# Solving a linear system with block-banded matrix

It is required to solve a linear system $$Ax=b$$, where the matrix $$A$$ is symmetric, all the variables and coefficients are real. The structure of $$A$$ is $$$$A = \begin{pmatrix} A_{11} & A_{12} \\ A_{12}^T & A_{22} \end{pmatrix},$$$$ where $$A_{11}$$, $$A_{12}$$ and $$A_{22}$$ are banded square blocks of dimension $$N$$, which may be large. $$A_{12}$$ is not symmetric.

I would like to avoid solving the full-matrix system, and to exploit the banded structure of the submatrices, without inverting any of the blocks.

• It seems your symmetric $A$ matrix is just a normal symmetric matrix without any specific feature, so I'm not sure why you don't try any conventional linear system solver either direct one like MUMPS or iterative one like GMRES or specifically MINRES which is developed for real symmetric matrices? Dec 18 '20 at 18:01
• For the same reason I would use a banded solver to solve $A_{11}x=b_1$. Full-matrix methods are much more costly. Dec 18 '20 at 19:15
• No, you don't need the full matrix $A$ to solve $Ax = b$ with MINRES or GMRES. You need just its action on a vector or define it as a linear operator. Dec 18 '20 at 19:16
• So, you're looking for a method using factorization rather than an iterative method? Dec 19 '20 at 4:33
• Have you considered using a sparse symmetric indefinite factorization rather than trying to use a banded factorization? Dec 19 '20 at 6:16