Efficient Computation of Sparse Elements of the Inverse of a Sparse Near-Tridiagonal Matrix with Application to the Nerve Equation

Standard algorithms for computing the inverse of a tridiagonal matrix (or more generally, any Hines matrix) compute the entire inverse, which is not sparse. For some problems, only the elements of the inverse at locations corresponding to nonzero elements in the original matrix are required. We present an algorithm that efficiently computes only these elements in O(n) time and memory. This algorithm is useful in solving discretized systems of partial differential equations that arise when computing electrical flow along a branched structure, such as a neuron’s dendritic arbor.