Generalized Capacitance Matrix Theorems and Algorithm for Solving Linear Systems

The capacitance matrix method has been widely used as an efficient numerical tool for solving the boundary value problems on irregular regions. Initially, this method was based on the Sherman–Morrison–Woodbury formula, an expression for the inverse of the matrix (A + UV ) with A ∈ <n×n and U,V ∈ <n×p. Extensions of this method reported in literature have made restrictive assumptions on the matrices A and (A + UV ). In this paper, we present several theorems which are generalizations of the capacitance matrix theorem in [4] and are suited for very general matrices A and (A + UV ). A generalized capacitance matrix algorithm is developed from these theorems and holds the promise of being applicable to more general problems; in addition, it gives ample freedom to choose the matrix A for developing very efficient numerical algorithms. We demonstrate the usefulness of our generalized capacitance matrix algorithm by applying it to efficiently solve large sparse linear systems in several computer vision problems, namely, the surface reconstruction and interpolation, and the shape from orientation problem.