This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Lecture 24 Complexity of the simplex method UCSD Math 171A: Numerical Optimization Philip E. Gill http://ccom.ucsd.edu/~peg/math171 Wednesday, March 11th, 2009 Recap: complexity of the simplex method Given a particular linear program, can we predict how long it will take to be solved? There are two issues: How much time does it take to do one iteration? How many iterations will be required overall? UCSD Center for Computational Mathematics Slide 2/26, Wednesday, March 11th, 2009 Recap: Column replacement To replace column t (i.e., a t ) of B by zeros , we form B a t e T t To add column a in place of the zeros we form B a t e T t + ae T t We can combine these steps as B = B + ( a a t ) e T t which means that column t of B is replaced by a . UCSD Center for Computational Mathematics Slide 3/26, Wednesday, March 11th, 2009 Recap: The ShermanMorrison formula The ShermanMorrison formula is useful in proving theoretical results (it is unreliable for practical computation). First, consider the case in which A is a general rankone modification of the nonsingular matrix A : A = A + uv T where u and v are nvectors. Then A 1 = ( A + uv T ) 1 = A 1 1 A 1 uv T A 1 , with = 1 + v T A 1 u A is nonsingular if and only if 1 + v T A 1 u 6 = 0, and its inverse is a rankone modification of the inverse of A : UCSD Center for Computational Mathematics Slide 4/26, Wednesday, March 11th, 2009 From the previous slide: A 1 = A 1 1 A 1 uv T A 1 , with = 1 + v T A 1 u The crucial feature is that, given A 1 , only n 2 operations are needed to update A 1 . Then solutions of Ax = b and A T y = c are x = A 1 b and y = ( A 1 ) T c Column replacement is a special case of rankone modification. UCSD Center for Computational Mathematics Slide 5/26, Wednesday, March 11th, 2009 B = B + ( a a t ) e T t The inverse of B is given by the ShermanMorrison formula with u = a a t and v = e t By definition, Be t = a t , so that B 1 a t = e t ....
View
Full
Document
This note was uploaded on 10/23/2010 for the course MATH 171a taught by Professor Staff during the Winter '08 term at UCSD.
 Winter '08
 staff
 Math

Click to edit the document details