TIDES: a Taylor Integrator for Differential Equations

TIDES is a free software to integrate numerically Ordinary Differential Equations by using a Taylor Series method. This software, developed by A. Abad, R Barrio, F. Blesa, M. Rodríguez, (GME), consists on a C (Fortran) library, libTIDES, and a Mathematica package, MathTIDES. (MathTIDES requires Mathematica version >= 7.0)

The main features of TIDES are the following:

• TIDES permits to integrate numerically ODE problems with multiple precision, that means that you can solve ODE problems up to any precision level in a reasonable computer time.
• TIDES may solve directly sensitivity equations with respect to initial conditions or parameters up to any any order.
• TIDES integrates by using the Taylor Series method with an optimized variable-stepsize and variable-order formulation, and extended formulas for variational equations.
• The software has been done to be extremely easy to use: with MathTIDES we write, in a natural way, the ODE and their parameters, together with the parameters of the integration. Then, MathTIDES writes the C (Fortran) code, that, compiled and linked with libTIDES, integrates the ODE.
• The derivatives and partial derivatives are obtained by using Automatic Differentiation (AD) techniques.
• There are four different versions of the code generated by MathTIDES:
  • Two minimal versions (in C or Fortran language). These are faster than standard versions and easy to use as subprograms, but they do not have most of the possibilities of the standard versions.
  • A standard version (only in C language) for double precision computation.
  • A standard version (only in C language) for multiple precision computations. This version uses MPFR, and GMP libraries to integrate ODEs with any arbitrary precision.
• MathTIDES writes automatically the code to compute partial derivatives of the solution of the ODE with respect to any variable or parameter (using AD and avoiding the use of any variational equation or sensitivity with respect to the parameters).
• TIDES may detect events of ODEs, i. e. points where a function of the solution of the ODE satisfies an event function, like it becomes zero or reaches an extremum.

References

• A. Abad, R. Barrio, F. Blesa, M. Rodríguez, 2012. Algorithm 924: TIDES, a Taylor series Integrator for Differential EquationS, ACM Transactions on Mathematical Software. 39, no. 1, article 5. (Use this reference for TIDES citation.)
  
  
• A. Abad, R. Barrio, F. Blesa, M. Rodríguez, 2011. TIDES tutorial: Integrating ODEs by using the Taylor Series Method., Monografías de la Academia de Ciencias de la Universidad de Zaragoza. 36 pp. 1 - 116.
• R. Barrio, 2005. Performance of the Taylor series method for ODEs/DAEs, Applied Mathematics and Computation 163, no. 2, 525--545.
• R. Barrio, F. Blesa, M. Lara, 2005. VSVO formulation of the Taylor method for the numerical solution of ODEs, Computers and Mathematics and Applications 50 (1-2) 93--111.
• R. Barrio, 2006. Sensitivity analysis of ODE's/DAE's using the Taylor series method, SIAM J. on Scientific Computing 27 (6) 1929--1947.

Related references

The following papers use TIDES. If you use TIDES, please send a reference of your paper to tides.taylor@gmail.com and it will be added to this list

• A. Abad, R. Barrio and A. Dena, 2011. Computing periodic orbits with arbitrary precision, Physical Review E. 84 - 016701.
  In this paper we compute four periodic orbits with 1000 digits of precision. The periodic orbits and the programs to check that they are really periodic orbits may be found here
• R. Barrio, F. Blesa, M. Lara, M. 2003. High-precision numerical solution of ODE with high-order Taylor methods in parallel. Analytic and numerical techniques in orbital dynamics (in Spanish), Monografías de la Real Academia de Ciencias de Zaragoza 22 pp 67-74.
• R. Barrio,F. Blesa, S. Serrano, 2010. Bifurcations and chaos in Hamiltonian systems. International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 20, no. 5, pp 1293-1319.
• R. Barrio, M. Rodríguez, A. Abad, S. Serrano, 2011. Uncertainty propagation or box propagation, Mathematical and Computer Modeling. 54 (11-12) 2602 - 2615.
• R. Barrio, M. Rodríguez, A. Abad, F. Blesa, 2011. Breaking the limits: the Taylor series method, Applied Mathematics and Computation 217 (20) 7940 - 7954.
• R. Barrio, A. Shilnikov, 2011. Parameter-sweeping techniques for temporal dynamics of neuronal systems: case study of Hindmarsh-Rose model. The Journal of Mathematical Neuroscience. 1, Art. 6, 20 pp.
• R. Barrio, A. Shilnikov, L. Shilnikov, 2012. Kneadings, symbolic dynamics and painting Lorenz chaos. International Journal of Bifurcations and Chaos 22, no. 4, 1230016.
• A. Dena, M. Rodríguez, S. Serrano, R. Barrio. 2012 High-precision continuation of periodic orbits. Abstract and Applied Analysis 2012, Art. ID 716024, 12pp.
• E. Gerlach, Ch. Skokos, 2011. Comparing the efficiency of numerical techniques for the integration of variational equations. in "Dynamical Systems, Differential Equations and Applications", Discr.Cont. Dyn. Sys.-Supp. 2011 (dedicated to the 8th AIMS Conference), eds. Feng W., Feng Z., Grasselli M., Ibragimov A., Lu X, Siegmund S. & Voirt J., AIMS, 475-484.
• E. Gerlach, S. Eggl and Ch. Skokos, 2012. Efficient integration of the variational equations of multi-dimensional Hamiltonian systems: Application to the Fermi-Pasta-Ulam lattice. International Journal of Bifurcations and Chaos 22, no. 9, 1250216.
• M. Rodríguez, R. Barrio, 2012. Reducing rounding errors and achieving Brouwer's law with Taylor Series Method. Applied Numerical Mathematics, 62, no. 8, pp 1014-1024.

How to obtain the software

To obtain the software fill the following form. The data of the form is just to create a data base of the people interested in the software and to communicate new versions and publications. When we receive the form we will send the software as soon as possible. If you have problems to send the form load and fill this file and send it by e-mail to tides.taylor@gmail.com.

Since April 2012 we distribute version 2.0 of TIDES. If you are a previously registered user and you have not received this version please ask us for it.