Efficient Evaluation of Sparse Jacobians by Matrix Compression Part II: Implementation and Experiments

The accurate and efficient calculations of Jacobians matrices at a sequence of arguments is a key ingre- dient of numerical methods for nonlinear problems in scientific computing. It has been known since the seminal work of Curtis Powell and Reid (1) that once their sparsity pattern is known Jacobians can be estimated on the basis of divided differences for a set carefully chosen directions. The number p of such seed directions and thus extra function evaluations can often be limited a priori to a smallish number, which is typically much smaller than the number of independent variables and unaffected by grid sizes and other discretization parameters. The cost factorp is bounded below by the maximal number of nonzeros per row, which is actually sufficient for Jacobian estimation using Newsam-Ramsdell compression. This NR approach is numerically less stable than the CPR method, which was therefore preferred in practice as divided differences are strongly affected by truncation and round off errors. However now, using automatic or algorithmic differentiation, one obtains directional derivatives with working ac- curacy and can thus utilize the more economical NR approach.