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Unformatted text preview: 9/5/11 LU Decomposition Three Basic Steps (1) Factor (decompose) [ A ] into [ L ] and [ U ] (2) given { b }, determine { d } from [ L ] { d } = { b } (3) using [ U ]{ x } = { d } and back 9/5/11 Solving x using LU Solve {D} given [L] and {B} Solve { x } given [U] and {D} {D}=intermediate variables 9/5/11 Matrix Norm • For n n matrix, the Frobenius norm • Frobenius norm provides a single value to quantify the size of matrix [ A ] ∑∑ = = = n 1 i n 1 j 2 ij e a A p 1 n 1 i p i p x X / = ∑ = p = 2 : Euclidean norm 9/5/11 Ø Condit ion num be r de fine d in t e rm s of m a t rix norm s Ø Cond[ A] is gre at t ha n or e qual t o 1 Ø M at rix A is illcondit ione d if Cond[ A] > > 1 Ø If [ A] has 1 0* 7 rounding e rror a nd Cond[ A] = 1 0 5 , t he n t he solut ion [ X] m a y be valid t o only 1 0 2 1 A A A Cond 1 ≥ ⋅ = ] [ Matrix Condition Number A A A Cond X X ∆ = ∆ ] [ Relative error of the norm 9/5/11 10 9 5 7 9 10 6 8 5 6 5 7 7 8 7 10 Example of illconditioned system 8 Consider the following 4 x 4 system: A x = x = = b !!! Scary !!! BAD approximation x ** to the true solution can almost satisfy the relevant equations! WHY??? Cond[ A] = 2984 31 33 23 32 * Exact solution: x = (1, 1, 1, 1) T • Let A x * = ( 3 1 .9 , 2 3 .1 , 3 2 .9 , 3 1 .1 ) T x * = (7.2, 14.6, 2.5, 3.1) T b differs by (0.1, 0.1,0.1, 0.1) T large change in x. • Let A x ** = ( 3 1 .9 9 , 2 3 .0 1 , 3 2 .9 9 , 3 1 .0 1 )T x ** = (0.18, 2.36, 0.65, 1.21) T b differs by (0.01, 0.01, 0.01, 0.01) T large change in x 9/5/11 10 9 5 7 9 10 6 8 5 6 5 7 7 8 7 10 Example of illconditioned system ˆ Consider the following 4 x 4 system: A x = x = = b !!! Scary !!! BAD approximation x ** to the true solution can almost satisfy the relevant equations! WHY??? Cond[ A] = 2984 31 33 23 32 * Exact solution: x = (1, 1, 1, 1) T • Let A x * = ( 3 1 .9 , 2 3 .1 , 3 2 .9 , 3 1 .1 ) T x * = (7.2, 14.6, 2.5, 3.1) T b differs by (0.1, 0.1,0.1, 0.1) T large change in x. • Let A x ** = ( 3 1 .9 9 , 2 3 .0 1 , 3 2 .9 9 , 3 1 .0 1 )T x ** = (0.18, 2.36, 0.65, 1.21) T b differs by (0.01, 0.01, 0.01, 0.01) T large change in x 9/5/11 10 9 5 7 9 10 6 8 5 6 5 7 7 8 7 10 Example of illconditioned system H Consider the following 4 x 4 system: A x = x = = b !!! Scary !!! BAD approximation x ** to the true solution can almost satisfy the relevant equations! WHY??? Cond[ A] = 2984 31 33 23 32 * Exact solution: x = (1, 1, 1, 1) T • Let A x * = ( 3 1 .9 , 2 3 .1 , 3 2 .9 , 3 1 .1 ) T x * = (7.2, 14.6, 2.5, 3.1) T b differs by (0.1, 0.1,0.1, 0.1) T large change in x....
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 Fall '09
 RAPHAELHAFTKA
 Numerical Analysis, Method, Runge–Kutta methods, large change, BAD approximation x**

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