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This article presents the first extended set of results from ELIAD, a source-transformation implementation of the vertex-elimination AutomaticDifferentiation approach to calculating the Jacobians of functions defined byFortran code (Griewank and Reese, Automatic Differentiation of Algorithms:Theory, Implementation, and Application, 1991, pp. 126-135). We introduce thenecessary theory in terms of well known algorithms of numerical linear algebraapplied to the linear, extended Jacobian system that prescribes the relationshipbetween the derivatives of all variables in the function code. Using an example,we highlight the potential for numerical instability in vertex-elimination. Wedescribe the source transformation implementation of our tool ELIAD and presentresults from five test cases, four of which are taken from the MINPACK-2 collection (Averick et al, Report ANL/MCS-TM-150, 1692) and for which hand-coded Jacobian codes are available. On five computer/compiler platforms, we showthat the Jacobian code obtained by ELIAD is as efficient as hand-coded Jacobiancode. It is also between 2 to 20 times more efficient than that produced bycurrent, state of the art, Automatic Differentiation tools even when such toolsmake use of sophisticated techniques such as sparse Jacobian compression. Wedemonstrate the effectiveness of reverse-ordered pre-elimination from the(successively updated) extended Jacobian system of all intermediate variablesused once. Thereafter, the monotonic forward/reverse ordered eliminations of allother intermediates is shown to be very efficient. On only one test case wereorderings determined by the Markowitz or related VLR heuristics found superior.A re-ordering of the statements of the Jacobian code, with the aim of reducingreads and writes of data from cache to registers, was found to have mixedeffects but could be very beneficial

Jacobian code generated by source transformation and vertex elimination can be as efficient as hand-coding