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Unformatted text preview: Conte 5] suggested examining the \angle" between pairs of columns of Uj (xj ) and
re-orthogonalizing whenever this angle became too small. Again, we'll omit the spatial
argument xj and compute
cos ik = (u(ji) u(jk) ) i k = 1 2 ::: r ;l ju(ji) j2ju(jk)j2 (7.2.15a) where
p (u w) = vT w juj2 = (u u): (7.2.15b) The matrix Uj is re-orthogonalized whenever
1 minr l j ik j <
ik (7.2.15c) ; Example 7.2.1. Solve Example 7.1.2 with R = 3600 using the multiple shooting
procedure with orthogonalization described above with = 10 . The result for the
maximum error in the component y1 of the solution is 2:0 10 7. Four orthogonalization
points were used during the integration.
; 15 Let us conclude this section with a brief discussion of the QR algorithm to reduce
an m n (m n) matrix A to upper triangular form using orthogonal transformations,
i.e., for nding an m m orthogonal matrix Q such that QA = T
0 (7.2.16) where T is n n.
The procedure can be done in many ways, but we'll focus on the use of plane re ections
or Householder transformations
H(!) = I ; 2 !!! :
!T (7.2.17) Householder transf...
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- Spring '14