EE103 Section 5

EE103 Section 5 - SECTION 5: ERROR ANALYSIS FOR SYSTEMS OF...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
SECTION 5: ERROR ANALYSIS FOR SYSTEMS OF LINEAR EQUATIONS .............................. 105 Vector and Matrix Norms .................................................................................................................... 105 Error Bounds ........................................................................................................................................ 107
Background image of page 1

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

View Full DocumentRight Arrow Icon
EE103 Lecture Notes, Winter 2009, Prof S. Jacobsen Section 5 105 SECTION 5: ERROR ANALYSIS FOR SYSTEMS OF LINEAR EQUATIONS Let A be an n x n matrix. We have seen examples where solving A xb = may lead to solutions that look good but are actually poor. We now develop an analysis of how such may occur. Consider Ax b = Let e x denote the exact solution and let c x denote a computed solution. Let c rA x b =− be the residual vector. Then 1 ce c e x A x x x xA r ⎛⎞ ⎜⎟ ⎝⎠ =−= ⇒−= [We are assuming that 1 A exists.] Therefore, even though r may be “small,” it may be that 1 A is “big” and thus it may be that 1 Ar is “big”. So, we need to develop what we mean by “big” and “small” for vectors and matrices. Vector and Matrix Norms Definition: A vector norm, , is a function with the following properties. () 12 ,,, T n xx x x = 1. 0 n x xR ≥∀ 2. 00 =⇔= 3. , x α αα =∀ 4. x yxy +≤ + Ex: 22 2 21 2 || T n x x x x =+ + + = " , the Euclidean norm or measure of distance. Ex: 1 l norm (“street walking”) 1 1 n j j x x = =
Background image of page 2
EE103 Lecture Notes, Winter 2009, Prof S. Jacobsen Section 5 106 Ex: l norm (“sup norm”) { } 12 max , , , n x xx x =
Background image of page 3

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

View Full DocumentRight Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 03/30/2009 for the course EE 103 taught by Professor Vandenberghe,lieven during the Winter '08 term at UCLA.

Page1 / 7

EE103 Section 5 - SECTION 5: ERROR ANALYSIS FOR SYSTEMS OF...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online