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higher order permits a longer time step without loss of accuracy, which improves efficiency.) The order and step size can be easily changed from one step to the next. It is possible to calculate a guaranteed error bound on the solution. Arbitrary precision floating point libraries allow this method to compute arbitrarily accurate solutions.
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error bound for a single step with the Parker–Sochacki method. This allows a Parker–Sochacki program to calculate the step size that guarantees that the error is below any non-zero given tolerance. Using this calculated step size with an error tolerance of less than half of the machine epsilon yields
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With the Parker–Sochacki method, information between integration steps is developed at high order. As the Parker–Sochacki method integrates, the program can be designed to save the power series coefficients that provide a smooth solution between points in time. The coefficients can be saved and used
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The method requires only addition, subtraction, and multiplication, making it very convenient for high-speed computation. (The only divisions are inverses of small integers, which can be precomputed.) Use of a high order—calculating many coefficients of the power series—is convenient. (Typically a
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The end result is a high order piecewise solution to the original ODE problem. The order of the solution desired is an adjustable variable in the program that can change between steps. The order of the solution is only limited by the floating point representation on the machine running the program.
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Most methods for numerically solving ODEs require only the evaluation of derivatives for chosen values of the variables, so systems like MATLAB include implementations of several methods all sharing the same calling sequence. Users can try different methods by simply changing the name of the
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so that polynomial evaluation provides the high order solution between steps. With most other classical integration methods, one would have to resort to interpolation to get information between integration steps, leading to an increase of error.
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If the ODEs do not have the required form, it is nearly always possible to find an expanded set of equations that do have the required form, such that a subset of the solution is a solution of the original
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And in some cases can be either extended by using arbitrary precision floating point numbers, or for special cases by finding solution with only integer or rational coefficients.
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Several coefficients of the power series are calculated in turn, a time step is chosen, the series is evaluated at that time, and the process repeats.
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function called. The Parker–Sochacki method requires more work to put the equations into the proper form, and cannot use the same calling sequence.
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132:"Explicit a-priori error bounds and adaptive error control for approximation of nonlinear initial value differential systems"
224:. A demonstration of the theory and usage of the Parker–Sochacki method, including a solution for the classical Newtonian
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solutions to systems of differential equations, with the coefficients in either algebraic or numerical form.
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Joseph W. Rudmin (1998), "Application of the Parker–Sochacki Method to
Celestial Mechanics",
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193:. A thorough explanation of the paradigm and application of the Parker–Sochacki method
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P.G. Warne; D.P. Warne; J.S. Sochacki; G.E. Parker; D.C Carothers (2006).
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The Parker–Sochacki method rests on two simple observations:
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Algorithm for solving ordinary differential equations
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can be used to find their solution in the form of a
228:-body problem with mutual gravitational attraction.
179:Polynomial ODEs – Examples, Solutions, Properties
63:If a set of ODEs has a particular form, then the
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245:. A collection of papers and some Matlab code.
139:Computers & Mathematics with Applications
47:Mathematics Department. The method produces
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199:Journal of Computational Neuroscience
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234:The Modified Picard Method.
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104:a symplectic integration.
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221:10.1007/s10827-008-0131-5
45:James Madison University
33:differential equations
25:Parker–Sochacki method
259:Mathematical analysis
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87:Advantages
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55:Summary
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