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Lecture 22 - Forward Substitution[A]cfw_x = cfw_b[L[U]cfw_x...

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Forward Substitution [ ] { } = + + + + + + = = 4 3 2 1 4 44 3 43 2 42 1 41 3 33 2 32 1 31 2 22 1 21 1 11 4 3 2 1 44 43 42 41 33 32 31 22 21 11 b b b b d l d l d l d l d l d l d l d l d l d l d d d d l l l l 0 l l l 0 0 l l 0 0 0 l d L - - - = - - = - = = 44 3 43 2 42 1 41 4 4 33 2 32 1 31 3 3 22 1 21 2 2 11 1 1 l d l d l d l b d l d l d l b d l d l b d l b d / ) ( / ) ( / ) ( / Very efficient for large matrices ! Once [ L ] is formed, we can use forward substitution instead of forward elimination for different { b }’s [L] [U]{x} = {b} [L] {d} = {b} [A] {x} = {b}
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Back Substitution [ ] { } = + + + + + + = = 4 3 2 1 4 44 4 34 3 33 4 24 3 23 2 22 4 14 3 13 2 12 1 11 4 3 2 1 44 34 33 24 23 22 14 13 12 11 d d d d x u x u x u x u x u x u x u x u x u x u x x x x u 0 0 0 u u 0 0 u u u 0 u u u u x U - - - = - - = - = = 11 4 14 3 13 2 12 1 1 22 4 24 3 23 2 2 33 4 34 3 3 44 4 4 u x u x u x u d x u x u x u d x u x u d x u d x / ) ( / ) ( / ) ( / Identical to Gauss elimination
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Forward Substitution 33 14(2) 6 1 14d d 6d 1 d 2 (1/2)d 2 d 0 1 1 d 1 d 1 d 3 2 1 4 2 3 1 2 1 - = - - = - + - = = - = = + - = + - = = [ ] { } { } b 1 2 1 1 d d d d 1 14 1 6 0 1 1/2 0 0 0 1 1 0 0 0 1 d L 4 3 2 1 = - = - = { } - = 33 2 0 1 d Example:
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Back-Substitution - = - - = = - = - = - = = - - = 13/70 x 3 x 2 1 x 8/35 x 2 x 4/35 2 x 4 x 33/70 70 33/ x 4 3 1 3 2 4 3 4 - - = 33/70 4/35 8/35 13/70 x } { [ ] { } 33 2 0 1 x x x x 70 0 0 0 4 1 0 0 0 4 2 0 3 2 0 1 x U 4 3 2 1 - = - - = {d}
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Forward and Back Substitutions Forward-substitution Back-substitution (identical to Gauss elimination) - = = - = l i 1 j j ij i i n , 1,2, i for d l b d 1 2 3 2 n 1 n i for u x u d x a d x ii n 1 i j j ij i i nn n n , , , , , / - - = - = = + =
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Forward and Back Substitutions
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Pivoting in LU Decomposition Still need pivoting in LU decomposition Messes up order of [ L ] What to do? Need to pivot both [ L ] and a permutation matrix [ P ] Initialize [ P ] as identity matrix and pivot when [ A ] is pivoted Also pivot [ L ]
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LU Decomposition with Pivoting Permutation matrix [ P ] - permutation of identity matrix [ I ] Permutation matrix performs “bookkeeping” associated with the row exchanges Permuted matrix [ P ] [ A ] LU factorization of the permuted matrix [ P ] [ A ] = [ L ] [ U ] Solution [ L ] [ U ] {x} = [ P ] {b}
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Permutation Matrix Bookkeeping for row exchanges Example: [ P ] interchanges row 1 and 3 Multiple permutations [ P ] 11 12 13 14 31 32 33 34 21 22 23 24
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