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Topic: Condition number of matrices
Replies: 5   Last Post: Jul 5, 2006 3:36 PM

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Peter Spellucci

Posts: 2,760
Registered: 12/7/04
Re: Condition number of matrices
Posted: Jul 3, 2006 10:06 AM
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In article <e89djl$7fg$1@mailhub227.itcs.purdue.edu>,
"Fijoy George" <tofijoy@yahoo.co.in> writes:
>Hi all,
>
>I have the following question regarding the sensitivity analysis of linear
>systems.
>
>In my numerical methods course, I have learned theorems which give upper
>bounds for the relative change in the solution of the linear system Ax=f.
>For example, if only f is changed, relative change in x = K(A)*relative
>change in f, where K(A) is the condition number of the matrix A.
>
>Now, for such theorems to be useful in practice, we need the condition
>number of A which is defined as ||A||*||A_inverse||.
>
>So how does one calculate the condition number of a matrix? Given that real
>world systems are large, can we precisely calculate K(A)? Or, can we only
>hope to obtain a upper bound for K(A)?
>
>Thank you very much
>Fijoy
>


no, upper bounds exist, but these are much too pessimistic (e.g. a bound
of Hadamard, involving the Frobenius norm of the matrix and the determinant)
normally, using the LU-decomposition one computes lower bounds for it, and
thetrick consists in finding realistic lower bounds cheaply.
there is a discussion of this in Highams book: accuracy and stability of
numerical algorithms (SIAM).
hth
peter



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