#LARGE SYSTEM OF EQUATIONS SOLVER HOW TO#
#LARGE SYSTEM OF EQUATIONS SOLVER CODE#

Find a free or commercial package that can handle such factorization (examples are Eigen, Intel MKL, SuitSpare) and have optimized code also support parallel computation.

Choose best Direct method for you matrix.

Identify the type of your matrix (SPD, Symmetric or general).

Also there are GPU based solvers available for both direct and iterative methods and highly parallelized, but maybe not suitable for your big problem where matrix being factorized is more than 8GB itself. I think Direct methods suits better for such large system (because of rounding errors). Based on type of your matrix (whether it is Symmetric Positive Definite or Non-symetric etc.) you need to choose one schema (i mean either Iterative or Direct). Matlab also uses one of these methods behind the scene (not sure but I've read somewhere that it have used SuiteSparse code underneath the A/b operator). Each one meet specific matrix types to solve. For example Cholesky Decomposition have a very good performance and low memory usage, but only works on SPD (symmetric Positive Definite) matrices. As i know mainly there are two methods for solving a sparse linear system, Direct and Iterative. Only thing you can do is to get more specific in it and use a more customizable solution, as Matlab do not allow you to do so and see it's source code and see if you can do any improvement. The sparse matrix computation is a big area itself. It's closed source and you have no idea whats happening in there - probably magic. In my experience, using MKL in my bottlenecks always increases performance. That will cut your memory usage in half and might make things a bit faster (moving 16 GB worth of numbers is slower than 8GB).ДЇinally, I can't find an answer on whether Matlab supports the Intel MKL Inspector-executor sparse BLAS routines. In either case, try using a routine that only requires you to pass half of the matrix (I don't think matlab supports this functionality). If it's literally a symmetric complex matrix (Aij = Aji), then that is very weird but maybe still better than a general complex matrix. While the solution is different from a direct decomposition (how could it not be with a gazillion coefficients), its probably not that different. Additionally, this allows you to use a guess solution (for instance, from a previous time step) which can be beneficial. That is almost always the fastest method if you have a single right hand side and is guaranteed to converge. If your matrix is Hermitian (Aij = Aji*) and positive definite, then you can't go wrong with a conjugate gradient approach.

#LARGE SYSTEM OF EQUATIONS SOLVER HOW TO#

However, I fear it will still take days.ДЄny suggestions on how to tackle this problem? Are there any commercial codes that can do this faster/more efficient? And how commercial FEA packages solve this problem when a model has over 1m nodes? I have converted the x=A part into a mexa64-format to potentially reduce the time. Also in some cases convergence was not obtained. I tried iterative methods however these did not yield the same results as the backslash operator. Therefore, 15days is not an acceptable amount of time. However, I need to perform solve the system of equations up to 10x per simulations. Technically speaking I can solve the system as is, and just wait for approximately 15days.

I have two HPC clusters to my disposal, one with 5.7TB of RAM and the other with 1.5TB (but faster cores). In addition, A is symmetric and mostly diagonal. The size of A is (n,n) and b is (n,1) where n is 850000 and can increase to up 2000000. Both A and b are sparse, complex, double precision matrix and vector respectively.

I have reached my limit on the following problem:ДЄs part of my FEA code (in MATLAB) I need to find x, x=A.

Large system of equations solver