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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