Inversion of Vandermonde-Like Matrices

Let {x }_=0,...,n be a set of distinct nodes, and let {p _n } _n0 be a sequence of polynomials, degp _n =n, whose elements satisfy a three-term recurrence relation. We derive explicit representations for the inverse of a Vandermonde-like matrix V _n =(p (x ))_,=0,...,n and present some numerical results.     

A New Method for Computing the Weighted Generalized Inversion of Partitioned Matrices

In this paper, we shall give a new method for computing the weighted generalized inversion of partitioned matrices, i.e. solve the weighted problems with the aid of the unweighted ones, which is quite efficient and convenient for either dealing with the inconsistent equations or computing the generalized inverses, now we discuss some problems about the latter. The theorem I below is an extension of the main result in [2], especially, the proof is quite succinct here.     

有理对称矩阵的准正交相似

对一个有理对称矩阵A,用线性代数的理论直接证明了存在准正交矩阵P使得P~(-1)AP成为A的有理标准形,并给出了P的算法.     

无限广义块Toeplitz和Hankel矩阵求逆的统一方法

利用 Sylvester位移方程的统一办法给出所谓的无限广义块Toeplitz和 Hankel矩阵的求逆公式 .     

Inversion of Toeplitz Matrices which are not Strongly Non-singular

A type of patterned matrix called r-Toeplitz is introduced.This is more general than a block Toeplitz matrix, which resultswhen the order n of the matrix is a multiple of r.A well-knownalgorithm for inverting Toeplitz matrices is extended to dealwith r-Toeplitz matrices, involving O(rn²) operations.This enablesan algorithm to be produced for inverting Toeplitz matriceswhich are not strongly non-singular, when the standard techniquesbreak down.     

Equidistribution of Rational Matrices in their Conjugacy Classes

Let G be a connected simply connected almost -simple algebraic group with non-compact and a cocompact congruence subgroup. For any homogeneous manifold of finite volume, and a , we show that the Hecke orbit T _a (x ₀ H) is equidistributed on as , provided H is a non-compact commutative reductive subgroup of G. As a corollary, we generalize the equidistribution result of Hecke points ([COU], [EO1]) to homogeneous spaces G/H. As a concrete application, we describe the equidistribution result in the rational matrices with a given characteristic polynomial. The second author partially supported by DMS 0333397. Received: May 2005 Revision: March 2006 Accepted: June 2006     

Newton's Iteration for Inversion of Cauchy-Like and Other Structured Matrices

We specify some initial assumptions that guarantee rapid refinement of a rough initial approximation to the inverse of a Cauchy-like matrix, by means of our new modification of Newton's iteration, where the input, output, and all the auxiliary matrices are represented with their short generators defined by the associated scaling operators. The computations are performed fast since they are confined to operations with short generators of the given and computed matrices. Because of the known correlations among various structured matrices, the algorithm is immediately extended to rapid refinement of rough initial approximations to the inverses of Vandermonde-like, Chebyshev–Vandermonde-like, and Toeplitz-like matrices, where again the computations are confined to operations with short generators of the involved matrices.     

The Rank-$k$ Updating Algorithm for the Exact Inversion of Matrices with Integer Elements

In this paper, the numerical solution of the matrix problems over a ring of integers is discussed. The rank-$k$ updating algorithm for the exact inversion of a matrix is proposed. This algorithm is generally more effective than Jordan elimination. The common divisor of the numbers involved is reduced to avoid over-swelling of intermediate numbers.     

Rational Procedures in the Problem of Common Invariant Subspaces of Two Matrices

The paper discusses criteria for the existence of common invariant subspaces of two complex matrices verifiable in a finite number of arithmetic operations. Bibliography: 20 titles.     

Enumerating Rational Curves: The Rational Fibration Method

A new method to approach enumerative questions about rational curves on algebraic varieties is described. The idea is to reduce the counting problems to computations on the Néron-Severi group of a ruled surface. Applications include a short proof of Kontsevich's formula for plane curves and the solution of the analogous problem for the Hirzebruch surface F₃.     

k-循环矩阵的最佳SOR方法

本文研究 K-循环矩阵的SOR迭代,提出一种确定最佳松弛因子的方法，应用和改进了Young和Eidson等人的结果，同时给出了计算实例．     

The Polynomial Method for Random Matrices

We define a class of “algebraic” random matrices. These are random matrices for which the Stieltjes transform of the limiting eigenvalue distribution function is algebraic, i.e., it satisfies a (bivariate) polynomial equation. The Wigner and Wishart matrices whose limiting eigenvalue distributions are given by the semicircle law and the Marčenko–Pastur law are special cases. Algebraicity of a random matrix sequence is shown to act as a certificate of the computability of the limiting eigenvalue density function. The limiting moments of algebraic random matrix sequences, when they exist, are shown to satisfy a finite depth linear recursion so that they may often be efficiently enumerated in closed form. In this article, we develop the mathematics of the polynomial method which allows us to describe the class of algebraic matrices by its generators and map the constructive approach we employ when proving algebraicity into a software implementation that is available for download in the form of the RMTool random matrix “calculator” package. Our characterization of the closure of algebraic probability distributions under free additive and multiplicative convolution operations allows us to simultaneously establish a framework for computational (noncommutative) “free probability” theory. We hope that the tools developed allow researchers to finally harness the power of infinite random matrix theory.     

非负矩阵最大特征值的估计法

通过构造一个新的矩阵,从而得到一个非负矩阵最大特征值的估计法,该方法将适用范围推广到一般非负矩阵,并通过实例验证了这种新方法精确度更高.     

P-循环矩阵和块AOR方法(英)

本文首先导出P-循环矩阵的块Jacobi迭代矩阵和相应的块AOR迭代矩阵的特征值间的关系式．然后，我们确定块AOR方法用于最小二乘问题时的收敛和发散区域．     

A New Method for Constructing Williamson Matrices

For every prime power q 1 (mod 4) we prove the existence of (q; x, y)-partitions of GF(q) with q=x²+4y² for some x, y, which are very useful for constructing SDS, DS and Hadamard matrices. We discuss the transformations of (q; x,y)-partitions and, by using the partitions, construct generalized cyclotomic classes which have properties similar to those of classical cyclotomic classes. Thus we provide a new construction for Williamson matrices of order q².The research supported by NSF of China (No. 10071029).     

块Toeplitz三角阵求逆及块Toeplitz三角线性方程组求解的复杂性

In this paper, it is showed that the computational complexity of inversion of block triangular Toeplitz matrix U= (U₀, U₁,…, U_n-1 ) is O (m²nlogn + m³), as well as solution of block triangla r Toeplitz linear systems, where U 's are m×m mat rices. By using this results, we reduce arithmetic operations of division of polynomials from O(nlog²n) to O(nlogn).     

一种新的互补判断矩阵排序方法

排序方法问题是决策分析中的重要问题,具有广泛的应用价值.面对某一项选拔,快速而有效的将备选项目进行排序进而择优和决策显得尤为重要.但由于备选项在各个方面的相近或相同,很难将其排序进行决策,将这些方面以数据来衡量并组合起来,形成判断矩阵.针对互补判断矩阵的乘性一致性定义和加性一致性定义,尝试提出一个新的函数,并分别建立最优化模型,得到了相应的两个排序公式,并从理论上研究了它们的一些性质,通过仿真实例说明该方法的可行性,简便性.     

A New Numerical Method for the Inversion of the Laplace Transform

This paper presents a new numerical method for the inversionof the Laplace transform, where only real values of the transformare known. This method, based on the approximation of the transformby truncated series of orthogonal functions, related to theJacobi or Laguerre polynomials, avoids the loss of significantfigures inherent in earlier methods.