Computing Sparse Jacobian Matrices Optimally

Abstract

A restoration procedure based on a priori knowledge of sparsity patterns of the compressed Jacobian matrix rows is proposed. We show that if the rows of the compressed Jacobian matrix contain certain sparsity patterns the unknown entries can essentially be restored with cost at most proportional to substitution while the number of matrix-vector products to be calculated still remains optimal. We also show that the conditioning of the reduced linear system can be improved by employing a combination of direct and indirect methods of computation. Numerical test results are presented to demonstrate the effectiveness of our proposal.