A software package for the numerical integration of ODEs by means of high-order Taylor methods. This paper revisits the Taylor method for the numerical integration of initial value problems of ordinary differential equations (ODEs). The main goal is to present a computer program that outputs a specific numerical integrator for a given set of ODEs. The generated code includes a function to compute the jet of derivatives of the solution up to a given order plus adaptive selection of order and step size at run time. The package provides support for several extended precision arithmetics, including user-defined types. par The authors discuss the performance of the resulting integrator in some examples, showing that it is very competitive in many situations. This is especially true for integrations that require extended precision arithmetic. The main drawback is that the Taylor method is an explicit method, so it has all the limitations of these kind of schemes. For instance, it is not suitable for stiff systems.

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  1. Ollé, Merce; Rodríguez, Oscar; Soler, Jaume: Transit regions and ejection/collision orbits in the RTBP (2021)
  2. Castejón, Oriol; Guillamon, Antoni: Phase-amplitude dynamics in terms of extended response functions: invariant curves and Arnold tongues (2020)
  3. Dolgakov, I.; Pavlov, D.: Landau: a language for dynamical systems with automatic differentiation (2020)
  4. Jorba, Àngel; Jorba-Cuscó, Marc; Rosales, José J.: The vicinity of the Earth-Moon (L_1) point in the bicircular problem (2020)
  5. Jorba, Àngel; Nicolás, Begoña: Transport and invariant manifolds near L(_3) in the Earth-Moon bicircular model (2020)
  6. Ollé, Merce; Rodríguez, Oscar; Soler, James: Analytical and numerical results on families of (n)-ejection-collision orbits in the RTBP (2020)
  7. Pérez-Cervera, Alberto; M-Seara, Tere; Huguet, Gemma: Global phase-amplitude description of oscillatory dynamics via the parameterization method (2020)
  8. van den Berg, Jan Bouwe; Sheombarsing, Ray: Validated computations for connecting orbits in polynomial vector fields (2020)
  9. Belbruno, Edward; Frauenfelder, Urs; van Koert, Otto: A family of periodic orbits in the three-dimensional lunar problem (2019)
  10. Burgos-García, Jaime; Lessard, Jean-Philippe; James, J. D. Mireles: Spatial periodic orbits in the equilateral circular restricted four-body problem: computer-assisted proofs of existence (2019)
  11. Gómez-Serrano, Javier: Computer-assisted proofs in PDE: a survey (2019)
  12. Perdomo, Oscar M.: The round Taylor method (2019)
  13. Schaumburg, Herman D.; Al Marzouk, Afnan; Erdelyi, Bela: Picard iteration-based variable-order integrator with dense output employing algorithmic differentiation (2019)
  14. Vasile, Massimiliano; Absil, Carlos Ortega; Riccardi, Annalisa: Set propagation in dynamical systems with generalised polynomial algebra and its computational complexity (2019)
  15. Walawska, Irmina; Wilczak, Daniel: Validated numerics for period-tupling and touch-and-go bifurcations of symmetric periodic orbits in reversible systems (2019)
  16. Al Khawaja, U.; Al-Mdallal, Qasem M.: Convergent power series of (\operatornamesech(x)) and solutions to nonlinear differential equations (2018)
  17. Al Sakkaf, Laila Y.; Al-Mdallal, Qasem M.; Al Khawaja, U.: A numerical algorithm for solving higher-order nonlinear BVPs with an application on fluid flow over a shrinking permeable infinite long cylinder (2018)
  18. Baeza, A.; Boscarino, S.; Mulet, P.; Russo, G.; Zorío, D.: Reprint of: “Approximate Taylor methods for ODEs” (2018)
  19. Breden, Maxime; Lessard, Jean-Philippe: Polynomial interpolation and a priori bootstrap for computer-assisted proofs in nonlinear ODEs (2018)
  20. Castelli, Roberto; Lessard, Jean-Philippe; James, Jason D. Mireles: Parameterization of invariant manifolds for periodic orbits. II: A posteriori analysis and computer assisted error bounds (2018)

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