The Computable Differential Equation Lecture ... - Bruce E. Shapiro

160 CHAPTER 8. BOUNDARY VALUE PROBLEMS
where Y is the fundamental matrix. This makes the Newton iteration formul
c i+1 = c i − Q −1 H(c i ) (8.55)
Since inverting a matrix is expensive, it is usually more reasonable to solve the linear
system
Q∆ i+1 = H(c i ) (8.56)
for ∆ and then perform the update
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Algorithm 8.1. Shooting
To solve the boundary value problem
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c i+1 = c i + ∆ i+1 (8.57)
y ′ = f(t, y)
g(y(a), y(b)) = 0
1. Input: f, f y , g(u, v), g u , g v , c (0) , and a convergence tolerance;
2. Repeat the following until the desired tolerance in c is reached
(a) Solve the initial value problem y ′ = f(t, y) with the initial
value at a with y(c i ) obtaining a mesh of solution values
y (i)
0 , . . . , y(i) N .
(b) Form the vector H = g(c (i) , y (i)
N )
(c) Integrate the fundamental matrix Y n on the same mesh
using A j = f y .
(d) Calculate Q and solve the linear system Q∆ = H(c (i) ).
(e) Update c (i) = c (i−1) + ∆.
3. Solve the initial value problem for y(0) = c, where c is the final
iteration in the preceding step.
Shooting has a number of problems, among them
• Roundoff error;
• Propagation error – for linear systems ≈ e L(b−a) where L = max ‖A‖;
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• It assumes that the solution to the IVP exists globally, which might not be
true.
and consequently this “simple” shooting algorithm is rarely used.
Math 582B, Spring 2007
California State University Northridge
c○2007, B.E.Shapiro
Last revised: May 23, 2007