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

This preview shows pages 1–8. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## 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.

### Page1 / 25

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

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online