Optimal direct determination of sparse Jacobian matrices

It is well known that a sparse Jacobian matrix can be determined with fewer function evaluations using finite differencing or forward automatic differentiation (AD) passes than the number of independent variables of the underlying function. In this paper, we show that by grouping together rows into blocks and partitioning the resulting column segments, one can reduce this number further. We suggest a graph colouring technique for row partitioned Jacobian matrices to efficiently determine the nonzero entries using direct determination. We characterize optimal direct determination and derive results on the optimality of any direct determination technique based on column computation. Previously published computational results by the authors [S. Hossain and T. Steihaug, Graph colouring in the estimation of sparse derivative matrices: Instances and applications, Discrete Appl. Math. 156(2) (2008), pp. 280–288; S. Hossain, CsegGraph: A graph colouring instance generator, Int. J. Comput. Math. 86(10–11) (2009), pp. 1956–1967] have demonstrated that the row partitioned direct determination can yield considerable savings in function evaluations or AD passes over methods based on the Curtis, Powell, and Reid technique.