solves the system of a functional equation and an end condition. The use of the definition of f
∞ for its actual computing proves to be impractical because of
slow convergence of the sequence fn, especially when u0 is close to its upper limit. This observation warrants a closer look at the system
x ( G ( x(v) , v
)) = F
( x(v) , v
) & x(0) = u
induced by the generator (7.1) and solved by the function f ∞ . Except for some
simplified generators as in (2.40), (3.36), (5.7), (5.21) , the induced systems of the type (7.2) can be solved only numerically using some iterative procedure. In the present chapter, we try to develop such a procedure. But first we have to get a constructive description of the convergence domain of the generator (7.1). In other words, we have to answer the question: precisely which starters (u0 , v0) cause the sequence (un, vn), generated by the generator from (u0 , v0), to converge.
