The author introduces 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 \(Ay= b\) for \(y\) corresponds to the adjoint operations \(\overline b:=\overline b+v\) and \(\overline A:=\overline A-yv\) where \(v\) is the solution to the adjoint equations \(vA=\overline y\). A more sophisticated construction applies in the nonlinear case. The author applies then techniques to obtain automatic numerical error estimates for calculated function values. These error estimates include the effects of inaccurate equation solutions as well as rounding errors.
This basic technique 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 solutions as part of their construction, the algorithm involves 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.