1) A well-conditioned boundary value problem of the form (104) may easily lead to anill-posed initial value problem (104);2) A bounded solution to the initial value problem (104) may exist only for s in a smallneighbourhood of y(s) (which, of course, in unknown).The idea behind the multiple shooting method is to divide the interval [a, b] intosmaller subintervalsa = x0 < x1 < · · · < xK−1 < xK = b ; (107)the problem then is to find a vector ST = (sT0, . . . , sTK−1) such that the solutions uk(x; sk)of the initial value problemsu′k(x; sk) = f(x, uk(x; sk)) , xk < x ≤ xk+1 ,(108)uk(xk; sk) = sk , k = 0, . . . , K − 1 ,satisfy the conditionsuk(xk+1; sk) − sk+1 = 0 , k = 0, . . . , K − 2 , g(s0, uK−1(b; sK−1)) = 0 . (109)The equations (109) can be written in the compact form G(S) = 0. A clear advantageof the multiple shooting method over simple shooting is that the growth of the solutionsto the initial value problems (108) and the related linear initial value problems for the∂uk(x; sk)/∂sk, k = 0, . . . , K−1, can be approximated accurately by selecting a sufficientlyfine subdivision (107) of the interval [a, b].Indeed, it is possible to prove that, under reasonable assumptions, the multiple shootingmethod leads to an exponential increase of the size of the domain of initial valuesfor which the first iteration of the root-finding procedure is defined. This is consistentwith the practical observation that multiple shooting is less sensitive to the choice of thestarting values than simple shooting.656.2 Matrix methodsIn this section, rather than attempting to convert the bound
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