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I am working on an integral-equation problem in the field of electromagnetics, which gives me a very large linear system that needs to be solved. I use a fairly recent method for this, namely the fast multipole method, which allows me to calculate a matrix-vector product in a fast way, so that an iterative solution of the linear system becomes possible.

Anyway, the integral equation I use is known to be relatively ill-conditioned and I want to find out the extent of the problem by taking the condition number of my matrix. However, the elements are not all explicitely calculated, FMM allows for the calculation of the matrix-vector product without having to calculate all the elements. I do ofcourse have written a number of routines to explicetely calculate the elements. The dimension of the matrix easily exceeds 80000, so that it is impossible to store it entirely in the memory.

My question is therefore if anyone knows a method to calculate or estimate the condition number of a very large matrix, of which all the elements are not a priori calculated but can be calculated, altho not all simultaneously stored. A fairly rough estimation would already be satisfactory.

In any case, thanks.

Joris

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# Condition number (or estimate) of a very large matrix

Can you offer guidance or do you also need help?

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