Normal Form Defect

(2D Projection)


A six dimensional function with up to a million terms that all cancel nearly completely.


Finding exact bounds of the function allows prediction of long-term stability of particle accelerators and other dynamical systems.


Solved rigorously for the first time with global optimization techniques based on Taylor Model methods.


Recent Developments

Upcoming: Version 10 of COSY will be released, featuring major enhancements in polynomial engines, including built-in high precision in TM and DA data types and rigorous, non-interval based treatment of errors.

Summer 2013: Numerous updates related to 3D graphics drivers, including new interfaces to OpenGL, significant updates to pdf, ps and various other drivers, as well as low level support for color shading and lighting.

June 2011: Version 9.1 of COSY is released. New features include an interface to JAVA, and one of the first applications of it, native support of GUIs from within COSY. Also included are tools that enable parallelization from within COSY based on MPI.


January 2010: Over the last three years, COSY user registrations have increased to about 200 per year, pushing the total number of registrations over 2000.

September 2008: Kyoko Makino and Martin Berz were awarded the R.E. Moore Prize for the development of some of the newest additions to COSY, which allow the rigorous treatment of remainder bounds of Taylor expansion, in particular those of the solutions of flows or maps of ODEs.

April 2008: Martin Berz was awarded an Honorary Doctorate from Saint Petersburg University in Russia “For His Eminent Contributions to the Solution of Important Problems of Applied Mathematics”. Much of this work forms the mathematical basis of the algorithms in COSY.

August 2007: Version 9.0 of COSY is released. New features include enhancements to the treatment of remainder bounds, many updates to existing data types, and external tools for the computation of field expansions from surface data.


What Is COSY?

COSY is a system for the use of various advanced concepts of modern scientific computing. COSY currently has more than 2000 registered users and has been extensively cross-checked and  verified. The COSY system consists of the following parts.

1) A collection of advanced highly optimized Data Types. In particular:

- The Differential Algebraic types for high-order multivariate study of ODEs, Flows, and PDEs. It also allow high-order multivariate automatic differentiation.

- The Taylor Model type for rigorous high-order computing with often far-reaching suppression of dependency. Tools for range bounding, derivative-based box rejection, constraint satisfaction, ODEs and PDEs.

2) The COSYScript environment for the use of these types. It is object oriented and supports polymorphism, has a compact and simple syntax, and is compiled and executed on the fly. It has built-in optimization and parallelization constructs, and is used for high turn around simulation.

3) Interfaces for C, F77, C++ and F90 to seamlessly use the types in external programs.

4) Various application packages using the COSY data types, including beam physics.


To download the source code and installers for most common platforms, obtain a personal license asserting indemnification of MSU and non-commercial use; commercial licenses are also available upon request. Print the License Agreement on the official letterhead of your organization, and fax a signed copy of that document to +1-815- 301-9725. (These instructions will be repeated after using the online registration form.)


Download COSY Installers and Sources

COSY INFINITY is available for
download by registered users. If you have not registered but want to obtain the code, register now. If you are a registered user but have forgotten your password or want a customized password, please contact us at


The Beam Physics Package


COSY INFINITY is an arbitrary order beam dynamics simulation and analysis code. It allows the study of accelerator lattices, spectrographs, beamlines, electron microscopes, and many other devices. It can determine high-order maps of combinations of particle optical elements of arbitrary field configurations. The elements can either be based on a large library of existing elements with realistic field configurations including fringe fields, or described in detail by measured data.


Analysis options include computation of high-order nonlinearities; analysis of properties of repetitive motion via chromaticities, normal form analysis, and symplectic tracking; analysis of single-pass systems resolutions, reconstructive aberration correction, and consideration of detector errors; and analysis of spin dynamics via computation of spin maps, spin normal form and spin tracking.


Beam Physics Input Converters

There are various tools to convert input files from the MAD, SXF, GICOSY and OptiM formats into COSY-compatible input directly on the web.

MAD Converter  SXF Converter  GICOSY Converter  OptiM Converter




Michigan State University - B149 Biomedical & Physical Sciences Building, East Lansing, MI 48824

Web page maintained by Kyoko Makino