2007-06-17 TPC DD LinearSystemsConditioning
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Transcript of 2007-06-17 TPC DD LinearSystemsConditioning
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Under-Determined, Over-Determined, Ill-conditioned Linear
Systems
August 2005
Dr. K.S.Ravichandran
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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.
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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
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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.
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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.
Linear Systems Ill-Conditioning
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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.
Linear Systems Ill-Conditioning
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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
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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.