Probabilistic evolution theory for explicit autonomous ODEs: Simplifying the factorials, Cauchy product folding and Kronecker product decomposition

摘要

Probabilistic evolution theory forms a framework for the solution of explicit ODEs. The squarification concept and the recursion between the squarified telescope matrices (or the images of initial vectors under the squarified telescope matrices) make the method efficient in terms of time and space necessities. This work is designed for further improvements. The new concepts in this work are "simplifying the factorials", "Cauchy product folding", the use of Kronecker product decomposition within Probabilistic evolution theory (PREVTH) and "condensation". "Simplifying the factorials" is about embedding the factorial that is appearing in the series expansion into the recursion so that number of calculations is reduced and numerical stability is improved. "Cauchy product folding" is about the Cauchy Kronecker product of two vectors. If the vectors change places in Kronecker product, the new result is a permutation of the original result. This property is used so that the number of terms in the series is halved. Also, the possibilities of the use of Kronecker product decomposition on the rectangular matrix are investigated in detail. Lastly, in "condensation", the effect of Cauchy product folding which causes equivalent columns in the rectangular matrix is utilized so that smaller matrices and vectors may be used in the recursion.