Frequently of use in optimization problems, automatic differentiation may be used to generate Taylor coefficients. Specialized software tools generate Taylor series approximations, one term at a time, more efficiently than the general AD software used to compute (partial) derivatives. Through the use of operator overloading, these tools provide a relatively easy-to-use interface that minimizes the complications of working with both point and interval operations.
Introduction
First, we briefly survey the tools of automatic differentiation and operator overloading used to compute point- and interval-valued Taylor coefficients. We assume that f is an analytic function f : R → R. Automatic differentiation (AD or computational differentiation) is the process of computing the derivatives of a function f at a point t = t 0 by applying rules of calculus for differentiation [9], [10], [17], [18]. One way to implement AD uses overloaded operators.
Operator Overloading...
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References
Benary, J.: ‘Parallelism in the reverse mode’, in M. Berz, Ch. Bischof, G. Corliss, and A. Griewank (eds.): Computational Differentiation: Techniques, Applications, and Tools, SIAM, 1996, pp. 137–147.
Berz, M., Bischof, Ch., Corliss, G., and Griewank, A. (eds.): Computational differentiation: Techniques, applications, and tools, SIAM, 1996.
Bischof, Ch., Carle, A., Khademi, P.M., Mauer, A., and Hovland, P.: ‘ADIFOR: 2.0 user’s guide’, Techn. Memorandum Math. and Computer Sci. Div. Argonne Nat. Lab.ANL/MCS-TM-192 (1994).
Bischof, Ch., and Haghighat, M.R.: ‘Hierarchical approaches to automatic differentiation’, in M. Berz, Ch. Bischof, G. Corliss, and A. Griewank (eds.): Computational Differentiation: Techniques, Applications, and Tools, SIAM, 1996, pp. 83–94.
Chang, Y.F., and Corliss, G.F.: ‘ATOMFT: Solving ODEs and DAEs using Taylor series’, Comput. Math. Appl.28 (1994), 209–233.
Corliss, G.F.: ‘Overloading point and interval Taylor operators’, in A. Griewank, and G.F. Corliss (eds.): Automatic Differentiation of Algorithms: Theory, Implementation, and Application, SIAM, 1991, pp. 139–146.
Corliss, G.F., and Rall, L.B.: ‘Adaptive, self-validating quadrature’, SIAM J. Sci. Statist. Comput.8, no. 5 (1987), 831–847.
Griewank, A.: ‘On automatic differentiation’, in M. Iri, and K. Tanabe (eds.): Mathematical Programming: Recent Developments and Applications, Kluwer Acad. Publ., 1989, pp. 83–108.
Griewank, A.: ‘The chain rule revisited in scientific computing’, SIAM News24, no. 3 (1991), 20 Also: Issue 4, page 8.
Griewank, A., and Corliss, G.F. (eds.): Automatic differentiation of algorithms: Theory, implementation, and application, SIAM, 1991.
Griewank, A., Juedes, D., and Utke, J.: ‘ADOL-C: A package for the automatic differentiation of algorithms written in C/C++’, ACM Trans. Math. Software22, no. 2 (1996), 131–167.
Griewank, A., and Reese, S.: ‘On the calculation of Jacobian matrices by the Markowitz rule’, in A. Griewank, and G.F. Corliss (eds.): Automatic Differentiation of Algorithms: Theory, Implementation, and Application, SIAM, 1991, pp. 126–135.
Kearfott, R.B.: ‘A Fortran 90 environment for research and prototyping of enclosure algorithms for nonlinear equations and global optimization’, ACM Trans. Math. Software21, no. 1 (1995), 63–78.
Kubota, K.: ‘PADRE2-Fortran precompiler for automatic differentiation and estimates of rounding error’, in M. Berz, Ch. Bischof, G. Corliss, and A. Griewank (eds.): Computational Differentiation: Techniques, Applications, and Tools, SIAM, 1996, 367–374.
Lohner, R.J.: ‘Enclosing the solutions of ordinary initial and boundary value problems’, in E.W. Kaucher, U.W. Kulisch, and C. Ullrich (eds.): Computer Arithmetic: Scientific Computation and Programming Languages, Wiley and Teubner, 1987, pp. 255–286.
Pryce, J.D., and Reid, J.K.: ‘AD01: A Fortran 90 code for automatic differentiation’, Techn. Report Rutherford-Appleton Lab.RAL-TR-97xxx (1997).
Rall, L.B.: Automatic differentiation: Techniques and applications, Vol. 120 of Lecture Notes Computer Sci., Springer, 1981.
Rall, L.B., and Corliss, G.F.: ‘An introduction to automatic differentiation’, in M. Berz, Ch. Bischof, G. Corliss, and A. Griewank (eds.): Computational Differentiation: Techniques, Applications, and Tools, SIAM, 1996, pp. 1–17.
Rostaing, N., Dalmas, S., and Galligo, A.: ‘Automatic differentiation in Odyssée’, Tellus45A (1993), 558–568.
Shiriaev, D.: ‘ADOL-F: Automatic differentiation of Fortran codes’, in M. Berz, Ch. Bischof, G. Corliss, and A. Griewank (eds.): Computational Differentiation: Techniques, Applications, and Tools, SIAM, 1996, pp. 375–384.
Yang, W., and Corliss, G.: ‘Bibliography of computational differentiation’, in M. Berz, Ch. Bischof, G. Corliss, and A. Griewank (eds.): Computational Differentiation: Techniques, Applications, and Tools, SIAM, 1996, pp. 393–418.
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Walters, J.B., Corliss, G.F. (2001). Automatic Differentiation: Point and Interval Taylor Operators . In: Floudas, C.A., Pardalos, P.M. (eds) Encyclopedia of Optimization. Springer, Boston, MA. https://doi.org/10.1007/0-306-48332-7_22
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DOI: https://doi.org/10.1007/0-306-48332-7_22
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