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Mathematical background

Let $P$ be a polynomial and be maxroot the maximal modulus of the root of $P$. From $P$ we may derive a the unitary polynomial $Q$ such that the roots of $Q$ have a modulus lower or equal to 1 and if $w$ is a root of $Q$ then maxroot$w$ is a root of $P$.

Let $Q=sum_{i=0}^{i=n} a_i x^i$ which may also be written as $sum_{i=0}^{i=n} b_i(x-a)^i$ where $a$ is some fixed point.

Let a range $[a,b]$ for $x$ and let $z_0$ be the mid point of the range. We consider the square in the complex plane centered at $z_0$ and whose edge length is $b-a$. Let $\delta$ be the length of the half-diagonal of this square. If

\begin{displaymath}
\vert b_0\vert> \sum_{j=1}^{j=n} \vert b_j\vert\delta^j
\end{displaymath}

then the polynomial has no root in the square [1]. This procedure may be used for univariate polynomial, with polynomial with interval coefficients or with parametric poynomial. For example it is very efficient for solging the Wilkinson polynomial.


next up previous contents
Next: Implementation Up: Weyl filter Previous: Weyl filter   Contents
Jean-Pierre Merlet 2012-12-20