Probabilistic evolution theory (PREVTH) solves a system of ODEs
with purely second degree polynomial right-hand sides (each
additive term of each of the right-hand sides is exactly second
degree). Quadratization takes a system of ODEs with polynomial
right-hand sides and makes it into an equivalent system of ODEs
with more equations and unknowns such that the new system has
right-hand sides which are at most second degree polynomials.
Quadratization may also be used as polynomialization +
quadratization so that the method is applicable to more
problems. Polynomialization can get rid of negative integer
powers, and in many cases, other function families such as
trigonometric and exponential. Going from at most second degree
terms on the right-hand sides to exactly (purely) second degree
terms on the right-hand sides is always possible by constancy
adding space extension (CASE). Therefore polynomialization +
quadratization + CASE + PREVTH is a highly efficient sequence
of steps to provide a solution for many ODE systems. Since
PREVTH sums the Taylor terms of the solution, polynomialization
+ quadratization + CASE + PREVTH is especially effective for
solutions in small $t$ intervals. Up to now, the quadratization
within the framework of PREVTH was either done manually or by
a naive algorithm that does not provide optimal quadratization.
Optimal quadratization (smallest number of equations and unknowns)
is important since having redundancy in the equations greatly
increases the necessary computational effort. In this work,
an ODE solver is implemented that successfully combines
optimal quadratization and PREVTH. Also, it is important to note
that the Taylor series representation of the solution by PREVTH is
exact. The convergence of the Taylor series depends on the system.
The paper also focuses on different system of ODEs to show
how the ODE solver can provide a Taylor series solution in powers of
$t$, from which a numerical result can be obtained by substituting
$t$ with the desired value.