e continue
considerations of the previous section.
We are solving a slightly nonlinear 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
RungeKutta 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 RungeKutta step, then improve
using Newton and then do RungeKutta again and so fourth.
The system of equations is
with
The continuous parameter that we need for our Cauchy problem is reversed time.
Let
Then
and
solves the equation for
.
We rewrite the
equation:
and differentiate with respect to
:
or
and we obtain a Cauchy problem. We now have the function
that solves both Cauchy problem and algebraic problem for all
.
We apply RungeKutta and Newton interchangeably, as discussed above.
Next, we calculate the components.
The term
was considered in the previous section.
According to the summary
(
Summary for mean
reverting equation in case
q=1
),
Thus
However, such approach requires calculation of
for variety of
.
Instead, we do exponentiation. We select a
,
a power
and
calculate
This
way
for reasonably large
.
Such
has multiplicative
property
which is especially useful in present situation. We
have
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.
