17 research outputs found
A deterministic algorithm to compute approximate roots of polynomial systems in polynomial average time
We describe a deterministic algorithm that computes an approximate root of n
complex polynomial equations in n unknowns in average polynomial time with
respect to the size of the input, in the Blum-Shub-Smale model with square
root. It rests upon a derandomization of an algorithm of Beltr\'an and Pardo
and gives a deterministic affirmative answer to Smale's 17th problem. The main
idea is to make use of the randomness contained in the input itself
Condition Numbers for the Cube. I: Univariate Polynomials and Hypersurfaces
The condition-based complexity analysis framework is one of the gems of
modern numerical algebraic geometry and theoretical computer science. One of
the challenges that it poses is to expand the currently limited range of random
polynomials that we can handle. Despite important recent progress, the
available tools cannot handle random sparse polynomials and Gaussian
polynomials, that is polynomials whose coefficients are i.i.d. Gaussian random
variables.
We initiate a condition-based complexity framework based on the norm of the
cube that is a step in this direction. We present this framework for real
hypersurfaces and univariate polynomials. We demonstrate its capabilities in
two problems, under very mild probabilistic assumptions. On the one hand, we
show that the average run-time of the Plantinga-Vegter algorithm is polynomial
in the degree for random sparse (alas a restricted sparseness structure)
polynomials and random Gaussian polynomials. On the other hand, we study the
size of the subdivision tree for Descartes' solver and run-time of the solver
by Jindal and Sagraloff (arXiv:1704.06979). In both cases, we provide a bound
that is polynomial in the size of the input (size of the support plus logarithm
of the degree) for not only on the average, but all higher moments.Comment: 34 pages. Version 1, conference version; from version 2, journal
versio
Condition Numbers for the Cube. I: Univariate Polynomials and Hypersurfaces
International audienceThe condition-based complexity analysis framework is one of the gems of modern numerical algebraic geometry and theoretical computer science. One of the challenges that it poses is to expand the currently limited range of random polynomials that we can handle. Despite important recent progress, the available tools cannot handle random sparse polynomials and Gaussian polynomials, that is polynomials whose coefficients are i.i.d. Gaussian random variables. We initiate a condition-based complexity framework based on the norm of the cube, that is a step in this direction. We present this framework for real hypersurfaces. We demonstrate its capabilities by providing a new probabilistic complexity analysis for the Plantinga-Vegter algorithm, which covers both random sparse (alas a restricted sparseness structure) polynomials and random Gaussian polynomials. We present explicit results with structured random polynomials for problems with two or more dimensions. Additionally, we provide some estimates of the separation bound of a univariate polynomial in our current framework
Rigid continuation paths I. Quasilinear average complexity for solving polynomial systems
How many operations do we need on the average to compute an approximate root
of a random Gaussian polynomial system? Beyond Smale's 17th problem that asked
whether a polynomial bound is possible, we prove a quasi-optimal bound
. This improves upon the previously known
bound.
The new algorithm relies on numerical continuation along \emph{rigid
continuation paths}. The central idea is to consider rigid motions of the
equations rather than line segments in the linear space of all polynomial
systems. This leads to a better average condition number and allows for bigger
steps. We show that on the average, we can compute one approximate root of a
random Gaussian polynomial system of~ equations of degree at most in
homogeneous variables with continuation steps. This is a
decisive improvement over previous bounds that prove no better than
continuation steps on the average
The average condition number of most tensor rank decomposition problems is infinite
The tensor rank decomposition, or canonical polyadic decomposition, is the
decomposition of a tensor into a sum of rank-1 tensors. The condition number of
the tensor rank decomposition measures the sensitivity of the rank-1 summands
with respect to structured perturbations. Those are perturbations preserving
the rank of the tensor that is decomposed. On the other hand, the angular
condition number measures the perturbations of the rank-1 summands up to
scaling.
We show for random rank-2 tensors with Gaussian density that the expected
value of the condition number is infinite. Under some mild additional
assumption, we show that the same is true for most higher ranks as
well. In fact, as the dimensions of the tensor tend to infinity, asymptotically
all ranks are covered by our analysis. On the contrary, we show that rank-2
Gaussian tensors have finite expected angular condition number.
Our results underline the high computational complexity of computing tensor
rank decompositions. We discuss consequences of our results for algorithm
design and for testing algorithms that compute the CPD. Finally, we supply
numerical experiments
New data structure for univariate polynomial approximation and applications to root isolation, numerical multipoint evaluation, and other problems
We present a new data structure to approximate accurately and efficiently a
polynomial of degree given as a list of coefficients. Its properties
allow us to improve the state-of-the-art bounds on the bit complexity for the
problems of root isolation and approximate multipoint evaluation. This data
structure also leads to a new geometric criterion to detect ill-conditioned
polynomials, implying notably that the standard condition number of the zeros
of a polynomial is at least exponential in the number of roots of modulus less
than or greater than .Given a polynomial of degree with
for , isolating all its complex roots or
evaluating it at points can be done with a quasi-linear number of
arithmetic operations. However, considering the bit complexity, the
state-of-the-art algorithms require at least bit operations even for
well-conditioned polynomials and when the accuracy required is low. Given a
positive integer , we can compute our new data structure and evaluate at
points in the unit disk with an absolute error less than in
bit operations, where means
that we omit logarithmic factors. We also show that if is the absolute
condition number of the zeros of , then we can isolate all the roots of
in bit operations. Moreover, our
algorithms are simple to implement. For approximating the complex roots of a
polynomial, we implemented a small prototype in \verb|Python/NumPy| that is an
order of magnitude faster than the state-of-the-art solver \verb/MPSolve/ for
high degree polynomials with random coefficients