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By G. A. Freiman

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The adjusting can readily be done by Newton's method and we note that both an eigenvalue and eigenfunction approximation are simultaneously obtained. 1b): where The p vectors 1V(A) e E" are assumed linearly independent as are the q vectors r v (A)  E". Thus rank Ca = p and rank Cb = q. 14a) by where e p + 1 is the (p + l)st unit vector in E p + 1 and Here we have dropped the last or qth condition at t = b; any other choice would have done at this stage, say, drop the vector rv(A) and retain all the others.

16a,b) is not isolated. This has not been treated extensively and is one of the important areas in need of study. Of course many successful calculations have been done on such problems, in particular for bifurcation phenomena (see, for example, Keller and Wolfe [32], Bauer, Keller and Reiss [4]). It should be noted that many other normalizations of the eigenfunction can be used which avoid the need for introducing zn+l(t). 14a,b). That is, a new linearly independent inhomogeneous linear constraint is simply adjoined at t = a.

When this is so we can define y (v) (f) as On the other hand if Y(v\t) exists, then it must have this representation. Now suppose BV(Q) has a unique solution. Then Y(0\t) = ^(OCo 1 exists and #d>y«»(f) = ^^Y(t)Qol = QiQo1. Thus ^ (1) y (0) (r) is nonsingular if and only if Q! is nonsingular. lb) are simply taken over as The net function uh = (u,}o is to approximate yh = {y(t/)}o, the exact solution on the net. The n x n coefficient matrices {Cjk(h)} and inhomogeneous terms (F/Ji, f)} define a very general family of difference schemes for which we develop a complete stability theory.

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