Fixed versus variable order Runge-Kutta
ACM Transactions on Mathematical Software (TOMS) - The MIT Press scientific computation series
Two FORTRAN packages for assessing initial value methods
ACM Transactions on Mathematical Software (TOMS)
An efficient method for the numerical evaluation of partial derivatives of arbitrary order
ACM Transactions on Mathematical Software (TOMS)
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
A Fortran 90-based multiprecision system
ACM Transactions on Mathematical Software (TOMS)
The Mathematica book (4th edition)
The Mathematica book (4th edition)
Evaluating higher derivative tensors by forward propagation of univariate Taylor series
Mathematics of Computation
On Taylor Series and Stiff Equations
ACM Transactions on Mathematical Software (TOMS)
Solving Ordinary Differential Equations Using Taylor Series
ACM Transactions on Mathematical Software (TOMS)
Pracniques: further remarks on reducing truncation errors
Communications of the ACM
Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations
Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations
Accuracy and Stability of Numerical Algorithms
Accuracy and Stability of Numerical Algorithms
High-Order Stiff ODE Solvers via Automatic Differentiation and Rational Prediction
WNAA '96 Proceedings of the First International Workshop on Numerical Analysis and Its Applications
SUNDIALS: Suite of nonlinear and differential/algebraic equation solvers
ACM Transactions on Mathematical Software (TOMS) - Special issue on the Advanced CompuTational Software (ACTS) Collection
Sensitivity Analysis of ODES/DAES Using the Taylor Series Method
SIAM Journal on Scientific Computing
N-body simulations: The performance of some integrators
ACM Transactions on Mathematical Software (TOMS)
Robust and reliable defect control for Runge-Kutta methods
ACM Transactions on Mathematical Software (TOMS)
MPFR: A multiple-precision binary floating-point library with correct rounding
ACM Transactions on Mathematical Software (TOMS)
Introduction to Interval Analysis
Introduction to Interval Analysis
VSVO formulation of the taylor method for the numerical solution of ODEs
Computers & Mathematics with Applications
Choosing a stepsize for Taylor series methods for solving ODE'S
Journal of Computational and Applied Mathematics
Reducing rounding errors and achieving Brouwer's law with Taylor Series Method
Applied Numerical Mathematics
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This article introduces the software package TIDES and revisits the use of the Taylor series method for the numerical integration of ODEs. The package TIDES provides an easy-to-use interface for standard double precision integrations, but also for quadruple precision and multiple precision integrations. The motivation for the development of this package is that more and more scientific disciplines need very high precision solution of ODEs, and a standard ODE method is not able to reach these precision levels. The TIDES package combines a preprocessor step in Mathematica that generates Fortran or C programs with a library in C. Another capability of TIDES is the direct solution of sensitivities of the solution of ODE systems, which means that we can compute the solution of variational equations up to any order without formulating them explicitly. Different options of the software are discussed, and finally it is compared with other well-known available methods, as well as with different options of TIDES. From the numerical tests, TIDES is competitive, both in speed and accuracy, with standard methods, but it also provides new capabilities.