Squarification of telescope matrices in the probabilistic evolution theoretical approach to the two particle classical mechanics as an illustrative implementation

Probabilistic evolution theory (PREVTH) forms a framework for the solution of explicit ODEs. The purpose of the paper is two-fold: (1) conversion of multinomial right-hand sides of the ODEs to purely second degree multinomial right-hand sides by space extension; (2) decrease the computational burden of probabilistic evolution theory by using the condensed Kronecker product. A first order ODE set with multinomial right-hand side functions may be converted to a first order ODE set with purely second degree multinomial right-hand side functions at the expense of an increase in the number of equations and unknowns. Obtaining purely second degree multinomial right-hand side functions is important because the solution of such equation set may be approximated by probabilistic evolution theory. A recent article by the authors states that the ODE set with the smallest number of unknowns can be found by searching. This paper gives the details of a way to search for the optimal space extension. As for the second purpose of the paper, the computational burden can be reduced by considering the properties of the Kronecker product of vectors and how the Kronecker product appears within the recursion of PREVTH: as a Cauchy product structure.

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“…where r(t) is a positional entity and p r (t) is the radial momentum [15,18]. Equation 14can be rewritten using the basis functions:…”

Section: Gravitational Two-body Problem As An Illustrative Examplementioning

confidence: 99%

“…Squarification is a way to prevent the rapid growth of matrices and vectors in probabilistic evolution theory [14]. PREVTH is successfully applied to certain systems [15,16]. Squarification is facilitated by the recursion of squarification [9,17,18].…”

Section: Introductionmentioning

confidence: 99%