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snooze_bear

Null space of a matrix

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Hello,

I'm trying to find a solution of the linear system Ax=0, where A is a (very large) singular matrix with null space of dimension 1. Most of the numerical methods I see for solving linear equations of the form Ax = b assume a nonsingular A and nonzero b. Does anyone know a fast and numerically stable way to find solutions for b = 0? Thanks very much! smile.png

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I'm not an expert but according to wikipedia it looks like singular value decomposition is your best bet (if you want a numerically accurate algorithm). As it happens, there are no exact algorithms to compute the SVD though, instead you have to use an iterative approach. LAPACK provides routines to compute the SVD and I suggest using them instead of implementing your own (unless you want to do it for educational purposes).

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The method of solving the system likely depends on the specifics of your problem. For example, this paper http://www.ima.umn.edu/preprints/apr99/1611.pdf has a subproblem that involves solving a large sparse linear system whose matrix has null space of dimension 1. The authors show that using the conjugate gradient method leads to a solution that is unique among values when you project out the null space. The iterations always keep you on the projection space, so numerically the solver is quite robust.

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