  Presented in this paper is a new sparse linear solver methodology motivated
by multigrid principles and based around general local transformations that
diagonalize a matrix while maintaining its sparsity. These transformations are
approximate, but the error they introduce can be systematically reduced. The
cost of each transformation is independent of matrix size but dependent on the
desired accuracy and a spatial error decay rate governed by local properties of
the matrix. We test our method by applying a single transformation to the 2D
Helmholtz equation at various frequencies, which illustrates the success of
this approach.
