Interpolation algorithm of Leverrier–Faddev type for polynomial matrices

We investigated an interpolation algorithm for computing outer inverses of a given polynomial matrix, based on the Leverrier–Faddeev method. This algorithm is a continuation of the finite algorithm for computing generalized inverses of a given polynomial matrix, introduced in [11]. Also, a method for estimating the degrees of polynomial matrices arising from the Leverrier–Faddeev algorithm is given as the improvement of the interpolation algorithm. Based on similar idea, we introduced methods for computing rank and index of polynomial matrix. All algorithms are implemented in the symbolic programming language MATHEMATICA , and tested on several different classes of test examples.     

二元对角向量值有理插值的算法

首先提出了二元对角向量值有理插值问题,它包括主对角和副对角两种向量值有理插值,并分别给出了主对角线和副对角线上向量值有理插值的两种算法,即直接求系数bi,j的算法和基于Samelson广义逆所定义的特殊初等变换的矩阵算法.然后构造了在预给极点情况下求主对角线和副对角线上向量值有理插值的矩阵算法.最后给出多个数值例子说明上述算法的有效性.     

Fast block diagonalization of k-tridiagonal matrices

In the present paper, we give a fast algorithm for block diagonalization of k-tridiagonal matrices. The block diagonalization provides us with some useful results: e.g., another derivation of a very recent result on generalized k-Fibonacci numbers in [M.E.A. El-Mikkawy, T. Sogabe, A new family of k-Fibonacci numbers, Appl. Math. Comput. 215 (2010) 4456-4461]; efficient (symbolic) algorithm for computing the matrix determinant.     

Pivoting strategy for rank-one modification ofLDM t-like factorization

The rank-one modification algorithm of theLDM ^t factorization was given by Bennett [1]. His method, however, could break down even when the matrix is nonsingular and well-conditioned. We introduce a pivoting strategy for avoiding possible break-down as well as for suppressing error growth in the modification process. The method is based on a symbolic formula of the rank-one modification of the factorization of a possibly singular nonsymmetric matrix. A new symbolic formula is also obtained for the inverses of the factor matrices. Repeated application of our method produces theLDM ^t-like product form factorization of a matrix. A numerical example is given to illustrate our pivoting method. An incomplete factorization algorithm is also introduced for updating positive definite matrix useful in quasi-Newton methods, in which the Fletcher and Powell algorithm [2] and the Gill, Murray and Saunders algorithm [4] are usually used.This paper is presented at the Japan SIAM Annual Meeting held at University of Tokyo, Japan, October 7–9, 1991.     

Computing generalized inverses of a rational matrix and applications

In this paper we investigate symbolic implementation of two modifications of the Leverrier-Faddeev algorithm, which are applicable in computation of the Moore-Penrose and the Drazin inverse of rational matrices. We introduce an algorithm for computation of the Drazin inverse of rational matrices. This algorithm represents an extension of the papers [11] and [14]. and a continuation of the papers [15, 16]. The symbolic implementation of these algorithms in the package mathEmatica is developed. A few matrix equations are solved by means of the Drazin inverse and the Moore-Penrose inverse of rational matrices.     

Inversion of general tridiagonal matrices

In the current work, the authors present a symbolic algorithm for finding the inverse of any general nonsingular tridiagonal matrix. The algorithm is mainly based on the work presented in [Y. Huang, W.F. McColl, Analytic inversion of general tridiagonal matrices, J. Phys. A 30 (1997) 7919–7933] and [M.E.A. El-Mikkawy, A fast algorithm for evaluating nth order tridiagonal determinants, J. Comput. Appl. Math. 166 (2004) 581–584]. It removes all cases where the numeric algorithm in [Y. Huang, W.F. McColl, Analytic inversion of general tridiagonal matrices, J. Phys. A 30 (1997) 7919–7933] fails. The symbolic algorithm is suited for implementation using Computer Algebra Systems (CAS) such as MACSYMA, MAPLE and MATHEMATICA. An illustrative example is given.     

Symbolic algorithm for solving cyclic penta-diagonal linear systems

In this paper, by using a special matrix factorization, a symbolic computational algorithm is developed to solve the cyclic penta-diagonal linear system. The algorithm is suitable for implementation using Computer Algebra Systems (CASs) such as MATLAB, MATHEMATICA and MAPLE. In addition, an efficient way of evaluating the determinant of a cyclic penta-diagonal matrix is also discussed. Two numerical examples are given for the purpose of illustration.     

Computing some classes of Cauchy type singular integrals with Mathematica software

We constructed an algorithm, [SInt], for computing some classes of Cauchy type singular integrals on the unit circle. The design of [SInt] was focused on the possibility of implementing on a computer all the extensive symbolic and numeric calculations present in the algorithm. Furthermore, we show how the factorization algorithm described in Conceição et al. (2010) allowed us to construct and implement the [SIntAFact] algorithm for calculating several interesting singular integrals that cannot be computed by [SInt]. All the above techniques were implemented using the symbolic computation capabilities of the computer algebra system Mathematica. The corresponding source code of [SInt] is made available in this paper. Several examples of nontrivial singular integrals computed with both algorithms are presented.     

基于广义逆的多元矩阵有理插值

本文借助于文[5]给出的一种矩阵广义逆，构造了二元Stieltjes型矩阵连分式的截断连分式，以此首次定义了平面上拟三角形网格上的二元矩阵有理插道值函数。文中给出了存在性的一个有用的判别条件。重要的特征定理和唯一性定理得到证明，并借助了实例说明了本文的结果。     

离散点集上的向量有理插值算法与特征性质

1引言 给出R~2中离散点集     

A parallel balance scheme for banded linear systems

A parallel algorithm is proposed for the solution of narrow banded non‐symmetric linear systems. The linear system is partitioned into blocks of rows with a small number of unknowns common to multiple blocks. Our technique yields a reduced system defined only on these common unknowns which can then be solved by a direct or iterative method. A projection based extension to this approach is also proposed for computing the reduced system implicitly, which gives rise to an inner–outer iteration method. In addition, the product of a vector with the reduced system matrix can be computed efficiently on a multiprocessor by concurrent projections onto subspaces of block rows. Scalable implementations of the algorithm can be devized for hierarchical parallel architectures by exploiting the two‐level parallelism inherent in the method. Our experiments indicate that the proposed algorithm is a robust and competitive alternative to existing methods, particularly for difficult problems with strong indefinite symmetric part. Copyright © 2001 John Wiley & Sons, Ltd.     

Symbolic computing

It is shown that symbolic computing can be effectively applied to derive actuarial formulae and used to prove theorems. Illustrating examples are given through applying the computer package MACSYMA to problems in deriving prob- ability functions for fractional ages, deriving the power series of an annuity coefficient, calculating annuities-certain, invert- ing a matrix containing a variable in graduation, and providing Jensen's inequality. Using symbolic computing effectively is a first step to building expert systems in actuarial science.     

解非对称矩阵特征值问题的一种并行分治算法

１引言考虑矩阵特征值问题其中Ａ是非对称矩阵．通过正交变换（如Ｈｏｕｓｅｈｏｌｄｅｒ变换或Ｇｉｖｅｎｓ变换）,Ａ可化为上Ｈｅｓｓｅｎｂｅｒｇ形．因而,本文假设Ａ为上Ｈｅｓｓｅｎｂｅｒｇ矩阵,表示如下：不失一般性,进一步假设所有的（ｊ＝２,…,ｎ）,即认为Ａ是不可约的关于如何求解上述问题,人们进行了不懈的努力,提出了许多行之有效的算法［１－８］．其中分治算法因具有良好的并行性而引人注目．分治算法的典型代表是基于同伦连续的分治算法［２,３,４］和基于Ｎｅｗｔｏｎ迭代的分治算法［１］．本文提出一种新的分…     

Determinantal formulae and nonsymmetric gaussian perfect elimination

Based upon certain key principal minors, a formula is given for the determinant of a nonsingular matrix in the case its inverse has a perfect elimination directed graph. The main result of [8] may be deduced from this formula. Our method allows us to construct a counterexample to a conjecture made in [8].     

A parallel quasi-Newton method for partially separable large scale minimization

The parallel quasi-Newton method based on updating conjugate subspaces proposed in [4] can be very effective for large-scale sparse minimization because conjugate subspaces with respect to sparse Hessians are usually easy to obtain. We demonstrate this point in this paper for the partially separable case with matrices updated by a quasi-Newton scheme ofGriewank andToint [2,3]. The algorithm presented is suitable for parallel computation and economical in computer storage. Some testing results of the algorithm on an Alliant FX/8 minisupercomputer are reported.The material is based on work supported in part by the National Science Foundation under Grant No. DMS 8602419 and by the Center for Supercomputing Research and Development at the University of Illinois.     

Parallelization Strategies for Rollout Algorithms

Rollout algorithms are innovative methods, recently proposed by Bertsekas et al. [3], for solving NP-hard combinatorial optimization problems. The main advantage of these approaches is related to their capability of magnifying the effectiveness of any given heuristic algorithm. However, one of the main limitations of rollout algorithms in solving large-scale problems is represented by their computational complexity. Innovative versions of rollout algorithms, aimed at reducing the computational complexity in sequential environments, have been proposed in our previous work [9]. In this paper, we show that a further reduction can be accomplished by using parallel technologies. Indeed, rollout algorithms have very appealing characteristics that make them suitable for efficient and effective implementations in parallel environments, thus extending their range of relevant practical applications.We propose two strategies for parallelizing rollout algorithms and we analyze their performance by considering a shared-memory paradigm. The computational experiments have been carried out on a SGI Origin 2000 with 8 processors, by considering two classical combinatorial optimization problems. The numerical results show that a good reduction of the execution time can be obtained by exploiting parallel computing systems.     

Equivalence-free exhaustive generation of matroid representations

In this paper we present an algorithm for the problem of exhaustive equivalence-free generation of 3-connected matroids which are represented by a matrix over some finite (partial) field, and which contain a given minor. The nature of this problem is exponential, and it appears to be much harder than, say, isomorph-free generation of graphs. Still, our algorithm is very suitable for practical use, and it has been successfully implemented in our matroid computing package MACEK [http://www.mcs.vuw.ac.nz/research/macek, 2001-05].     

A new algorithm for computing the inverse and the determinant of a Hessenberg matrix

In this paper, a new recursive symbolic computational Hessenberg matrix algorithm is presented to compute the inverse and the determinant of a Hessenberg matrix.     

Computing hyperbolic choreographies

An algorithm is presented for numerical computation of choreographies in spaces of constant negative curvature in a hyperbolic cotangent potential, extending the ideas given in a companion paper [14] for computing choreographies in the plane in a Newtonian potential and on a sphere in a cotangent potential. Following an idea of Diacu, Pérez-Chavela and Reyes Victoria [9], we apply stereographic projection and study the problem in the Poincaré disk. Using approximation by trigonometric polynomials and optimization methods with exact gradient and exact Hessian matrix, we find new choreographies, hyperbolic analogues of the ones presented in [14]. The algorithm proceeds in two phases: first BFGS quasi-Newton iteration to get close to a solution, then Newton iteration for high accuracy.