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Unformatted text preview: EE103 (Fall 200910) 6. The LU factorization • factorsolve method • nonsingular matrices • LU factorization • solving Ax = b with A nonsingular • the inverse of a nonsingular matrix • LU factorization algorithm • effect of rounding error • sparse LU factorization 61 Factorsolve approach to solve Ax = b , first write A as a product of ‘simple’ matrices A = A 1 A 2 ··· A k then solve ( A 1 A 2 ··· A k ) x = b by solving k equations A 1 z 1 = b, A 2 z 2 = z 1 , . . . , A k 1 z k 1 = z k 2 , A k x = z k 1 Examples • Cholesky factorization (for positive definite A ) k = 2 , A = LL T • sparse Cholesky factorization (for sparse positive definite A ) k = 4 , A = PLL T P The LU factorization 62 Complexity of factorsolve method # flops = f + s • f is cost of factoring A as A = A 1 A 2 ··· A k (factorization step) • s is cost of solving the k equations for z 1 , z 2 , . . . z k 1 , x (solve step) • usually f ≫ s Example: positive definite equations using the Cholesky factorization f = (1 / 3) n 3 , s = 2 n 2 The LU factorization 63 Multiple righthand sides two equations with the same matrix but different righthand sides Ax = b, A ˜ x = ˜ b • factor A once ( f flops) • solve with righthand side b ( s flops) • solve with righthand side ˜ b ( s flops) cost: f + 2 s instead of 2( f + s ) if we solve second equation from scratch Conclusion: if f ≫ s , we can solve the two equations at the cost of one The LU factorization 64 Nonsingular matrices a square matrix with a zero nullspace is called nonsingular a square matrix with nonzero vectors in its nullspace is called singular (note: the terms ‘nonsingular/singular’ only apply to square matrices) Examples A = bracketleftbigg 1 − 1 1 2 bracketrightbigg , B = bracketleftbigg 1 − 1 − 2 2 bracketrightbigg • A is nonsingular: x = 0 is the only vector that satisfies Ax = 0 • B is singular: ( x 1 , x 2 ) = (1 , 1) satisfies Bx = 0 The LU factorization 65 Vandermonde matrix A = 1 t 1 t 2 1 ··· t n 1 1 1 t 2 t 2 2 ··· t n 1 2 . . . . . . . . . . . . . . . 1 t n t 2 n ··· t n 1 n (with t i negationslash = t j for i negationslash = j ) A is nonsingular (has a zero nullspace): Ax = 0 means p ( t 1 ) = p ( t 2 ) = ··· = p ( t n ) = 0 where p ( t ) = x 1 + x 2 t + ··· + x n t n 1 this is only possible if x = 0 because a polynomial of degree n − 1 or less cannot have n distinct real roots The LU factorization 66 LU factorization LU factorization without pivoting A = LU • L unit lower triangular, U upper triangular • does not always exist (even if A is nonsingular) LU factorization (with row pivoting) A = PLU • P permutation matrix, L unit lower triangular, U upper triangular...
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This note was uploaded on 10/15/2009 for the course EE 103 taught by Professor Vandenberghe,lieven during the Spring '08 term at UCLA.
 Spring '08
 VANDENBERGHE,LIEVEN

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