Structural Packing and Dyadic Factorization of Sparse Positive Definite Matrices

Abstract

Efficient inversion of large sparse positive definite matrices requires exploiting sparsity patterns beyond those captured by conventional bandwidth reduction. In this work, we recast nested dissection, a prominent alternative, as a two-stage framework. The matrix was first packed into block-tridiagonal or dyadic form, followed by sparse Gram-Schmidt orthogonalization. This decomposition provided a unified perspective on sparse matrix factorization and inversion and identified dyadic structure as a fundamental component of sparse Cholesky factorization. For the first stage, we introduced a packing algorithm that recovered block-tridiagonal and dyadic patterns using a novel $\ell_1$ criterion. Using approximate distances obtained through classical multidimensional scaling, the method was effective when the target structure was sufficiently represented among the nonzero entries. Iterative application could also remove structural noise and reveal hidden dyadic organization, corresponding to separator identification in nested dissection. For the second stage, we developed the theory of dyadically structured matrices. We derived sparse factorization and inversion procedures, analyzed their computational complexity, and obtained an efficient inversion algorithm. A modified version reduced the cost of inverting block-tridiagonal matrices, demonstrating the benefit of exploiting their structure directly rather than treating them as generic band matrices.