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Unformatted text preview: xample 7.1.1. Find the stability constant. Show that this problem is stable and has exponential dichotomy.
8 y Yj (x) x
a=x 0 x1 x j-1 xj xN Figure 7.2.1: Geometry and nomenclature for multiple shooting. 7.2 Multiple Shooting
As seen in Section 7.1, errors may accumulate as the solution is generated from the
initial to the terminal point. Since round-o errors increase in proportion to the number
of operations, they can be reduced by integrating over shorter distances. With this
procedure, called parallel or multiple shooting, we create IVPs that start at several points
on a b]. Let us illustrate the procedure with a linear m-dimensional BVP of the form y = Ay + b a<x<b 0 (7.2.1a) Ly(a) = l (7.2.1b) Ry(b) = r (7.2.1c) with L of dimension l m and R of dimension r m. Let us further divide a b] into
N subintervals as shown in Figure 7.2.1. Multiple shooting procedures that are based on
the strategies (7.1.2 - 7.1.3) and (7.1.7 - 7.1.10) have been developed. The rst approach
is the si...
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- Spring '14