lu - EE103 (Fall 2009-10) 6. The LU factorization...

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Unformatted text preview: EE103 (Fall 2009-10) 6. The LU factorization factor-solve 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 6-1 Factor-solve 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 6-2 Complexity of factor-solve 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 6-3 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 6-4 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 6-5 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 6-6 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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lu - EE103 (Fall 2009-10) 6. The LU factorization...

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