Get inverse of a huge matrix by inverting its submatrices

Hi,

I need to calculate the inverse of a matrix i.e. 85000x85000 (later bigger), which I need to break down to submatrices to be able to store them (memory problem). Therefore, I would need to work with inverses of the submatrices to calculate the inverse of the matrix.

1) Is there a more efficient way to think about this problem in Matlab?

2) Are parts of these submatrix inversion problems already solved in available codes?

Thanks for your advice

Best

[John D'Errico]

The first question is IF any WHY you think you need to invert the matrix.

Solving a linear system of equations is rarely a good reason to form a matrix inverse.

[nash]

Basically, I would like to solve linear systems of the type of Ax=b. So far my problems have been small enough to go the direct way. This time I work with bigger data such that I am stuck in this dimension. The gmres package as proposed by Steven might do the job. Other proposals are still welcome.

[Walter Roberson]

The \ operator is better than inverting for solving those kinds of problems.

[nash]

Thanks. The problem I search for is, how to organize a huge matrix optimally (breakdown to submatrices due to memory constraint) and perform with these submatrices operations without having to program all algorithms myself. Everything should be possible without ImageProcessingToolbox, while being fast. In this topic I have no experience and also didn't find much helpful suggestions so far.

[Walter Roberson]

I am not aware of any mathematical transformation that allows the calculation of an inverse of a matrix in terms of the inverses of submatrices -- only of calculations of the inverse in terms of the adjunct matrices.

[nash]

Juarez Ruiz, Cortes Maldonado, Perez Rodriguez "Relationship between the Inverses of a Matrix and a Submatrix" (2016) seems to do the job. However, my question was more like about alternatives to organize such a problem. Thanks so far Walter. I need to think about it and just try different setups.

Answer 1

[Steven Lord]

If you're inverting the matrix to try to solve a system of linear equations, consider using the iterative solvers like gmres and friends instead. If the problem is sparse and structured, you may not need to explicitly create the 85000-by-85000 matrix but instead write a function that computes the product of that matrix (and/or its transpose) with a vector.

[nash]

Thanks, this might be a promising solution.