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At least 289 records · Page 16Linked to original sources
arXiv2011

Quality Up in Polynomial Homotopy Continuation by Multithreaded Path Tracking

Speedup measures how much faster we can solve the same problem using many cores. If we can afford to keep the execution time fixed, then quality up measures how much better the solution will be computed using many cores. In this paper we describe our multithreaded implementation to track one solution path defined by a polynomial homotopy. Limiting quality to accuracy and confusing accuracy with precision, we strive to offset the cost of multiprecision arithmetic running multithreaded code on many cores.

cs.DC↗
arXiv2011

Solving large linear algebraic systems in the context of integrable non-abelian Laurent ODEs

The paper reports on a computer algebra program LSSS (Linear Selective Systems Solver) for solving linear algebraic systems with rational coefficients. The program is especially efficient for very large sparse systems that have a solution in which many variables take the value zero. The program is applied to the symmetry investigation of a non-abelian Laurent ODE introduced recently by M. Kontsevich. The computed symmetries confirmed that a Lax pair found for this system earlier generates all first integrals of degree at least up to 14.

cs.SC↗
arXiv2012

Computing the Hermite Form of a Matrix of Ore Polynomials

Let R=F[D;sigma,delta] be the ring of Ore polynomials over a field (or skew field) F, where sigma is a automorphism of F and delta is a sigma-derivation. Given a an m by n matrix A over R, we show how to compute the Hermite form H of A and a unimodular matrix U such that UA=H. The algorithm requires a polynomial number of operations in F in terms of both the dimensions m and n, and the degree of the entries in A. When F=k(z) for some field k, it also requires time polynomial in the degree in z, and if k is the rational numbers Q, it requires time polynomial in the bit length of the coefficients as well. Explicit analyses are provided for the complexity, in particular for the important cases of differential and shift polynomials over Q(z). To accomplish our algorithm, we apply the Dieudonne determinant and quasideterminant theory for Ore polynomial rings to get explicit bounds on the degrees and sizes of entries in H and U.

cs.SC↗
arXiv2011

On the complexity of computing with zero-dimensional triangular sets

We study the complexity of some fundamental operations for triangular sets in dimension zero. Using Las-Vegas algorithms, we prove that one can perform such operations as change of order, equiprojectable decomposition, or quasi-inverse computation with a cost that is essentially that of modular composition. Over an abstract field, this leads to a subquadratic cost (with respect to the degree of the underlying algebraic set). Over a finite field, in a boolean RAM model, we obtain a quasi-linear running time using Kedlaya and Umans' algorithm for modular composition. Conversely, we also show how to reduce the problem of modular composition to change of order for triangular sets, so that all these problems are essentially equivalent. Our algorithms are implemented in Maple; we present some experimental results.

cs.SC↗
arXiv2011

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 Schönhage (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.

cs.SC↗
arXiv2011

Asymptotic Methods of ODEs: Exploring Singularities of the Second Kind

We develop symbolic methods of asymptotic approximations for solutions of linear ordinary differential equations and use to them stabilize numerical calculations. Our method follows classical analysis for first-order systems and higher-order scalar equations where growth behavior is expressed in terms of elementary functions. We then recast our equations in mollified form - thereby obtaining stability.

cs.SC↗
arXiv2012

Affine solution sets of sparse polynomial systems

This paper focuses on the equidimensional decomposition of affine varieties defined by sparse polynomial systems. For generic systems with fixed supports, we give combinatorial conditions for the existence of positive dimensional components which characterize the equidimensional decomposition of the associated affine variety. This result is applied to design an equidimensional decomposition algorithm for generic sparse systems. For arbitrary sparse systems of n polynomials in n variables with fixed supports, we obtain an upper bound for the degree of the affine variety defined and we present an algorithm which computes finite sets of points representing its equidimensional components.

math.AG↗
arXiv2011

Improvement Of Barreto-Voloch Algorithm For Computing $r$th Roots Over Finite Fields

Root extraction is a classical problem in computers algebra. It plays an essential role in cryptosystems based on elliptic curves. In 2006, Barreto and Voloch proposed an algorithm to compute $r$th roots in ${F}_{q^m} $ for certain choices of $m$ and $q$. If $r\,||\,q-1$ and $ (m, r)=1, $ they proved that the complexity of their method is $\widetilde{\mathcal {O}}(r(\log m+\log\log q)m\log q) $. In this paper, we extend the Barreto-Voloch algorithm to the general case that $r\,||\,q^m-1$, without the restrictions $r\,||\,q-1$ and $(m, r)=1 $. We also specify the conditions that the Barreto-Voloch algorithm can be preferably applied.

cs.SC↗
arXiv2012

Stability of Triangular Decomposition and Comprehensive Triangular Decomposition

A new concept, decomposition-unstable (DU) variety of a parametric polynomial system, is introduced in this paper and the stabilities of several triangular decomposition methods, such as characteristic set decomposition, relatively simplicial decomposition and regular chain decomposition, for parametric polynomial systems are discussed in detail. The concept leads to a definition of weakly comprehensive triangular decomposition (WCTD) and a new algorithm for computing comprehensive triangular decomposition (CTD) which was first introduced in [4] for computing an analogue of comprehensive Groebner systems for parametric polynomial systems. Our algorithm takes advantage of a hierarchical solving strategy and a self-adaptive order of parameters. The algorithm has been implemented with Maple 15 and experimented with a number of benchmarks from the literature. Comparison with the Maple package RegularChains, which contains an implementation of the algorithm in [4], is provided and the results illustrate that the time costs by our program for computing CTDs of most examples are no more than those by RegularChains.

cs.SC↗
arXiv2011

Generating Loop Invariants by Computing Vanishing Ideals of Sample Points

Loop invariants play a very important role in proving correctness of programs. In this paper, we address the problem of generating invariants of polynomial loop programs. We present a new approach, for generating polynomial equation invariants of polynomial loop programs through computing vanishing ideals of sample points. We apply rational function interpolation, based on early termination technique, to generate invariants of loop programs with symbolic initial values. Our approach avoids first-order quantifier elimination and cylindrical algebraic decomposition(CAD). An algorithm for generating polynomial invariants is proposed and some examples are given to illustrate the algorithm. Furthermore, we demonstrate on a set of loop programs with symbolic initial values that our algorithm can yield polynomial invariants with degrees high up to 15.

cs.SC↗
arXiv2012

Sparse Differential Resultant for Laurent Differential Polynomials

In this paper, we first introduce the concept of Laurent differentially essential systems and give a criterion for Laurent differentially essential systems in terms of their supports. Then the sparse differential resultant for a Laurent differentially essential system is defined and its basic properties are proved. In particular, order and degree bounds for the sparse differential resultant are given. Based on these bounds, an algorithm to compute the sparse differential resultant is proposed, which is single exponential in terms of the number of indeterminates, the Jacobi number of the system, and the size of the system.

cs.SC↗
arXiv2011

Adleman-Manders-Miller Root Extraction Method Revisited

In 1977, Adleman, Manders and Miller had briefly described how to extend their square root extraction method to the general $r$th root extraction over finite fields, but not shown enough details. Actually, there is a dramatic difference between the square root extraction and the general $r$th root extraction because one has to solve discrete logarithms for $r$th root extraction. In this paper, we clarify their method and analyze its complexity. Our heuristic presentation is helpful to grasp the method entirely and deeply.

cs.SC↗
arXiv2011

Mixed Discriminants

The mixed discriminant of n Laurent polynomials in n variables is the irreducible polynomial in the coefficients which vanishes whenever two of the roots coincide. The Cayley trick expresses the mixed discriminant as an A-discriminant. We show that the degree of the mixed discriminant is a piecewise linear function in the Plucker coordinates of a mixed Grassmannian. An explicit degree formula is given for the case of plane curves.

math.AG↗
arXiv2011

Lie algebra conjugacy

We study the problem of matrix Lie algebra conjugacy. Lie algebras arise centrally in areas as diverse as differential equations, particle physics, group theory, and the Mulmuley--Sohoni Geometric Complexity Theory program. A matrix Lie algebra is a set L of matrices such that $A, B\in L$ implies $AB - BA \in L$. Two matrix Lie algebras are conjugate if there is an invertible matrix $M$ such that $L_1 = M L_2 M^{-1}$. We show that certain cases of Lie algebra conjugacy are equivalent to graph isomorphism. On the other hand, we give polynomial-time algorithms for other cases of Lie algebra conjugacy, which allow us to essentially derandomize a recent result of Kayal on affine equivalence of polynomials. Affine equivalence is related to many complexity problems such as factoring integers, graph isomorphism, matrix multiplication, and permanent versus determinant. Specifically, we show: Abelian Lie algebra conjugacy is equivalent to the code equivalence problem, and hence is as hard as graph isomorphism. Abelian Lie algebra conjugacy of $n \times n$ matrices can be solved in poly(n) time when the Lie algebras have dimension O(1). Semisimple Lie algebra conjugacy is equivalent to graph isomorphism. A Lie algebra is semisimple if it is a direct sum of simple Lie algebras. Semisimple Lie algebra conjugacy of $n \times n$ matrices can be solved in polynomial time when the Lie algebras consist of only $O(\log n)$ simple direct summands. Conjugacy of completely reducible Lie algebras---that is, a direct sum of an abelian and a semisimple Lie algebra---can be solved in polynomial time when the abelian part has dimension O(1) and the semisimple part has $O(\log n)$ simple direct summands.

cs.CC↗
arXiv2012

The fundamental invariants of 3 x 3 x 3 arrays

We determine the three fundamental invariants in the entries of a $3 \times 3 \times 3$ array over $\mathbb{C}$ as explicit polynomials in the 27 variables $x_{ijk}$ for $1 \le i, j, k \le 3$. By the work of Vinberg on $θ$-groups, it is known that these homogeneous polynomials have degrees 6, 9 and 12; they freely generate the algebra of invariants for the Lie group $SL_3(\mathbb{C}) \times SL_3(\mathbb{C}) \times SL_3(\mathbb{C})$ acting irreducibly on its natural representation $\mathbb{C}^3 \otimes \mathbb{C}^3 \otimes \mathbb{C}^3$. These generators have respectively 1152, 9216 and 209061 terms; we find compact expressions in terms of the orbits of the finite group $(S_3 \times S_3 \times S_3) \rtimes S_3$ acting on monomials of weight zero for the action of the Lie algebra $\mathfrak{sl}_3(\mathbb{C}) \oplus \mathfrak{sl}_3(\mathbb{C}) \oplus \mathfrak{sl}_3(\mathbb{C})$.

math.AC↗
arXiv2011

Note on fast division algorithm for polynomials using Newton iteration

The classical division algorithm for polynomials requires $O(n^2)$ operations for inputs of size $n$. Using reversal technique and Newton iteration, it can be improved to $O({M}(n))$, where ${M}$ is a multiplication time. But the method requires that the degree of the modulo, $x^l$, should be the power of 2. If $l$ is not a power of 2 and $f(0)=1$, Gathen and Gerhard suggest to compute the inverse,$f^{-1}$, modulo $x^{\lceil l/2^r\rceil}, x^{\lceil l/2^{r-1}\rceil},..., x^{\lceil l/2\rceil}, x^l$, separately. But they did not specify the iterative step. In this note, we show that the original Newton iteration formula can be directly used to compute $f^{-1}\,{mod}\,x^{l}$ without any additional cost, when $l$ is not a power of 2.

cs.SC↗