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A Necessary and Sufficient Condition for Local Maxima of Polynomial Modulus Over Unit Disc | | Bahman Kalantari
; | | Date: |
2 May 2016 | | Abstract: | An important quantity associated with a complex polynomial $p(z)$ is $Vert p
Vert_infty$, the maximum of its modulus over the unit disc $D$. We prove,
$z_* in D$ is a local maximum of $|p(z)|$ if and only if $a_*$ satisfies,
$z_*=p(z_*)|p’(z_*)|/p’(z_*)|p(z_*)|$, i.e. it is proportional to its
corresponding Newton direction. This explicit formula gives rise to novel
iterative algorithms for computing $Vert p Vert_infty$. We describe two such
algorithms, including a Newton-like method and present some visualization of
their performance. | | Source: | arXiv, 1605.0621 | | Services: | Forum | Review | PDF | Favorites | |
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