21,372 research outputs found
Computing all roots of the likelihood equations of seemingly unrelated regressions
Seemingly unrelated regressions are statistical regression models based on
the Gaussian distribution. They are popular in econometrics but also arise in
graphical modeling of multivariate dependencies. In maximum likelihood
estimation, the parameters of the model are estimated by maximizing the
likelihood function, which maps the parameters to the likelihood of observing
the given data. By transforming this optimization problem into a polynomial
optimization problem, it was recently shown that the likelihood function of a
simple bivariate seemingly unrelated regressions model may have several
stationary points. Thus local maxima may complicate maximum likelihood
estimation. In this paper, we study several more complicated seemingly
unrelated regression models, and show how all stationary points of the
likelihood function can be computed using algebraic geometry.Comment: To appear in the Journal of Symbolic Computation, special issue on
Computational Algebraic Statistics. 11 page
Exact Symbolic-Numeric Computation of Planar Algebraic Curves
We present a novel certified and complete algorithm to compute arrangements
of real planar algebraic curves. It provides a geometric-topological analysis
of the decomposition of the plane induced by a finite number of algebraic
curves in terms of a cylindrical algebraic decomposition. From a high-level
perspective, the overall method splits into two main subroutines, namely an
algorithm denoted Bisolve to isolate the real solutions of a zero-dimensional
bivariate system, and an algorithm denoted GeoTop to analyze a single algebraic
curve.
Compared to existing approaches based on elimination techniques, we
considerably improve the corresponding lifting steps in both subroutines. As a
result, generic position of the input system is never assumed, and thus our
algorithm never demands for any change of coordinates. In addition, we
significantly limit the types of involved exact operations, that is, we only
use resultant and gcd computations as purely symbolic operations. The latter
results are achieved by combining techniques from different fields such as
(modular) symbolic computation, numerical analysis and algebraic geometry.
We have implemented our algorithms as prototypical contributions to the
C++-project CGAL. They exploit graphics hardware to expedite the symbolic
computations. We have also compared our implementation with the current
reference implementations, that is, LGP and Maple's Isolate for polynomial
system solving, and CGAL's bivariate algebraic kernel for analyses and
arrangement computations of algebraic curves. For various series of challenging
instances, our exhaustive experiments show that the new implementations
outperform the existing ones.Comment: 46 pages, 4 figures, submitted to Special Issue of TCS on SNC 2011.
arXiv admin note: substantial text overlap with arXiv:1010.1386 and
arXiv:1103.469
Computational linear algebra over finite fields
We present here algorithms for efficient computation of linear algebra
problems over finite fields
An Elimination Method for Solving Bivariate Polynomial Systems: Eliminating the Usual Drawbacks
We present an exact and complete algorithm to isolate the real solutions of a
zero-dimensional bivariate polynomial system. The proposed algorithm
constitutes an elimination method which improves upon existing approaches in a
number of points. First, the amount of purely symbolic operations is
significantly reduced, that is, only resultant computation and square-free
factorization is still needed. Second, our algorithm neither assumes generic
position of the input system nor demands for any change of the coordinate
system. The latter is due to a novel inclusion predicate to certify that a
certain region is isolating for a solution. Our implementation exploits
graphics hardware to expedite the resultant computation. Furthermore, we
integrate a number of filtering techniques to improve the overall performance.
Efficiency of the proposed method is proven by a comparison of our
implementation with two state-of-the-art implementations, that is, LPG and
Maple's isolate. For a series of challenging benchmark instances, experiments
show that our implementation outperforms both contestants.Comment: 16 pages with appendix, 1 figure, submitted to ALENEX 201
NumGfun: a Package for Numerical and Analytic Computation with D-finite Functions
This article describes the implementation in the software package NumGfun of
classical algorithms that operate on solutions of linear differential equations
or recurrence relations with polynomial coefficients, including what seems to
be the first general implementation of the fast high-precision numerical
evaluation algorithms of Chudnovsky & Chudnovsky. In some cases, our
descriptions contain improvements over existing algorithms. We also provide
references to relevant ideas not currently used in NumGfun
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