14,431 research outputs found
Computation of multiple eigenvalues and generalized eigenvectors for matrices dependent on parameters
The paper develops Newton's method of finding multiple eigenvalues with one
Jordan block and corresponding generalized eigenvectors for matrices dependent
on parameters. It computes the nearest value of a parameter vector with a
matrix having a multiple eigenvalue of given multiplicity. The method also
works in the whole matrix space (in the absence of parameters). The approach is
based on the versal deformation theory for matrices. Numerical examples are
given. The implementation of the method in MATLAB code is available.Comment: 19 pages, 3 figure
Rational Solutions of the Painlev\'e-II Equation Revisited
The rational solutions of the Painlev\'e-II equation appear in several
applications and are known to have many remarkable algebraic and analytic
properties. They also have several different representations, useful in
different ways for establishing these properties. In particular,
Riemann-Hilbert representations have proven to be useful for extracting the
asymptotic behavior of the rational solutions in the limit of large degree
(equivalently the large-parameter limit). We review the elementary properties
of the rational Painlev\'e-II functions, and then we describe three different
Riemann-Hilbert representations of them that have appeared in the literature: a
representation by means of the isomonodromy theory of the Flaschka-Newell Lax
pair, a second representation by means of the isomonodromy theory of the
Jimbo-Miwa Lax pair, and a third representation found by Bertola and Bothner
related to pseudo-orthogonal polynomials. We prove that the Flaschka-Newell and
Bertola-Bothner Riemann-Hilbert representations of the rational Painlev\'e-II
functions are explicitly connected to each other. Finally, we review recent
results describing the asymptotic behavior of the rational Painlev\'e-II
functions obtained from these Riemann-Hilbert representations by means of the
steepest descent method
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