213b thus qt uj1 vj1 uj uj1 vj1 pj uj1pj 0 3

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 T H(!) = I ; 2 !!! : !T (7.2.17) Householder transf...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online