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Unformatted text preview: = N ; 1 N ; 2 : : : 1: (7.2.9) Several details still remain, including 1. the satisfaction of the initial conditions (7.2.5a,b), 2. the orthogonalization procedure (7.2.7a,b), and 3. the test for a linearly independent basis. Consider the solution of the under-determined system (7.2.5a,b) rst. The r columns of U1 (a) are calculated so that they are mutually orthogonal and span the null space of L. A procedure follows. 1. Reduce LT to upper triangular form using the QR algorithm ( 6], Chapter 3). This involves nding an m m orthogonal matrix Q such that QLT = T : 0 (7.2.10a) Recall that an orthogonal matrix is one where QT Q = I: (7.2.10b) The matrix T is an l l upper triangular matrix and the zero matrix is r l. Pivoting can frequently be ignored with orthogonal transformations and we'll assume this to be the case here. 13 2. Choose U1(a) = QT 0 I (7.2.11) where I is the r r identity matrix and the zero matrix is l r. Thus U1 is the last r columns of QT . Using (7.2.10) and (7.2.11) we verify that LU1(a) = LQT 0 = TT 0] 0 = 0: I I...
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