A recursive algorithm for constructing complicated Dixon matrices

Whether the determinant of the Dixon matrix equals zero or not is used for determining if a system of n + 1 polynomial equations in n variables has a common root, and is a very efficient quantifier elimination approach too. But for a complicated polynomial system, it is not easy to construct the Dixon matrix. In this paper, a recursive algorithm to construct the Dixon matrix is proposed by which some problems that cannot be tackled by other methods can be solved on the same computer platform. A dynamic programming algorithm based on the recursive formula is developed and compared for speed and efficiency to the recursive algorithm.     

Three kinds of extraneous factors in Dixon resultants

Dixon resultant is a basic elimination method which has been used widely in the high technology fields of automatic control, robotics, etc. But how to remove extraneous factors in Dixon resultants has been a very difficult problem. In this paper, we discover some extraneous factors by expressing the Dixon resultant in a linear combination of original polynomial system. Furthermore, it has been proved that the factors mentioned above include three parts which come from Dixon derived polynomials, Dixon matrix and the resulting resultant expression by substituting Dixon derived polynomials respectively. This work was supported by the National Key Basic Special Funds of China (Grant No. 2004CB318003), the Knowledge Innovation Project of the Chinese Academy of Sciences (Grant No. KJCX2-YW-S02), the National Natural Science Foundation of China (Grant No. 90718041), Shanghai Leading Academic Discipline Project (Grant No. B412) and the Doctor Startup Foundation of East China Normal University (Grant No. 790013J4)     

循环码译码的Dixon结式方法

针对纠错码译码就是非线性方程组的求解问题,提出利用Dixon结式方法对译码方程进行消元以得到接收数据中的错位多项式.首先,根据纠错码的纠错能力和接收数据得到伴随式矩阵并通过该矩阵的秩确定接收码字中错误位的个数.然后,根据错位个数和伴随多项式构造译码方程.译码时,将其中一个错位变元作为隐藏变元,利用Dixon结式方法进行消元.最后,得到的Dixon结式就是关于隐藏变元的多项式.该多项式去掉多余因子后就是错位多项式,利用Chien搜索法即可求解出错误位置.当错位较多时,采用逐次计算结式的方法以筛除计算过程中的多余因子和重因子.另外,根据不同错位个数得到的错位多项式,提出了构造一类循环码错位多项式符号解的猜想,该猜想可以大大提高译码效率.实验验证了结式理论在纠错码译码方面的应用是有效的且有助于降低对芯片性能的要求.     

On the complexity of the multivariate resultant

The multivariate resultant is a fundamental tool of computational algebraic geometry. It can in particular be used to decide whether a system of n$n$ homogeneous equations in n$n$ variables is satisfiable (the resultant is a polynomial in the system’s coefficients which vanishes if and only if the system is satisfiable). In this paper, we investigate the complexity of computing the multivariate resultant.     

多变元Sylvester结式与多余因子

经典的Sylvester结式方法是代数几何的一种基本消元方法,但它一次只能处理1个变元2个方程的多项式系统.本文将Sylvester结式扩展到n个变元n+1个方程的多项式系统,并且证明了新的多变元的Sylvester结式包含在原多项式系统的理想中.同时,给出了一种去掉其部分多余因子的方法.     

On generator and parity-check polynomial matrices of generalized quasi-cyclic codes

Generalized quasi-cyclic (GQC) codes have been investigated as well as quasi-cyclic (QC) codes, e.g., on the construction of efficient low-density parity-check codes. While QC codes have the same length of cyclic intervals, GQC codes have different lengths of cyclic intervals. Similarly to QC codes, each GQC code can be described by an upper triangular generator polynomial matrix, from which the systematic encoder is constructed. In this paper, a complete theory of generator polynomial matrices of GQC codes, including a relation formula between generator polynomial matrices and parity-check polynomial matrices through their equations, is provided. This relation generalizes those of cyclic codes and QC codes. While the previous researches on GQC codes are mainly concerned with 1-generator case or linear algebraic approach, our argument covers the general case and shows the complete analogy of QC case. We do not use Gröbner basis theory explicitly in order that all arguments of this paper are self-contained. Numerical examples are attached to the dual procedure that extracts one from each other. Finally, we provide an efficient algorithm which calculates all generator polynomial matrices with given cyclic intervals.     

Orthogonal matrix polynomials, scalar-type Rodrigues’ formulas and Pearson equations

Some families of orthogonal matrix polynomials satisfying second-order differential equations with coefficients independent of n have recently been introduced (see [Internat. Math. Res. Notices 10 (2004) 461–484]). An important difference with the scalar classical families of Jacobi, Laguerre and Hermite, is that these matrix families do not satisfy scalar type Rodrigues’ formulas of the type (ΦⁿW)⁽ⁿ⁾W^-1, where Φ is a matrix polynomial of degree not bigger than 2. An example of a modified Rodrigues’ formula, well suited to the matrix case, appears in [Internat. Math. Res. Notices 10 (2004) 482].In this note, we discuss some of the reasons why a second order differential equation with coefficients independent of n does not imply, in the matrix case, a scalar type Rodrigues’ formula and show that scalar type Rodrigues’ formulas are most likely not going to play in the matrix valued case the important role they played in the scalar valued case. We also mention the roles of a scalar-type Pearson equation as well as that of a noncommutative version of it.     

On dual minimum cost flow algorithms

We describe a new dual algorithm for the minimum cost flow problem. It can be regarded as a variation of the best known strongly polynomial minimum cost flow algorithm, due to Orlin. Indeed we obtain the same running time of O(m log m(m+n log n)), where n and m denote the number of vertices and the number of edges. However, in contrast to Orlin's algorithm we work directly with the capacitated network (rather than transforming it to a transshipment problem). Thus our algorithm is applicable to more general problems (like submodular flow) and is likely to be more efficient in practice.  Our algorithm can be interpreted as a cut cancelling algorithm, improving the best known strongly polynomial bound for this important class of algorithms by a factor of m. On the other hand, our algorithm can be considered as a variant of the dual network simplex algorithm. Although dual network simplex algorithms are reportedly quite efficient in practice, the best worst-case running time known so far exceeds the running time of our algorithm by a factor of n.     

Direct algorithm for the solution of two‐sided obstacle problems with M‐matrix

A direct algorithm for the solution to the affine two‐sided obstacle problem with an M‐matrix is presented. The algorithm has the polynomial bounded computational complexity O(n³) and is more efficient than those in (Numer. Linear Algebra Appl. 2006; 13 :543–551). Copyright © 2010 John Wiley & Sons, Ltd.     

Parametric Integer Programming Algorithm for Bilevel Mixed Integer Programs

We consider discrete bilevel optimization problems where the follower solves an integer program with a fixed number of variables. Using recent results in parametric integer programming, we present polynomial time algorithms for pure and mixed integer bilevel problems. For the mixed integer case where the leader’s variables are continuous, our algorithm also detects whether the infimum cost fails to be attained, a difficulty that has been identified but not directly addressed in the literature. In this case, it yields a “better than fully polynomial time” approximation scheme with running time polynomial in the logarithm of the absolute precision. For the pure integer case where the leader’s variables are integer, and hence optimal solutions are guaranteed to exist, we present an algorithm which runs in polynomial time when the total number of variables is fixed.     

A resultant system as the set of coefficients of a single resultant

Explicit expressions for polynomials forming a homogeneous resultant system of a set of m+1 homogeneous polynomial equations in n+1<m+1 variables are given. These polynomials are obtained as coefficients of a homogeneous resultant for an appropriate system of n+1 equations in n+1 variables, which is explicitly constructed from the initial system. Similar results are obtained for mixed resultant systems of sets of n + 1 sections of line bundles on a projective variety of dimension n < m. As an application, an algorithm determining whether one of the orbits under an action of an affine irreducible algebraic group on a quasi-affine variety is contained in the closure of another orbit is described.     

Eigenvalues of a symmetric tridiagonal matrix: A divide-and-conquer approach

Summary The Symmetric Tridiagonal Eigenproblem has been the topic of some recent work. Many methods have been advanced for the computation of the eigenvalues of such a matrix. In this paper, we present a divide-and-conquer approach to the computation of the eigenvalues of a symmetric tridiagonal matrix via the evaluation of the characteristic polynomial. The problem of evaluation of the characteristic polynomial is partitioned into smaller parts which are solved and these solutions are then combined to form the solution to the original problem. We give the update equations for the characteristic polynomial and certain auxiliary polynomials used in the computation. Furthermore, this set of recursions can be implemented on a regulartree structure. If the concurrency exhibited by this algorithm is exploited, it can be shown that thetime for computation of all the eigenvalues becomesO(nlogn) instead ofO(n ²) as is the case for the approach where the order is increased by only one at every step. We address the numerical problems associated with the use of the characteristic polynomial and present a numerically stable technique for the eigenvalue computation.     

Preconditioners for block Toeplitz systems based on circulant preconditioners

We study the numerical solution of a block system T _m,n x=b by preconditioned conjugate gradient methods where T _m,n is an m×m block Toeplitz matrix with n×n Toeplitz blocks. These systems occur in a variety of applications, such as two-dimensional image processing and the discretization of two-dimensional partial differential equations. In this paper, we propose new preconditioners for block systems based on circulant preconditioners. From level-1 circulant preconditioner we construct our first preconditioner q ₁(T _m,n ) which is the sum of a block Toeplitz matrix with Toeplitz blocks and a sparse matrix with Toeplitz blocks. By setting selected entries of the inverse of level-2 circulant preconditioner to zero, we get our preconditioner q ₂(T _m,n ) which is a (band) block Toeplitz matrix with (band) Toeplitz blocks. Numerical results show that our preconditioners are more efficient than circulant preconditioners.     

Fast rectangular matrix multiplication and some applications

We study asymptotically fast multiplication algorithms for matrix pairs of arbitrary di- mensions, and optimize the exponents of their arithmetic complexity bounds. For a large class of input matrix pairs, we improve the known exponents. We also show some applications of our results:(i) we decrease from O(n~2 n~(1 o)(1)logq)to O(n~(1.9998) n~(1 o(1))logq)the known arithmetic complexity bound for the univariate polynomial factorization of degree n over a finite field with q elements; (ii) we decrease from 2.837 to 2.7945 the known exponent of the work and arithmetic processor bounds for fast deterministic(NC)parallel evaluation of the determinant, the characteristic polynomial, and the inverse of an n×n matrix, as well as for the solution to a nonsingular linear system of n equations; (iii)we decrease from O(m~(1.575)n)to O(m~(1.5356)n)the known bound for computing basic solutions to a linear programming problem with m constraints and n variables.     

Polynomial dynamical systems and ordinary differential equations associated with the heat equation

We consider homogeneous polynomial dynamical systems in n-space. To any such system our construction matches a nonlinear ordinary differential equation and an algorithm for constructing a solution of the heat equation. The classical solution given by the Gaussian function corresponds to the case n = 0, while solutions defined by the elliptic theta-function lead to the Chazy-3 equation and correspond to the case n = 2. We explicitly describe the family of ordinary differential equations arising in our approach and its relationship with the wide-known Darboux-Halphen quadratic dynamical systems and their generalizations.     

A class of polynomially solvable linear complementarity problems

Although the general linear complementarity problem (LCP) is NP-complete, there are special classes that can be solved in polynomial time. One example is the type where the defining matrix is nondegenerate and for which the n-step property holds. In this paper we consider an extension of the property to the degenerate case by introducing the concept of an extended n-step vector and matrix. It is shown that the LCP defined by such a matrix is polynomially solvable as well.     

Resultants and contour integrals

Resultants are important special functions used to describe nonlinear phenomena. The resultant R_{r1 ?rn} R_{r_1 \ldots r_n } determines a consistency condition for a system of n homogeneous polynomials of degrees r ₁, ..., r _n in n variables in precisely the same way as the determinant does for a system of linear equations. Unfortunately, there is a lack of convenient formulas for resultants in the case of a large number of variables. In this paper we use Cauchy contour integrals to obtain a polynomial formula for resultants, which is expected to be useful in applications.     

Solution of systems of general-form nonilinear algebraic equations. Methods and algorithms. IV

This paper considers the solution of a system of m nonlinear equations in q>02 variables (SNAE-q). A method for finding all of the finite zero-dimensional roots of a given SNAE-q, which extends the method suggested previously for q=2 and q=3 to the case q≥2, is developed and theoretically justified. This method is based on the algorithm of the ΔW-q factorization of a polynomial q-parameter matrix and on the algorithm of relative factorization of a scalar polynomial in q variables. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 248, 1998, pp. 124–146. Translated by V. N. Kublanovskaya.     

Integer programming formulation of combinatorial optimization problems

This paper considers in a somewhat general setting when a combinatorial optimization problem can be formulated as an (all-integer) integer programming (IP) problem. The main result is that any combinatorial optimization problem can be formulated as an IP problem if its feasible region S is finite but there are many rather sample problems that have no IP formulation if their S is infinite. The approach used for finite S usually gives a formulation with a relatively small number of additional variables for example, an integer polynomial of n 0?1 variables requires at most n + 1 additional variables by our approach, whereas 2ⁿ - (n + 1) additional variables at maximum are required by other existing methods. Finally, the decision problem of deciding whether an arbitrarily given combinatorial optimization problem has an IP formulation is considered and it is shown by an argument closely related to Hilbert's tenth problem (drophantine equations) that no such algorithm exists.     

Efficient generation of random nonsingular matrices

We present an efficient algorithm for generating an n × n nonsingular matrix uniformly over a finite field. This algorithm is useful for several cryptographic and checking applications. Over GF[2] our algorithm runs in expected time M(n) + O(n²), where M(n) is the time needed to multiply two n × n matrices, and the expected number of random bits it uses is n² + 3. (Over other finite fields we use n² + O(1) random field elements on average.) This is more efficient than the standard method for solving this problem, both in terms of expected running time and the expected number of random bits used. The standard method is to generate random n × n matrices until we produce one with nonzero determinant. In contrast, our technique directly produces a random matrix guaranteed to have nonzero determinant. We also introduce efficient algorithms for related problems such as uniformly generating singular matrices or matrices with fixed determinant. © 1993 John Wiley & Sons, Inc.