8,339 research outputs found
New Acceleration of Nearly Optimal Univariate Polynomial Root-findERS
Univariate polynomial root-finding has been studied for four millennia and is
still the subject of intensive research. Hundreds of efficient algorithms for
this task have been proposed. Two of them are nearly optimal. The first one,
proposed in 1995, relies on recursive factorization of a polynomial, is quite
involved, and has never been implemented. The second one, proposed in 2016,
relies on subdivision iterations, was implemented in 2018, and promises to be
practically competitive, although user's current choice for univariate
polynomial root-finding is the package MPSolve, proposed in 2000, revised in
2014, and based on Ehrlich's functional iterations. By proposing and
incorporating some novel techniques we significantly accelerate both
subdivision and Ehrlich's iterations. Moreover our acceleration of the known
subdivision root-finders is dramatic in the case of sparse input polynomials.
Our techniques can be of some independent interest for the design and analysis
of polynomial root-finders.Comment: 89 pages, 5 figures, 2 table
Simultaneous core partitions: parameterizations and sums
Fix coprime . We re-prove, without Ehrhart reciprocity, a conjecture
of Armstrong (recently verified by Johnson) that the finitely many simultaneous
-cores have average size , and that the
subset of self-conjugate cores has the same average (first shown by
Chen--Huang--Wang). We similarly prove a recent conjecture of Fayers that the
average weighted by an inverse stabilizer---giving the "expected size of the
-core of a random -core"---is . We also prove
Fayers' conjecture that the analogous self-conjugate average is the same if
is odd, but instead if is even. In principle,
our explicit methods---or implicit variants thereof---extend to averages of
arbitrary powers.
The main new observation is that the stabilizers appearing in Fayers'
conjectures have simple formulas in Johnson's -coordinates parameterization
of -cores.
We also observe that the -coordinates extend to parameterize general
-cores. As an example application with , we count the number of
-cores for coprime , verifying a recent conjecture of
Amdeberhan and Leven.Comment: v4: updated references to match final EJC versio
Novel Approach to Real Polynomial Root-finding and Matrix Eigen-solving
Univariate polynomial root-finding is both classical and important for modern
computing. Frequently one seeks just the real roots of a polynomial with real
coefficients. They can be approximated at a low computational cost if the
polynomial has no nonreal roots, but typically nonreal roots are much more
numerous than the real ones. We dramatically accelerate the known algorithms in
this case by exploiting the correlation between the computations with matrices
and polynomials, extending the techniques of the matrix sign iteration, and
exploiting the structure of the companion matrix of the input polynomial. We
extend some of the proposed techniques to the approximation of the real
eigenvalues of a real nonsymmetric matrix.Comment: 17 pages, added algorithm
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