Automatic Hessians by reverse accumulation

Let n be the number of independent variables of a function f,and let W and S respectively be the time and space bounds forthe joint evaluation of {f, f} using automatic differentiationwith reverse accumulation. In this note, we examine an extensionof the technique of reverse accumulation which allows the automaticextraction of the Hessian of f. The method allows the parallelevaluation of all rows of the Hessian matrix in about 2W timeunits and 3S space units on each of n processors, or sequentialrow-by-row evaluation in about 2nW time units and 3S space unitson a single processor. The approach described here is intendedfor use with operator overloading (for example in Ada) and allowsthe conventional coding of the target function f.