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Re: Condition number of matrices
Posted:
Jul 3, 2006 10:06 AM


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 LUdecomposition 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



