na_lec_2

na_lec_2 - 5. The LU factorization 5.1.1 The factorization...

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1 5. The LU factorization 5.1.1 The factorization If the 1 - n leading principal minors of n n × A are all non-singular then there exist L and U such that LU A = , where L is a lower triangular matrix with unit diagonal elements = 1 1 1 2 1 21 L O M M n n l l l 0 L and U is an upper triangular matrix = nn n n u u u u u u 0 U M O L L 2 22 1 12 11 . 5.1.2 Algorithm for Factorization We will demonstrate the factorization algorithm for the case 3 = n . The extension to higher dimensions is obvious. We first write the factorization in component form = 33 32 31 23 22 21 13 12 11 33 23 22 13 12 11 32 31 21 1 1 1 a a a a a a a a a u u u u u u l l l . Form the first column of A we obtain 11 11 a u = 11 21 21 21 11 21 u a l a u l = = 11 31 31 31 11 31 u a l a u l = =
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2 Form the second column of A we obtain 12 12 a u = 12 21 22 22 22 22 12 21 u l a u a u u l - = = + 22 12 31 32 32 32 22 32 12 31 u u l a l a u l u l - = = + Form the third column of A we obtain 13 13 a u = 13 21 23 23 23 23 13 21 u l a u a u u l - = = + 23 32 13 31 33 33 33 33 23 32 13 31 u l u l a u a u u l u l - - = = + + which completes the factorization of A . Form inspection we find that the algorithm brakes down if, and only if, 0 11 = u or 0 22 = u . The condition 0 11 = u occurs whenever 0 11 = a , i.e., the 1 1 × leading principal minor of A is singular. The condition 0 22 = u occurs whenever 0 12 11 21 22 12 21 22 = - = - a a a a u l a , i.e., the 2 2 × leading principal minor of A is singular. If A is singular then 0 33 = u . 5.1.3 Solving b Ax = via the LU factorization Suppose b Ax = . The objective is to solve this problem without inverting matrices. With LU A = we have b LUx = , or b L Ux 1 - = . Hence the product
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3 b L z 1 - = can be evaluated by solving b Lz = for z using forward substitution. Then x is determined from z Ux = by backward substitution.
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This note was uploaded on 12/02/2011 for the course ME 7533 taught by Professor Staff during the Summer '11 term at LSU.

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na_lec_2 - 5. The LU factorization 5.1.1 The factorization...

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