Approximaths

Regularization III

Written by

approximaths

in

In this post, we justify the regularization procedure introduced in previous posts for singular polynomial problems. Consider:

\displaystyle P(x)=a_0+a_1x+\dots+a_nx^n
\displaystyle P(x)=1+A_1x+\dots+A_nx^n
\displaystyle P_{\epsilon}(x)=f_0(\epsilon)+f_1(\epsilon)\epsilon^{\alpha_1}A_1x+\dots+f_n(\epsilon)\epsilon^{\alpha_n}A_nx^n
\displaystyle P_{\epsilon}(x)=(1+b_0\epsilon+c_0\epsilon^2+\dots)+(1+b_1\epsilon+c_1\epsilon^2+\dots)\epsilon^{\alpha_1}A_1x+\dots+(1+b_n\epsilon+c_n\epsilon^2+\dots)\epsilon^{\alpha_n}A_nx^n

where f_0(\epsilon),f_1(\epsilon),\dots,f_n(\epsilon) admit power-series expansions in \epsilon, \alpha_i‘s are rational numbers and b_i‘s, c_i‘s are constants.

The last expression is therefore a generalized version of a perturbed polynomial

. We can rearrange this expression as follows:

\displaystyle P_{\epsilon}(x)=(1+b_0\epsilon+c_0\epsilon^2+\dots)
\displaystyle +(1+b_1\epsilon+c_1\epsilon^2+\dots)\epsilon^{\alpha_1}A_1x
\displaystyle +\dots
\displaystyle +(1+b_n\epsilon+c_n\epsilon^2+\dots)\epsilon^{\alpha_n}A_nx^n

Collecting terms according to their powers of \epsilon, we obtain:

\displaystyle P_{\epsilon}(x)=1+\epsilon^{\alpha_1}A_1x+\dots+\epsilon^{\alpha_n}A_nx^n
\displaystyle +b_0\epsilon+b_1\epsilon^{\alpha_1+1}A_1x+\dots+b_n\epsilon^{\alpha_n+1}A_nx^n
\displaystyle +\dots

Theorem (Newton–Puiseux). Each root of the polynomial above is of the form:

\displaystyle x(\epsilon)=\epsilon^pw(\epsilon),\quad w(0)\neq0

where w(\epsilon) is a continuous function near zero. For convenience, we simply write x=\epsilon^pw. If the theorem holds, the polynomial above becomes:

\displaystyle P_{\epsilon}(\epsilon^pw)=1+\epsilon^{\alpha_1}A_1\epsilon^pw+\dots+\epsilon^{\alpha_n}A_n(\epsilon^pw)^n
\displaystyle +b_0\epsilon+b_1\epsilon^{\alpha_1+1}A_1\epsilon^pw+\dots+b_n\epsilon^{\alpha_n+1}A_n(\epsilon^pw)^n
\displaystyle +\dots

We define Q_{\epsilon}(w):=1+\epsilon^{\alpha_1+p}A_1w+\dots+\epsilon^{\alpha_n+np}A_nw^n and have:

\displaystyle P_{\epsilon}(\epsilon^pw)=Q_{\epsilon}(w)+\epsilon(b_0+b_1\epsilon^{\alpha_1+p}A_1w+\dots+b_n\epsilon^{\alpha_n+np}A_nw^n)+\dots

If \epsilon^pw is a root of P_{\epsilon}(x), we must satisfy:

\displaystyle \lim_{\epsilon\to0}P_{\epsilon}(\epsilon^pw)=\lim_{\epsilon\to0}Q_{\epsilon}(w)=0

Since w(\epsilon) is a continuous function near zero, we inspect the structure of Q_{\epsilon}(w(0)) to determine the leading-order behavior:

\displaystyle Q_{\epsilon}(w(0))=1+\epsilon^{\alpha_1+p}A_1w(0)+\dots+\epsilon^{\alpha_n+np}A_nw(0)^n

Considering the set of its exponents of \epsilon:

\displaystyle E=\{0,\alpha_1+p,\dots,\alpha_k+kp,\dots,\alpha_n+np\}

In order for the condition \lim_{\epsilon\to0}Q_{\epsilon}(w)=0 to hold, the minimal value of the set E must be exactly 0, and this minimum must be shared by at least two distinct exponents. These identical minimal exponents define the dominant terms of Q_{\epsilon}(w(0)) which can then compensate each other to balance the equation at leading order.

Comments

Leave a comment

More posts