The Iso-Inverse: A Contravariant Sparse Approximate Inverse Matrix

Given a sparse matrix <b>A</b>, we define and discuss a matrix <b>B</b> that approximates <b>A</b>^-1, has exactly the same sparsity structure as <b>A</b>, and varies contravariantly with <b>A</b>. Unlike many other sparse inverse approximations, the iso-inverse <b>B</b> is not constructed by minimizing the residual ∥<b>AB</b> - <b>I</b>∥ but, instead, by stipulating that <b>AB</b> = <b>I</b> for a well-defined subset of the elements of <b>I</b>. The iso-inverse is potentially useful in electronic structure calculations with nonorthogonal localized molecular orbitals.