Automatic computation of partial derivatives and rounding error estimates with applications to large-scale systems of nonlinear equations
The problem of computing the Jacobian of functions is studied in some detail. The main idea is to evaluate formulas and its (first partial) derivatives by sequences of very simple steps of the form \(v_ i=\phi_ i(u_{i1},u_{i2},...,u_{im})\) where the chain rule is used repeatedly for the derivatives and where the \(\phi_ i\) are predefined basic computations. The order of the computation is described by a graph. The authors distinguish between bottom-up and top-down traversing this graph, where the classical method is represented by the bottom-up traversing. The authors develop absolute and probabilistic error estimates. They perform various tests with respect to running time and rounding errors using involved models from chemistry. They report that their algorithm for computing the accurate Jacobian is six times faster than that by numerical differentiation.
scientific article; zbMATH DE number 4109931
scientific article