A list of selected older papers on Taylor Models up to 2003. To avoid duplication of bookkeeping, copies of newer papers on Taylor Models are available only at our main reprint server.
Suppression of the Wrapping Effect by Taylor Model- based Validated Integrators K. Makino and M. Berz, MSU REPORT MSUHEP 40910
Study of Validated Inclusion Functions For a Test Example with Variable Dependency Y. Kim and M. Berz, Michigan State University Report MSU HEP 31909 (2003)
Taylor Models and Floating-Point Arithmetic: Proof that Arithmetic Operations are Validated in COSY N. Revol and K. Makino and M. Berz, Journal of Logic and Algebraic Programming, in print(2004), University of Lyon LIP Report RR 2003-11, MSU HEP report 30212
Testing COSY's INSRF G. Corliss, MSU Report MSUHEP 31228 (2003)
Interval Testing Strategies Applied to COSY’s Interval and Taylor Model Arithmetic G. F. Corliss and Jun Yu, Numerical Software with Result Verification, R. Alt et al. (Eds.), Springer LNCS 2991 (2004) 91–106, MSU Report MSUHEP
Taylor Models and Other Validated Functional Inclusion Methods K. Makino and M. Berz, International Journal of Pure and Applied Mathematics, 4(4), 379-456 (2003)
Verification of Invertibility of Complicated Functions over Large Domains J. Hoefkens and M. Berz, Reliable Computing, 8(1), 1-16 (2002).
The Method of Shrink Wrapping For the Validated Solution of ODEs, K. Makino and M. Berz, Michigan State University Report MSU HEP 020510 (2002)
Implementation of Taylor Model Arithmetic, K. Makino and M. Berz, Michigan State University Report MSU HEP 020511 (2002)
Verified Integration of Dynamics in the Solar System M. Berz, K. Makino and J. Hoefkens, Nonlinear Analysis: Theory, Methods & Applications, 47, 179-190 (2001).
Verified High-Order Integration of DAEs and Higher-order ODEs J. Hoefkens, M. Berz and K. Makino, in "Scientific Computing, Validated Numerics and Interval Methods, W. Kraemer and J. W. v. Gudenberg (Eds.)" (Kluwer Academic Publishers, Dordrecht, Netherlands, 2001).
Verified High-Order Inversion of Functional Dependencies and Interval Newton Methods M. Berz and J. Hoefkens, Reliable Computing, 7(5), 379-398 (2001).
Efficient High-Order Methods for ODEs and DAEs J. Hoefkens, M. Berz and K. Makino, in "Automatic Differentiation: From Simulation to Optimization, G. Corliss, C. Faure, A. Griewank, L. Hascoet, and U. Naumann (Eds.)" (Springer, New York, 2001).
New Applications of Taylor Model Methods K. Makino and M. Berz, in Automatic Differentiation: From Simulation to Optimization, G. Corliss, C. Faure, A. Griewank, L. Hascoet, and U. Naumann (Eds.) (Springer, New York, 2001).
Towards a Universal Data Type for Scientific Computing M. Berz, in Automatic Differentiation: From Simulation to Optimization, G. Corliss, C. Faure, A. Griewank, L. Hascoet, and U. Naumann (Eds.) (Springer, New York, 2001).
Constructive Generation and Verification of Lyapunov Functions around Fixed Points of Nonlinear Dynamical Systems M. Berz and K. Makino, International Journal of Computer Research, in print (2001).
Higher Order Verified Inclusions of Multidimensional Systems by Taylor Models K. Makino and M. Berz, Nonlinear Analysis, 47, 3503-3514 (2001).
Differential Algebraic Techniques M. Berz, in "Handbook of Accelerator Physics and Engineering, M. Tigner, A.Chao (Eds.)" (World Scientific, 1998)
New Methods for High-Dimensional Verified Quadrature M. Berz and K. Makino, Reliable Computing, 5, 13-22 (1999).
Efficient Control of the Dependency Problem Based on Taylor Model Methods K. Makino and M. Berz, Reliable Computing, 5, 3-12 (1999)
Verified Integration of ODEs and Flows using Differential Algebraic Methods on High-Order Taylor Models M. Berz and K. Makino, Reliable Computing, 4, 361-369 (1998)
The use of Taylor Models is an approach to rigorously treat certain problems in scientific computing and numerical mathematics. It rests on the representation of any function given as a finite computer code list of basic binary operations and intrinsics in terms of a Taylor polynomial of a certain order n plus a rigorous interval enclosure of the Taylor remainder error. This bound for the approximation error is computed alongside with the Taylor coefficients using interval methods, and its width scales with the (n+1)st order of the domain size of the polynomial part.
Compared to other rigorous methods, for sufficiently small domains this leads to quite significant increases of sharpness in the remainder and avoids much of the dependency problem that commonly affects verified tools. The arithmetic and the computational implementation take into account all round-off and threshold cut-off errors due to finite precision of floating point arithmetic, so that the remainder enclosure is mathematically rigorous.
The most important characteristics of Taylor Models include
1) High-order scaling property of the remainder bound interval
2) Alleviation of the cancellation problem
3) Comparably inexpensive, i.e. sub-exponential, extension to higher-dimensional problems
4) The ability to determine high-order Taylor models for inverses and implicit equations
5) Direct availability of the antiderivative as an intrinsic, allowing treatment of ODEs and PDEs
Important validated applications include
1) Solution of ODEs under far-reaching avoidance of the wrapping effect
2) Global constrained optimization via Taylor model relaxation and under suppression of dependency and with modern domain reduction techniques
3) Solution of implicit algebraic equations and DAEs
4) Solution of various classes of PDEs
The methods were originally developed for a particular problem that exhibits all the above difficulties, and simultaneously requires the sharp validated solution of ODEs and validated global optimization. The problem of interest is the treatment of Normal Form Defect Functions which allow the estimation of long term stability of dynamics around elliptic fixed points like those occurring in modern particle accelerators.
They rest on the computation of the flow of the ODE, expressing it as a Poincare map, and measuring their non-integrability via the explicit construction of approximate invariants through normal form methods. These functions showcase an extremely complicated behavior with code lists consisting of several thousand terms and a significant cancellation problem.
Aside from these specific problems, the work of the MSU group is directed at developing general tools for verified integration of ODEs, verified global optimization, solution of implicit problems, verified PDEs, and improving the computational performance of self-verified methods in general. The tools are written in the COSY environment.
Taylor Models and