Lecture+22-LU+methods

# Lecture+22-LU+methods - LU Decomposition Decomposition...

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Unformatted text preview: LU Decomposition Decomposition Methods (more than ONE!) * Doolittle decomposition lii = 1 * Crout decomposition uii = 1 (omitted) * Cholesky decomposition (for symmetric matrices) uii = lii Doolittle LU Decomposition Doolittle Algorithm Get 1s on diagonal of [ L ] ( lii =1) Operate on rows and columns sequentially, narrowing down to single element Identical to LU decomposition based on Gauss elimination, but different approach Doolittle LU Decomposition [ ] = = 44 34 33 24 23 22 14 13 12 11 43 42 41 32 31 21 44 43 42 41 34 33 32 31 24 23 22 21 14 13 12 11 u u u u u u u u u u 1 l l l 1 l l 1 l 1 a a a a a a a a a a a a a a a a A st 21 21 11 31 31 11 41 41 11 1 column: / ; / ; / l a u l a u l a u = = = 14 14 13 13 12 12 11 11 st a u ; a u ; a u ; a u : row 1 = = = = [ ] u u l u l u l u l u l u l u l u l u l u u l u l u u l u l u l u l u l u u l u u l u u l u l u u u u A 44 34 43 24 42 14 41 33 43 23 42 13 41 22 42 12 41 11 41 34 24 32 14 31 33 23 32 13 31 22 32 12 31 11 31 24 14 21 23 13 21 22 12 21 11 21 14 13 12 11 + + + + + + + + + + + + + + = Doolittle LU Decomposition nd 31 12 32 22 32 41 12 42 22 42 32 32 31 12 22 42 42 41 12 22 2 column: ; ( ) / ( ) / l u l u a l u l u a l a l u u l a l u u + = + = =- =- nd 21 12 22 22 21 13 23 2 22 23 3 21 14 24 24 22 21 12 23 21 13 24 4 4 2 21 1 2 : ; ; row l u u a l u u a l u u a a l u a u u u l u a l u + = + = + = =- =- =- [ ] 11 12 13 14 21 11 21 12 21 13 21 14 31 11 31 12 32 22 31 13 32 23 33 31 14 32 24 34 41 11 41 12 42 22 41 13 42 23 43 33 41 14 42 24 43 34 4 22 23 4 24 u u u u l u l u l u l u A l u l u l u l u l u u l u l u u l u l u l u l u l u l u l u l u u l u u u u + + + = + + + + + + + + + + + 11 12 13 14 21 22 23 24 31 32 33 34 41 42 43 44 a a a a a a a a a a a a a a a a Doolittle LU Decomposition rd 43 43 41 13 42 23 3 column: ( ) l a l u l u =-- 31 13 32 23 33 33 31 14 32 24 34 34 33 33 31 13 32 23 34 34 31 14 32 24 3 : ; rd row l u l u u a l u l u u a u a l u l u u a l u l u + + = + + = =-- =-- th 44 44 41 14 42 24 43 34 4 row: u a l u l u l u =--- [ ] u u l u l u l u l u l u l u l u l u l u u l u l u u l u l u l u l u l u u l u u l u u l u l u u u u A 44 34 43 24 42 14 41 33 43 23 42 13 41 22 42 12 41 11 41 34 24 32 14 31 33 23 32 13 31 22 32 12 31 11 31 24 14 21 23 13 21 22 12 21 11 21 14 13 12 11 + + + + + + + + + + + + + + = 11 12 13 14 21 22 23 24 31 32 33 34 41 42 43 44 a a a a a a a a a a a a a a a a Forward Substitution Very efficient for large matrices !...
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## This note was uploaded on 11/07/2011 for the course EGM 3344 taught by Professor Raphaelhaftka during the Spring '09 term at University of Florida.

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Lecture+22-LU+methods - LU Decomposition Decomposition...

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