Reverse accumulation and implicit functions

We begin by introducing a simple technique for using reverse accumulation to obtain the first derivatives of target functions which include in their construction the solution of systems of linear or nonlinear equations. In the linear case solving Ay = b for y corresponds to the adjoint operations (b) over bar := (b) over bar + v and (A) over bar := (A) over bar - yv where v is the solution to the adjoint equations vA = (y) over bar. A more sophisticated construction applies in the nonlinear case. We apply these techniques to obtain automatic numerical error estimates for calculated function values. These error estimates include the effects of inaccurate equation solution as well as rounding error. Our basic techniques can be generalized to functions which contain several (linear or nonlinear) implicit functions in their construction, either serially or nested. In the case of scalar-valued target functions that include equation solution as part of their construction, our algorithms involve at most the same order of computational effort as the computation of the target function value, regardless of the number of independent variables or the size of the systems of equations.