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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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