# Lec6 - Solving many systems of linear equations LU...

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Solving many systems of linear equations LU decomposition

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Lecture 6 2 LU decomposition … … is used when you need to solve many systems of equations of the form: 1 2 3 4 [] {} {} A xb A A A = = = = M Each system of equations has the same matrix [A] but has different constants { b } on the right hand side. 1234 12 3 4 3 4 22521 32 6 52 7 4 3 53 4 1 xxxx xx x x x x + ++ = + = + ++= +− + = 3 4 2252 3 2 7 4 3 . 5 4 1 . 4 x x x x + = + = +++= + = 3 4 5 3 7 4 2 . 9 4 9 x x x x + = + = + = 3 4 8 0 7 4 6 4 1 x x x x + = + = + + =
Lecture 6 3 How to save some work 11 1 1 1 22 1 33 1 44 []{} {} [] []{} [] {} {} [] {} {} []{} A xb A Ax A b xA b b b b b −− = = = = = = = 1 2 3 4 {} {} A A A A = = = = M Find [ A ] -1 once and use it to solve as many equations as you want But, remember that computing an inverse is expensive ~2N 3 flops! Perhaps there is a better way…

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Lecture 6 4 LU decomposition: A more efficient alternative We want to solve a system of equations: [] {} {} A xb = Suppose we can write [ A ] as the product of 2 matrices [ L ] & [ U ] where: 11 12 13 1 21 22 2 31 32 33 3 123 [ ] [] 100 0 10 0 0 0 1 0 [ ] 0 0 0 0 n n n nn n n n LU A uuu u lu u Ll l U u u ll l u = ⎛⎞ ⎜⎟ == ⎝⎠ LL MMM O M M M M O M L ower triangular U pper triangular Choice of 1’s on diagonal arbitrary --> Doolittle vs. Crout
Lecture 6 5 () [] {} {} [ ] [ ] { } { } { }{ } {} [ ]{} {} A xb LU x b LUx b Ux y Ly b y y y x x = = =≡ = = Let Solve for by forward substitution Solve for by backward substitution LU decomposition: A more efficient alternative We want to solve a system of equations: A = With [ L ][ U ] = [ A ] , the problem becomes:

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Lecture 6 6 () [] {} {} [ ] [ ] { } { } { }{ } {} [ ]{} {} A xb LU x b LUx b Ux y Ly b y y y x x = = =≡ = = Let Solve for by forward substitution Solve for by backward substitution LU decomposition: A more efficient alternative We want to solve a system of equations: A = With [ L ][ U ] = [ A ] , the problem becomes:
Lecture 6 7 () [] {} {} [ ] [ ] { } { } { }{ } {} [ ]{} {} A xb LU x b LUx b Ux y Ly b y y y x x = = =≡ = = Let Solve for by forward substitution Solve for by backward substitution LU decomposition: A more efficient alternative We want to solve a system of equations: A = With [ L ][ U ] = [ A ] , the problem becomes: 21 31 32 123 100 0 10 0 1 0 1 nn n l Ll l lll ⎛⎞ ⎜⎟ = ⎝⎠ L L L MMM O M L

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Lecture 6 8 () [] {} {} [ ] [ ] { } { } { }{ } {} [ ]{} {} A xb LU x b LUx b Ux y Ly b y y y x x = = =≡ = = Let Solve for by forward substitution Solve for by backward substitution LU decomposition: A more efficient alternative We want to solve a system of equations: A = With [ L ][ U ] = [ A ] , the problem becomes: 21 31 32 123 100 0 10 0 1 0 1 nn n l Ll l lll ⎛⎞ ⎜⎟ = ⎝⎠ L L L MMM O M L
Lecture 6 9 LU decomposition: A more efficient alternative We want to solve a system of equations: [] {} {} A xb = With [ L ][ U ] = [ A ] , the problem becomes: () [ ] [ ] { } { } { }{ } {} [ ]{} {} A LU x b LUx b Ux y Ly b y y y x x = = =≡ = = Let Solve for by forward substitution Solve for by backward substitution 11 12 13 1 22 2 33 3 00 0 0 000 n n n nn uuu u uu Uu u u ⎛⎞ ⎜⎟ = ⎝⎠ L L L MMM O M L

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Lecture 6 10 LU decomposition: A more efficient alternative We want to solve a system of equations: [] {} {} A xb = With [ L ][ U ] = [ A ] , the problem becomes: () [ ] [ ] { } { } { }{ } {} [ ]{} {} A LU x b LUx b Ux y Ly b y y y x x = = =≡ = = Let Solve for by forward substitution Solve for by backward substitution Solution 11 12 13 1 22 2 33 3 00 0 0 000 n n n nn uuu u uu Uu u u ⎛⎞ ⎜⎟ =
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Lec6 - Solving many systems of linear equations LU...

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