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    <p>I assume that your matrix is dense. If it is sparse, you can find numerous specialised algorithms as already mentioned by <a href="https://stackoverflow.com/questions/1242190/c-memory-efficient-solution-for-axb-linear-algebra-system/1242210#1242210">DeusAduro</a> and <a href="https://stackoverflow.com/questions/1242190/c-memory-efficient-solution-for-axb-linear-algebra-system/1242217#1242217">duffymo</a>. </p> <p>If you don't have a (large enough) cluster at your disposal, you want to look at out-of-core algorithms. <a href="http://www.netlib.org/scalapack/" rel="nofollow noreferrer">ScaLAPACK</a> has a few out-of-core solvers as part of its <a href="http://www.netlib.org/scalapack/prototype/" rel="nofollow noreferrer">prototype package</a>, see the documentation <a href="http://www.netlib.org/scalapack/prototype/readme.outofcore" rel="nofollow noreferrer">here</a> and <a href="http://www.google.com/search?q=out-of-core+extension+to+ScaLAPACK" rel="nofollow noreferrer">Google</a> for more details. Searching the web for "out-of-core LU / (matrix) solvers / packages" will give you links to a wealth of further algorithms and tools. I am not an expert on those.</p> <p>For this problem, most people would use a cluster, however. The package you will find on almost any cluster is ScaLAPACK, again. In addition, there are usually numerous other packages on the typical cluster, so you can pick and choose what suits your problem (examples <a href="http://www.tacc.utexas.edu/resources/software/" rel="nofollow noreferrer">here</a> and <a href="http://www.netlib.org/utk/people/JackDongarra/la-sw.html" rel="nofollow noreferrer">here</a>).</p> <p>Before you start coding, you probably want to quickly check how long it will take to solve your problem. A typical solver takes about O(3*N^3) flops (N is dimension of matrix). If N = 100000, you are hence looking at 3000000 Gflops. Assuming that your in-memory solver does 10 Gflops/s per core, you are looking at 3 1/2 days on a single core. As the algorithms scale well, increasing the number of cores should reduce the time close to linearly. On top of that comes the I/O.</p>
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    1. COCaveat: the above O(3*N^3) assumes that you use complex numbers. For real numbers, divide everything by 6, i.e. somewhere around O(0.5 * N^3).
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