求置换因子循环矩阵的逆阵及广义逆阵的快速算法

1 引 言 循环矩阵由于其应用非常广泛而成为一类重要的特殊矩阵,如在图象处理、编码理论、自回归滤波器设计等领域中经常会遇到以这类矩阵为系数的线性系统的求解问题.而对称循环组合系统也具有广泛的实际背景,例如造纸机的横向控制系统,具有平行结     

A spectral element method using the modal basis and its application in solving second‐order nonlinear partial differential equations

We present a high‐order spectral element method (SEM) using modal (or hierarchical) basis for modeling of some nonlinear second‐order partial differential equations in two‐dimensional spatial space. The discretization is based on the conforming spectral element technique in space and the semi‐implicit or the explicit finite difference formula in time. Unlike the nodal SEM, which is based on the Lagrange polynomials associated with the Gauss–Lobatto–Legendre or Chebyshev quadrature nodes, the Lobatto polynomials are used in this paper as modal basis. Using modal bases due to their orthogonal properties enables us to exactly obtain the elemental matrices provided that the element‐wise mapping has the constant Jacobian. The difficulty of implementation of modal approximations for nonlinear problems is treated in this paper by expanding the nonlinear terms in the weak form of differential equations in terms of the Lobatto polynomials on each element using the fast Fourier transform (FFT). Utilization of the Fourier interpolation on equidistant points in the FFT algorithm and the enough polynomial order of approximation of the nonlinear terms can lead to minimize the aliasing error. Also, this approach leads to finding numerical solution of a nonlinear differential equation through solving a system of linear algebraic equations. Numerical results for some famous nonlinear equations illustrate efficiency, stability and convergence properties of the approximation scheme, which is exponential in space and up to third‐order in time. Copyright © 2014 John Wiley & Sons, Ltd.     

The black-box Niederreiter algorithm and its implementation over the binary field

The most time-consuming part of the Niederreiter algorithm for factoring univariate polynomials over finite fields is the computation of elements of the nullspace of a certain matrix. This paper describes the so-called ``black-box' Niederreiter algorithm, in which these elements are found by using a method developed by Wiedemann. The main advantages over an approach based on Gaussian elimination are that the matrix does not have to be stored in memory and that the computational complexity of this approach is lower. The black-box Niederreiter algorithm for factoring polynomials over the binary field was implemented in the C programming language, and benchmarks for factoring high-degree polynomials over this field are presented. These benchmarks include timings for both a sequential implementation and a parallel implementation running on a small cluster of workstations. In addition, the Wan algorithm, which was recently introduced, is described, and connections between (implementation aspects of) Wan's and Niederreiter's algorithm are given.

     

On Koetter's Algorithm and the Computation of Error Values

In this article, we show that Koetter's algorithm for decoding one-point codes can compute error evaluator polynomials as well. We also show that the error evaluators do not need to be computed. The updating functions used in Koetter's algorithm can be used to compute error values instead.     

Fast Algorithms to Generate Necklaces, Unlabeled Necklaces, and Irreducible Polynomials over GF(2)

Many applications call for exhaustive lists of strings subject to various constraints, such as inequivalence under group actions. A k-ary necklace is an equivalence class of k-ary strings under rotation (the cyclic group). A k-ary unlabeled necklace is an equivalence class of k-ary strings under rotation and permutation of alphabet symbols. We present new, fast, simple, recursive algorithms for generating (i.e., listing) all necklaces and binary unlabeled necklaces. These algorithms have optimal running times in the sense that their running times are proportional to the number of necklaces produced. The algorithm for generating necklaces can be used as the basis for efficiently generating many other equivalence classes of strings under rotation and has been applied to generating bracelets, fixed density necklaces, and chord diagrams. As another application, we describe the implementation of a fast algorithm for listing all degree n irreducible and primitive polynomials over GF(2).     

Simple algorithms for approximating all roots of a polynomial with real roots

We present a simple algorithm for approximating all roots of a polynomial p(x) when it has only real roots. The algorithm is based on some interesting properties of the polynomials appearing in the Extended Euclidean Scheme for p(x) and p′(x). For example, it turns out that these polynomials are orthogonal; as a consequence, we are able to limit the precision required by our algorithm in intermediate steps. A parallel implementation of this algorithm yields a P-uniform NC₂ circuit, and the bit complexity of its sequential implementation is within a polylog factor of the bit complexity of the best known algorithm for the problem.     

Continued Fractions for Rogers–Szegö Polynomials

We evaluate different Hankel determinants of Rogers–Szegö polynomials, and deduce from it continued fraction expansions for the generating function of RS polynomials. We also give an explicit expression of the orthogonal polynomials associated to moments equal to RS polynomials, and a decomposition of the Hankel form with RS polynomials as coefficients.     

Error analysis of corner cutting algorithms

Corner cutting algorithms are used in different fields and, in particular, play a relevant role in Computer Aided Geometric Design. Evaluation algorithms such as the de Casteljau algorithm for polynomials and the de Boor–Cox algorithm for B‐splines are examples of corner cutting algorithms. Here backward and forward error analysis of corner cutting algorithms are performed. The running error is also analyzed and as a consequence the general algorithm is modified to include the computation of an error bound. This revised version was published online in June 2006 with corrections to the Cover Date.     

A new formula for fractional integrals of Chebyshev polynomials: Application for solving multi-term fractional differential equations

A new explicit formula for the integrals of shifted Chebyshev polynomials of any degree for any fractional-order in terms of shifted Chebyshev polynomials themselves is derived. A fast and accurate algorithm is developed for the solution of linear multi-order fractional differential equations (FDEs) by considering their integrated forms. The shifted Chebyshev spectral tau (SCT) method based on the integrals of shifted Chebyshev polynomials is applied to construct the numerical solution for such problems. The method is then tested on examples. It is shown that the SCT yields better results.     

Constructions of general polynomial lattice rules based on the weighted star discrepancy

In this paper we study construction algorithms for polynomial lattice rules modulo arbitrary polynomials. Polynomial lattice rules are a special class of digital nets which yield well distributed point sets in the unit cube for numerical integration.Niederreiter obtained an existence result for polynomial lattice rules modulo arbitrary polynomials for which the underlying point set has a small star discrepancy and recently Dick, Leobacher and Pillichshammer introduced construction algorithms for polynomial lattice rules modulo an irreducible polynomial for which the underlying point set has a small (weighted) star discrepancy.In this work we provide construction algorithms for polynomial lattice rules modulo arbitrary polynomials, thereby generalizing the previously obtained results. More precisely we use a component-by-component algorithm and a Korobov-type algorithm. We show how the search space of the Korobov-type algorithm can be reduced without sacrificing the convergence rate, hence this algorithm is particularly fast. Our findings are based on a detailed analysis of quantities closely related to the (weighted) star discrepancy.     

Avoiding breakdown in the CGS algorithm

The conjugate gradient squared algorithm can suffer of similar breakdowns as Lanczos type methods for the same reason that is the non-existence of some formal orthogonal polynomials. Thus curing such breakdowns is possible by jumping over these non-existing polynomials and using only those of them which exist. The technique used is similar to that employed for avoiding breakdowns in Lanczos type methods. The implementation of these new methods is discussed. Numerical examples are given.     

The ERES method for computing the approximate GCD of several polynomials

The computation of the greatest common divisor (GCD) of a set of polynomials has interested the mathematicians for a long time and has attracted a lot of attention in recent years. A challenging problem that arises from several applications, such as control or image and signal processing, is to develop a numerical GCD method that inherently has the potential to work efficiently with sets of several polynomials with inexactly known coefficients. The presented work focuses on: (i) the use of the basic principles of the ERES methodology for calculating the GCD of a set of several polynomials and defining approximate solutions by developing the hybrid implementation of this methodology. (ii) the use of the developed framework for defining the approximate notions for the GCD as a distance problem in a projective space to develop an optimization algorithm for evaluating the strength of different ad-hoc approximations derived from different algorithms. The presented new implementation of ERES is based on the effective combination of symbolic–numeric arithmetic (hybrid arithmetic) and shows interesting computational properties for the approximate GCD problem. Additionally, an efficient implementation of the strength of an approximate GCD is given by exploiting some of the special aspects of the respective distance problem. Finally, the overall performance of the ERES algorithm for computing approximate solutions is discussed.     

Parallel collocation solution of index-1 BVP-DAEs arising from constrained optimal control problems

The indirect solution of constrained optimal control problems gives rise to two-point boundary value problems (BVPs) that involve index-1 differential-algebraic equations (DAEs) and inequality constraints. This paper presents a parallel collocation algorithm for the solution of these inequality constrained index-1 BVP-DAEs. The numerical algorithm is based on approximating the DAEs using piecewise polynomials on a nonuniform mesh. The collocation method is realized by requiring that the BVP-DAE be satisfied at Lobatto points within each interval of the mesh. A Newton interior-point method is used to solve the collocation equations, and maintain feasibility of the inequality constraints. The implementation of the algorithm involves: (i) parallel evaluation of the collocation equations; (ii) parallel evaluation of the system Jacobian; and (iii) parallel solution of a boarded almost block diagonal (BABD) system to obtain the Newton search direction. Numerical examples show that the parallel implementation provides significant speedup when compared to a sequential version of the algorithm.     

On the Key Equation Over a Commutative Ring

We define alternant codes over a commutative ring R and a corresponding key equation. We show that when the ring is a domain, e.g. the p-adic integers, the error-locator polynomial is the unique monic minimal polynomial (equivalently, the unique shortest linear recurrence) of the finite sequence of syndromes and that it can be obtained by Algorithm MR of Norton.WhenR is a local ring, we show that the syndrome sequence may have more than one (monic) minimal polynomial, but that all the minimal polynomials coincide modulo the maximal ideal ofR . We characterise the set of minimal polynomials when R is a Hensel ring. We also apply these results to decoding alternant codes over a local ring R: it is enough to find any monic minimal polynomial over R and to find its roots in the residue field. This gives a decoding algorithm for alternant codes over a finite chain ring, which generalizes and improves a method of Interlando et. al. for BCH and Reed-Solomon codes over a Galois ring.     

Hensel lifting and bivariate polynomial factorisation over finite fields

This paper presents an average time analysis of a Hensel lifting based factorisation algorithm for bivariate polynomials over finite fields. It is shown that the average running time is almost linear in the input size. This explains why the Hensel lifting technique is fast in practice for most polynomials.

     

A note on inverses of cyclotomic mapping permutation polynomials over finite fields

In this note, we give a shorter proof of the result of Zheng, Yu, and Pei on the explicit formula of inverses of generalized cyclotomic permutation polynomials over finite fields. Moreover, we characterize all these cyclotomic permutation polynomials that are involutions. Our results provide a fast algorithm (only modular operations are involved) to generate many classes of generalized cyclotomic permutation polynomials, their inverses, and involutions.     

Numerical investigation into the dependence of the Allen–Cahn equation on the free energy

In this paper, we study a direct parallel-in-time (PinT) algorithm for first- and second-order time-dependent differential equations. We use a second-order boundary value method as the time integrator. Instead of solving the corresponding all-at-once system iteratively, we diagonalize the time discretization matrix B, which yields a direct parallel implementation across all time steps. A crucial issue of this methodology is how the condition number (denoted by Cond₂(V )) of the eigenvector matrix V of B behaves as n grows, where n is the number of time steps. A large condition number leads to large roundoff error in the diagonalization procedure, which could seriously pollute the numerical accuracy. Based on a novel connection between the characteristic equation and the Chebyshev polynomials, we present explicit formulas for V and V^− 1, by which we prove that Cond\(_{2}(V)=\mathcal {O}(n^{2})\). This implies that the diagonalization process is well-conditioned and the roundoff error only increases moderately as n grows, and thus, compared to other direct PinT algorithms, a much larger n can be used to yield satisfactory parallelism. A fast structure-exploiting algorithm is also designed for computing the spectral diagonalization of B. Numerical results on parallel machine are given to support our findings, where over 60 times speedup is achieved with 256 cores.

     

Error Analysis of Clenshaw's Algorithm for Evaluating Derivatives of a Polynomial

A forward rounding error analysis is presented for the extended Clenshaw algorithm due to Skrzipek for evaluating the derivatives of a polynomial expanded in terms of orthogonal polynomials. Reformulating in matrix notation the three-term recurrence relation satisfied by orthogonal polynomials facilitates the estimate of the rounding error for the m-th derivative, which is recursively estimated in terms of the one for the (m – 1)-th derivative. The rounding errors in an important case of Chebyshev polynomial are discussed in some detail.This revised version was published online in October 2005 with corrections to the Cover Date.