Solving Ax=b where A is too big to be stored in a single array

Question

Problem: A is square, full rank, sparse and banded. It has way too many elements to be stored as a single matrix in Matlab (at least ~4.6*10¹⁸ and ideally ~10⁴⁰, both of which exceed max array size. EDIT: A is stored as sparse, and the problem is not with limited memory but with limited number of elements). Therefore I have to store it as a collection of smaller arrays (rows/diagonals/columns/blocks).

Looking for: a way to solve Ax=b, with A given as a collection of smaller arrays. Ideally in Matlab but not a must.
Alternatively, if not in Matlab: maybe there's a program that can store and solve such a big A?

Found so far: methods if A is tri/pentadiagonal, but my A has N diagonals. Also found something about partitioning A to blocks, but couldn't find a way to then solve a linear system with these blocks.

p.s. The system is 64-bit.

Thanks everyone!

Answer 1

Not using Matlab would allow you to store larger arrays. ROOT is an open source framework developed at CERN that has C++ and Python interfaces and a variety of solvers. It is also capable of handling huge datasets and has a variety of visualization and analysis tools as well.

If you are interested in writing C or Fortran BLAS(Basic Linear Algebra Subroutines) and CBLAS would be good options. There are many open source and proprietary implementations of BLAS that should be available for most Linux/UNIX distributions. There are also plenty of examples showing how to use the BLAS subroutines in C and Fortran code available online.