The Parker–Sochacki method is a mathematical technique used for solving systems of ordinary differential equations (ODEs) by computing Taylor series solutions iteratively. Developed by George E. Parker and James A. Sochacki in the 1990s, the method leverages Picard iteration and power series expansions to generate high-order approximations to solutions of initial value problems. It is particularly notable for its ability to handle nonlinear systems, often by introducing auxiliary variables to reformulate nonlinear terms into quadratic or polynomial forms amenable to recursive coefficient computation.
The method is deterministic and provides guaranteed error bounds over small time steps, making it useful in applications requiring high precision, such as celestial mechanics and validated numerics. One of its advantages is that it enables adaptive step-size control based on the convergence of the Taylor series.
The Parker–Sochacki method has been applied in fields including astrophysics, computational biology, and numerical analysis. It is compatible with interval arithmetic to produce verified computations, which helps ensure the reliability of the numerical results.
Accurate information on the method’s original publication dates and detailed derivation can be found in peer-reviewed journals such as "Neural, Parallel & Scientific Computation" and related applied mathematics literature.