2007-06-17 TPC DD LinearSystemsConditioning

8/13/2019 2007-06-17 TPC DD LinearSystemsConditioning

1/8

Tech-Pro Consultants

Under-Determined, Over-Determined, Ill-conditioned Linear

Systems

August 2005

Dr. K.S.Ravichandran


2/8


Linear Systems

Reference :

1. Lecture notes on optimization, Prof. Baldick, The University of Texas at

Austin

A non-singular matrix A can always be factorized in to

A=LU where L is lower triangular and U is strictly upper

triangular with unit entries on the diagonal.

The procedure for triangulation breaks down for a

singular matrix because at some stage of the procedure, a

non-zero pivot does not exist.


3/8


Linear Systems Ill-Conditioning

Example:Consider Ax = b,

Det[A] = - , 0 and the matrix A is nearly singular.Using infinite precision arithmetic, we can solve Ax = b forany vector b = [b1, b2]

t. The solution is


4/8


Linear Systems - Perturbations

Example: Take b = [1 , 1]t. .

Then ||b || = 2 and the solution x* = [1 , 0]t, || x* || = 1Perturb the RHS:Take b = [1 , 1]t.+ [, 0]t = b + bThen ||b || = and the solutionx** = x*+ x* = [1 , 0]t+ [0 , /]t, || x** || = {1+ (/)2}|| x* || =|/|.The relative change in solution is || x* || / || x* || =|/|.The relative change in b is || b || / || b || =|| / 2.


5/8


Ratio of the relative change is =|/| / (|| / 2) = 2/ ||The change in the solution vector is independent of the

change in the forcing vector and is large if is small.We say that the matrix is ill-conditioned.



6/8


Perturb the Matr ix :Take (A+ A)x = b. Let A =

Then ||A || = and the solution. With b = [1 , 1]tx** = x*+ x* = [1 , 0]t+ [0 , -/]t, || x* || =|/|.The relative change in solution is || x* || / || x* || =|/|.The relative change in A is || A || / || A || =|| / 2.Ratio of the relative change is =|/| / (|| / 2) = 2/ ||The change in the solution vector is independent of the change

in the matrix and is large if is small.Thus the matrix is ill-conditioned.



7/8


AnalysisCondition Number

Condition Number of a Matr ix :

Denote the L2norm of a vector x by ||x|| = = xtx

The L2norm of a matrix A is given by Max ||Ax=y|| for all x s.t.

||x|| = 1.

Typically the L2norm of a matrix is computed as

||A||2= { aij2}1/2 where the aijare the elements of AThe condition number is now defined as

C= ||A||2||A-1||2

A matrix for which C is large is said to be ill-conditioned


8/8Tech-Pro Consultants

AnalysisMain Result

Suppose A is a non-singular matrix , and consider Ax = b, Ax =

b + b and (A + A)x = b. We then have the following bounds:(i) If x*+ x* solves Ax = b + b we have

|| x*|| / || x*|| ||A||2||A-1||2||b|| / ||b|| = C ||b|| / ||b||(ii) If x*+ x* solves (A + A)x = b we have

|| x*|| / || x* + x*|| C ||A|| / ||A|| where Ax* = bIn either case, the amplification to the solution is large if the

condition number of the system is large.

2007-06-17 TPC DD LinearSystemsConditioning

Documents

Transcript of 2007-06-17 TPC DD LinearSystemsConditioning