Abstract
A new method to decide the invertibility of a given high-dimensional function over a domain is presented. The problem arises in the field of verified solution of differential algebraic equations (DAEs) related to the need to perform projections of certain constraint manifolds over large domains. The question of invertibility is reduced to a verified linear algebra problem involving first partials of the function under consideration. Different from conventional approaches, the elements of the resulting matrices are Taylor models for the derivatives of the functions.
The linear algebra problem is solved based on Taylor model methods, and it will be shown the method is able to decide invertibility with a conciseness that often goes substantially beyond what can be obtained with other interval methods. The theory of the approach is presented. Comparisons with three other interval-based methods are performed for practical examples, illustrating the applicability of the new method.
Similar content being viewed by others
References
Abraham, R., Marsden, J. E., and Ratiu, T.: Manifolds, Tensor Analysis, and Applications, Applied Mathematical Sciences 75, Springer Verlag, second edition, 1988.
Berz, M.: Differential Algebras with Remainder and Rigorous Proofs of Long-Term Stability, in: Fourth Computational Accelerator Physics Conference, AIP Conference Proceedings, vol. 391, 1996, p. 221.
Berz, M.: Modern Map Methods in Particle Beam Physics, Academic Press, San Diego, 1999.
Berz, M. and Hoffstätter, G.: Computation and Application of Taylor Polynomials with Interval Remainder Bounds, Reliable Computing 4 (1998), pp. 83–97.
Berz, M. and Hoffstätter, G.: Exact Bounds of the Long Term Stability of Weakly Nonlinear Systems Applied to the Design of Large Storage Rings, Interval Computations 2 (1994), pp. 68–89.
Berz, M. and Hoefkens, J.: Verified High-Order Inversion of Functional Dependencies and Interval Newton Methods, Reliable Computing 7 (5) (2001), pp. 379–398.
Berz, M. and Makino, K.: New Methods for High-Dimensional Verified Quadrature, Reliable Computing 5 (1) (1999), pp. 13–22.
Berz, M. and Makino, K.: Verified Integration of ODEs and Flows with Differential Algebraic Methods on Taylor Models, Reliable Computing 4 (4) (1998), pp 361–369.
Makino, K.: Rigorous Analysis of Nonlinear Motion in Particle Accelerators, PhD thesis, Michigan State University, East Lansing, Michigan, 1998, also MSUCL-1093.
Makino, K. and Berz, M.: Efficient Control of the Dependency Problem Based on Taylor Model Methods, Reliable Computing 5 (1) (1999), pp. 3–12.
Makino, K. and Berz, M.: Remainder Differential Algebras and Their Applications, in: Berz, M., Bischof, C., Corliss, G., and Griewank, A. (eds), Computational Differentiation: Techniques, Applications, and Tools, SIAM, Philadelphia, 1996, pp. 63–74.
Neumaier, A.: Interval Methods for Systems of Equations, Cambridge University Press, 1990.
Ortega, J. M. and Rheinboldt, W. C.: Iterative Solution of Nonlinear Equations in Several Variables, Computer Science and Applied Mathematics, Academic Press, New York and London, 1970.
Rump, S. M.: Ill-Conditioned Matrices Are Componentwise Near to Singularity, SIAM Review 41 (1) (1999), pp. 102–112.
Rump, S. M.: Private Communication, 2000.
Rump, S. M.: Validated Solution of Large Linear Systems, in: Albrecht, R., Alefeld, G., and Stetter, H. J. (eds), Computing Supplement 9, Springer-Verlag, Wien, 1993, pp. 191–212.
Rump, S. M.: Verification Methods forDense and Sparse Systems of Equations, in: Herzberger, J. (ed.), Topics in Validated Computations, North-Holland, Amsterdam, 1994, pp. 63–135.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hoefkens, J., Berz, M. Verification of Invertibility of Complicated Functions over Large Domains. Reliable Computing 8, 67–82 (2002). https://doi.org/10.1023/A:1014789619479
Issue Date:
DOI: https://doi.org/10.1023/A:1014789619479