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Algorithms for Higher Order Automatic Differentiation in Many Variables with Applications to Beam Physics

Abstract

Efficient algorithms for automatic differentiation with several variables and high orders are presented. The algorithms are geared towards sparse vectors, which is particularly important in this case and allows significant savings in computer time. Besides the mere computation of derivatives, algorithms for the efficient composition and inversion of functions with sparse derivatives are discussed.

The algorithms are implemented in a FORTRAN library. The library can be utilized by a precompiler that transforms FORTRAN into code to perform the desired automatic differentiation task. The precompiler allows passing of variables into subroutines and functions and allows the user to provide functions. Besides the use with the precompiler, the routines can be accessed from a dedicated language environment. The language has the flavor of PASCAL, but provides object oriented features and nonlinear optimization at the language level.

The tools have been used in numerous cases for the computation and correction of aberrations of beam physics systems and the simulation and analysis of nonlinear dynamics problems, including the simulation of large particle accelerators.

M. Berz, in: "Automatic Differentiation of Algorithms: Theory, Implementation and Application" (1991) SIAM

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