Approximaths

Regularize IV (scaling)

Written by

approximaths

in

Let us consider the following perturbed polynomial:

\displaystyle P(x)=x^3+x^2+x
\displaystyle P_{\epsilon}(x)=\epsilon^3x^3+x^2+\epsilon x
\displaystyle P_{\epsilon}(\epsilon^p w)=\epsilon^3(\epsilon^p w)^3+(\epsilon^p w)^2+\epsilon(\epsilon^p w)
\displaystyle =\epsilon^{3p+3}w^3+\epsilon^{2p}w^2+\epsilon^{p+1}w

In the figure below, we plot the graphs of the affine functions 3p+3, 2p, and p+1. Their intersection points determine the values of p for which two exponents coincide. Among these, we retain only the intersections where the common exponent is minimal. In this example, we obtain

\displaystyle (p_j,\nu_j)=(-3,-6)

We now compute T_{\epsilon}^{(j)}(w) defined as follows:

\displaystyle T_{\epsilon}^{(j)}(w):=\epsilon^{-\nu_j}P_{\epsilon}(\epsilon^{p_j}w)
\displaystyle T_{\epsilon}^{(j)}(w)=\epsilon^{6}P_{\epsilon}(\epsilon^{-3}w)
\displaystyle =\epsilon^{6}\bigl(\epsilon^{3(-3)+3}w^3+\epsilon^{2(-3)}w^2+\epsilon^{-3+1}w\bigr)
\displaystyle =\epsilon^{6}\bigl(\epsilon^{-6}w^3+\epsilon^{-6}w^2+\epsilon^{-2}w\bigr)
\displaystyle =\underbrace{w^3+w^2}_{T_{0}^{(j)}(w)}+\underbrace{\epsilon^{4}w}_{E_{\epsilon}^{(j)}(w)}

Observe that E_{0}^{(j)}(w)=0. Therefore, we may write

\displaystyle T_{\epsilon}^{(j)}(w)=T_{0}^{(j)}(w)+E_{\epsilon}^{(j)}(w)

The polynomial T_{0}^{(j)}(w) is the dominant part of T_{\epsilon}^{(j)}(w) as \epsilon\to0, since it is independent of \epsilon. In this example, the dominant polynomial is non-degenerate. Therefore, the scaling x=\epsilon^{p_j}w regularizes the perturbed polynomial P_{\epsilon}(x).

Graphs of the affine functions 3p+3, 2p, and p+1. The point (-3,-6) is the intersection of the lines 3p+3 and 2p. At p=-3, the minimum exponent is attained simultaneously by two monomials.

Comments

Leave a comment

More posts