Computer Science

The best method for computing the adjoint matrix of an order n matrix in an arbitrary commutative ring requires O( n β+1/3 lognloglogn) operations, provided the complexity of the algorithm for multiplying two matrices is γ n β +o( n β ) . For a commutative domain -- and under the same assumptions -- the complexity of the best method is 6γ n β /( 2 β −2)+o( n β ) . In the present work a new method is presented for the computation of the adjoint matrix in a commutative domain. Despite the fact that the number of operations required is now 1.5 times more, than that of the best method, this new method permits a better parallelization of the computational process and may be successfully employed for computations in parallel computational systems.

The p -curvature of a system of linear differential equations in positive characteristic p is a matrix that measures how far the system is from having a basis of polynomial solutions. We show that the similarity class of the p -curvature can be determined without computing the p -curvature itself. More precisely, we design an algorithm that computes the invariant factors of the p -curvature in time quasi-linear in p – √ . This is much less than the size of the p -curvature, which is generally linear in p . The new algorithm allows to answer a question originating from the study of the Ising model in statistical physics.

In perturbative calculations, e.g., in the setting of Quantum Chromodynamics (QCD) one aims at the evaluation of Feynman integrals. Here one is often faced with the problem to simplify multiple nested integrals or sums to expressions in terms of indefinite nested integrals or sums. Furthermore, one seeks for solutions of coupled systems of linear differential equations, that can be represented in terms of indefinite nested sums (or integrals). In this article we elaborate the main tools and the corresponding packages, that we have developed and intensively used within the last 10 years in the course of our QCD-calculations.

Differential (Ore) type polynomials with "approximate" polynomial coefficients are introduced. These provide an effective notion of approximate differential operators, with a strong algebraic structure. We introduce the approximate Greatest Common Right Divisor Problem (GCRD) of differential polynomials, as a non-commutative generalization of the well-studied approximate GCD problem. Given two differential polynomials, we present an algorithm to find nearby differential polynomials with a non-trivial GCRD, where nearby is defined with respect to a suitable coefficient norm. Intuitively, given two linear differential polynomials as input, the (approximate) GCRD problem corresponds to finding the (approximate) differential polynomial whose solution space is the intersection of the solution spaces of the two inputs. The approximate GCRD problem is proven to be locally well-posed. A method based on the singular value decomposition of a differential Sylvester matrix is developed to produce an initial approximation of the GCRD. With a sufficiently good initial approximation, Newton iteration is shown to converge quadratically to an optimal solution. Finally, sufficient conditions for existence of a solution to the global problem are presented along with examples demonstrating that no solution exists when these conditions are not satisfied.

We study the computation of canonical bases of sets of univariate relations ( p 1 ,?? p m )?�K[x ] m such that p 1 f 1 +?? p m f m =0 ; here, the input elements f 1 ,?? f m are from a quotient K[x ] n /M , where M is a K[x] -module of rank n given by a basis M?�K[x ] n?n in Hermite form. We exploit the triangular shape of M to generalize a divide-and-conquer approach which originates from fast minimal approximant basis algorithms. Besides recent techniques for this approach, we rely on high-order lifting to perform fast modular products of polynomial matrices of the form PFmodM . Our algorithm uses O ~ ( m ??? D+ n ? D/m) operations in K , where D=deg(det(M)) is the K -vector space dimension of K[x ] n /M , O ~ (?? indicates that logarithmic factors are omitted, and ? is the exponent of matrix multiplication. This had previously only been achieved for a diagonal matrix M . Furthermore, our algorithm can be used to compute the shifted Popov form of a nonsingular matrix within the same cost bound, up to logarithmic factors, as the previously fastest known algorithm, which is randomized.

A Chebyshev curve C(a,b,c,ϕ) has a parametrization of the form x(t)=T_a(t) ; \ y(t)=T_b(t) ; z(t)=T_c(t+ϕ) , where a,b,c are integers, T_n(t) is the Chebyshev polynomialof degree n and ϕ∈R . When C(a,b,c,ϕ) is nonsingular,it defines a polynomial knot. We determine all possible knot diagrams when ϕ varies. Let a,b,c be integers, a is odd, (a,b)=1 , we show that one can list all possible knots C(a,b,c,ϕ) in O ~ ( n 2 ) bit operations, with n=abc .

We develop a probabilistic algorithm for computing elimination ideals of likelihood equations, which is for larger models by far more efficient than directly computing Groebner bases or the interpolation method proposed in the first author's previous work. The efficiency is improved by a theoretical result showing that the sum of data variables appears in most coefficients of the generator polynomial of elimination ideal. Furthermore, applying the known structures of Newton polytopes of discriminants, we can also efficiently deduce discriminants of the elimination ideals. For instance, the discriminants of 3 by 3 matrix model and one Jukes-Cantor model in phylogenetics (with sizes over 30 GB and 8 GB text files, respectively) can be computed by our methods.

Differential (Ore) type polynomials with approximate polynomial coefficients are introduced. These provide a useful representation of approximate differential operators with a strong algebraic structure, which has been used successfully in the exact, symbolic, setting. We then present an algorithm for the approximate Greatest Common Right Divisor (GCRD) of two approximate differential polynomials, which intuitively is the differential operator whose solutions are those common to the two inputs operators. More formally, given approximate differential polynomials f and g , we show how to find "nearby" polynomials f ˜ and g ˜ which have a non-trivial GCRD. Here "nearby" is under a suitably defined norm. The algorithm is a generalization of the SVD-based method of Corless et al. (1995) for the approximate GCD of regular polynomials. We work on an appropriately "linearized" differential Sylvester matrix, to which we apply a block SVD. The algorithm has been implemented in Maple and a demonstration of its robustness is presented.

We present two algorithms for computing hypergeometric solutions of second order linear differential operators with rational function coefficients. Our first algorithm searches for solutions of the form exp(∫rdx)⋅ 2 F 1 ( a 1 , a 2 ; b 1 ;f) where r,f∈ Q(x) ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ , and a 1 , a 2 , b 1 ∈Q . It uses modular reduction and Hensel lifting. Our second algorithm tries to find solutions in the form exp(∫rdx)⋅( r 0 ⋅ 2 F 1 ( a 1 , a 2 ; b 1 ;f)+ r 1 ⋅ 2 F 1 ′ ( a 1 , a 2 ; b 1 ;f)) where r 0 , r 1 ∈ Q(x) ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ , as follows: It tries to transform the input equation to another equation with solutions of the first type, and then uses the first algorithm.

We present an algorithm for computing limits of quotients of real analytic functions. The algorithm is based on computation of a bound on the Lojasiewicz exponent and requires the denominator to have an isolated zero at the limit point.