considerations of the previous section.
We are solving a slightly non-linear system of algebraic equations using a
Newton technique and we are having problems because of limitations on area of
convergence. The next most effective relevant numerical technique is
Runge-Kutta technique. In order to apply it, perhaps in combination with the
Newton procedure, we need a continuous parameter that would produce a Cauchy
problem with smooth RHS. The idea is to make a Runge-Kutta step, then improve
using Newton and then do Runge-Kutta again and so fourth.
The system of equations is
The continuous parameter that we need for our Cauchy problem is reversed time.
solves the equation for
We rewrite the
and differentiate with respect to
and we obtain a Cauchy problem. We now have the function
that solves both Cauchy problem and algebraic problem for all
We apply Runge-Kutta and Newton interchangeably, as discussed above.
Next, we calculate the components.
was considered in the previous section.
According to the summary
Summary for mean
reverting equation in case
However, such approach requires calculation of
for variety of
Instead, we do exponentiation. We select a
for reasonably large
which is especially useful in present situation. We
The procedure is implemented in the script soCauchy2.py within the directory
OTSProjects/python/wavelet2. We are able to make reasonably small time steps
without stability and divergence issues. However, the procedure exhibits
uncomfortable level of sensitivity to the control parameters
and scale of probing functions. In the following section we discuss reasons of
such sensitivity and propose an improved procedure.