241 research outputs found
Polynomials that Sign Represent Parity and Descartes' Rule of Signs
A real polynomial sign represents if
for every , the sign of equals
. Such sign representations are well-studied in computer
science and have applications to computational complexity and computational
learning theory. In this work, we present a systematic study of tradeoffs
between degree and sparsity of sign representations through the lens of the
parity function. We attempt to prove bounds that hold for any choice of set
. We show that sign representing parity over with the
degree in each variable at most requires sparsity at least . We show
that a tradeoff exists between sparsity and degree, by exhibiting a sign
representation that has higher degree but lower sparsity. We show a lower bound
of on the sparsity of polynomials of any degree representing
parity over . We prove exact bounds on the sparsity of such
polynomials for any two element subset . The main tool used is Descartes'
Rule of Signs, a classical result in algebra, relating the sparsity of a
polynomial to its number of real roots. As an application, we use bounds on
sparsity to derive circuit lower bounds for depth-two AND-OR-NOT circuits with
a Threshold Gate at the top. We use this to give a simple proof that such
circuits need size to compute parity, which improves the previous bound
of due to Goldmann (1997). We show a tight lower bound of
for the inner product function over .Comment: To appear in Computational Complexit
When Newton meets Descartes: A Simple and Fast Algorithm to Isolate the Real Roots of a Polynomial
We introduce a new algorithm denoted DSC2 to isolate the real roots of a
univariate square-free polynomial f with integer coefficients. The algorithm
iteratively subdivides an initial interval which is known to contain all real
roots of f. The main novelty of our approach is that we combine Descartes' Rule
of Signs and Newton iteration. More precisely, instead of using a fixed
subdivision strategy such as bisection in each iteration, a Newton step based
on the number of sign variations for an actual interval is considered, and,
only if the Newton step fails, we fall back to bisection. Following this
approach, our analysis shows that, for most iterations, we can achieve
quadratic convergence towards the real roots. In terms of complexity, our
method induces a recursion tree of almost optimal size O(nlog(n tau)), where n
denotes the degree of the polynomial and tau the bitsize of its coefficients.
The latter bound constitutes an improvement by a factor of tau upon all
existing subdivision methods for the task of isolating the real roots. In
addition, we provide a bit complexity analysis showing that DSC2 needs only
\tilde{O}(n^3tau) bit operations to isolate all real roots of f. This matches
the best bound known for this fundamental problem. However, in comparison to
the much more involved algorithms by Pan and Sch\"onhage (for the task of
isolating all complex roots) which achieve the same bit complexity, DSC2
focuses on real root isolation, is very easy to access and easy to implement
A new computational approach to the synthesis of fixed order controllers
The research described in this dissertation deals with an open problem concerning
the synthesis of controllers of xed order and structure. This problem is encountered
in a variety of applications. Simply put, the problem may be put as the
determination of the set, S of controller parameter vectors, K = (k1; k2; : : : ; kl),
that render Hurwitz a family (indexed by F) of complex polynomials of the form
fP0(s; ) + Pl
i=1 Pi(s; )ki; 2 Fg, where the polynomials Pj(s; ); j = 0; : : : ; l
are given data. They are specied by the plant to be controlled, the structure of the
controller desired and the performance that the controllers are expected to achieve.
Simple examples indicate that the set S can be non-convex and even be disconnected.
While the determination of the non-emptiness of S is decidable and amenable
to methods such as the quantier elimination scheme, such methods have not been
computationally tractable and more importantly, do not provide a reasonable approximation
for the set of controllers. Practical applications require the construction of a
set of controllers that will enable a control engineer to check the satisfaction of performance
criteria that may not be mathematically well characterized. The transient
performance criteria often fall into this category. From the practical viewpoint of the construction of approximations for S, this
dissertation is dierent from earlier work in the literature on this problem. A novel
feature of the proposed algorithm is the exploitation of the interlacing property of
Hurwitz polynomials to provide arbitrarily tight outer and inner approximation to
S. The approximation is given in terms of the union of polyhedral sets which are
constructed systematically using the Hermite-Biehler theorem and the generalizations
of the Descartes' rule of signs
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