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**Additional resources for Algorithms and Theory in Filtering and Control: Proceedings of the Workshop on Numerical Techniques for Systems Engineering Problems, Part L**

**Sample text**

That is, the coefficients form a nontrivial solution of the h o m o g e n e o u s linear system (L~,o) ~, c,_jvj = f, u,, 0 <_ i <- m, ~=0 [0, m < i < - m + n , of m + n + 1 equations in m + n + 2 unknowns. (We put ck ~-0 for k < 0 . j=l (m > O, n -- 0). +2). ----0 if and only if Lm,~ has a nontrivial solution with Uo = v0 = 0. To see this, write the ' v - p a r t ' of Lr,,~ as CVo+ C~ = 0, with cm,, = det C, and note that Uo = CoVo, v = 0 ~ u = O. The power series f will be called regular if all the Hankel determinants, c~): = det(ci+j)7,iJo (n --- 0), do not vanish, and positive definite if {c~}~ is real and all c ~ ) > 0 .

This is the algorithm obtained from the nonsymmetric factorization mentioned in [13, p. 320]; also cf. [8, 13]. 2). 1). 3) again. 10) one step backward in time we get the following recursion for M. 1. Let A be nonsingular. 15) and M satisfies the recursion M ( t + 1) = M ( t ) + M ( t ) X ( t + 1 ) ' n ' R ( t ) - l H f ( ( t + 1)M(t), (4. 10) M(0) = X(1)'A(Po)X(1). Proof. 10) with t exchanged for t - 1 . 5). 10), since these relations hold for all Xo E R". 5). 2). A. 11) instead of M. 2. Let A be nonsingular.

C. in both variables x, v and bounded below. , and clearly positive. A s s u m e that u " -< if, let us check that it implies u n+l < ~. Indeed u"+~(x) <- f(x, v) + a O ~ ( x ) <- f(x, v) + aq~Vwo(x) = w(x, v) and since v is arbitrary, it follows that u"+~(x) <- ~v(x). Let us check that the sequence is m o n o t o n e decreasing. This is proved by induction. /n+l ~ un. It remains to c h e c k that u ~ <- u ~ which is true since u ~ if, by the a b o v e argument. N o w one has u" ~ t~, t~ is Borel.