On the Equivalence of Forward Mode Automatic Differentiation and Symbolic Differentiation

We show that forward mode automatic differentiation and symbolic differentiation are equivalent in the sense that they both perform the same operations when computing derivatives. This is in stark contrast to the common claim that they are substantially different. The difference is often illustrated by claiming that symbolic differentiation suffers from "expression swell" whereas automatic differentiation does not. Here, we show that this statement is not true. "Expression swell" refers to the phenomenon of a much larger representation of the derivative as opposed to the representation of the original function

The CPR Method and Beyond : Prologue

In $1974$ A.R. Curtis, M.J.D. Powell, and J.K.Reid published a seminal paper on the estimation of Jacobian matrices which was later coined as the CPR method. Central to the CPR method is the effective utilization of a priori known sparsity information. It is only recently that the optimal CPR method in its general form is characterized and the theoretical underpinning for the optimality is shown. In this short note we provide an overview of the development of computational techniques and software tools for the estimation of Jacobian matrices