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After p = min(m ; 1 n) steps, we have ... H(p)H(p 1) : : : H(1) A = T
; thus, Q = H(p)H(p 1) : : : H(1)
; and I
H(k) = 0 H(!0k 1) )
; 17 7
5 where the identity matrix is (k ; 1) (k ; 1).
At each stage of the procedure, the sign of k is chosen so that cancellation is avoided.
Thus, if a11 0 choose the positive sign for 1 , and if a11 < 0 choose the negative
sign. The reduction can be done e ciently in about 2n2(m ; n=3) multiplications plus
additions and approximately n square roots. Techniques to save storage and avoid explicit
multiplication to determine Q are described by Demmel 6]. 7.3 Multiple Shooting for Nonlinear Problems
We extend the multiple shooting procedures of Section 7.2 to nonlinear problems y = f (x y)
0 gL(y(a)) = 0 a<x<b gR (y(b)) = 0 (7.3.1) by linearization in function space. Thus, we assume y to be known and let y(x) = y(x) + y(x) (7.3.2) and use a Taylor's series expansion of f to obtain f (x y + y) = f (x y) + f (x y) y + O( y2):
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- Spring '14