求n阶实方阵A的全部模最大的特征值及其相应特征向量的幂法

本给出并论证了,当ｎ阶实方阵Ａ具有ｉ（１≤ｉ≤ｎ）个（即任意多个）模最大的特征值时,用幂法求出这些模最大的特征值及其相应特征向量的方法。该方法是对幂法理论的进一步完善。     

具有确定非零对角元个数的布尔矩阵广义幂敛指数的极矩阵

设A是周期为P的n阶布尔矩阵,1≤i≤n,A的广义幂敛指数k(A,i)是使得Ak和Ak+p有i行对应相等的最小非负整数k.本文刻画了恰含d(1≤d≤n)个非零对角元的n阶布尔矩阵的广义幂敛指数的极矩阵.     

求n阶实方阵的任意个模最大的特征值及其相应特征向量的规范化幂法

本文给出并论证了当 n阶实方阵 A具有 r( 1≤ r≤n)个模最大的特征值及其相应特征向量的方法 .实施规范化措施 ,使得行范数等于 1 ,在电子计算机上不会产生溢出停机 ,这是一种有实用价值的算法     

“数值分析”有关教材中幂法的一点注记

<正> 众所周知,幂法是求矩阵A的主特征值(按模最大的特征值)λ_1与对应特征向量的一种常用方法。当A有完备特征向量系X_1,X_2,…,X_n且对应特征值满足     

求n阶实方阵的任意个模最大的特征值及其相应特征向量的规范化幂法

本给出并论证了当n阶实方阵A具有r(1≤r≤n）个模最大的特征值及其相应特征向量的方法，实验规范化措施，使得行范数等于1，在电子计算机上不会产生溢出停机，这是一种有实用价值的算法。     

求解特征值问题的 AOR 算法的收敛性

用 AOR 方法求解线性方程组是众所周知的,我们将此方法应用到求解特征值问题方面.考虑下面特征值问题:(A—λI)x=0,(1.1)这里 A 是大型稀疏非奇异对称矩阵.显然,问题(1.1)有下面三条性质:i)其 n 个特征值都是实的,不妨设为λ_1≤λ_2≤…≤λ_n;(1.2)     

一种求方阵特征值的三按模幂法规范化方法

对于ｎ阶实方阵Ａ，用幂法求其一个或二个模最大的特征值（当Ａ只有一个或二个模最大的特征值时）及其相应的特征向量问题已得到解决，这里给出并论证了求一类实方阵Ａ的三个模最大的特征值及其相应的特征向量的规范化方法，实行规范化措施，使得迭代向量的行范数为１，在电子计算机上不会产生溢出停机，这是一种有实用价值的方法。     

任意阶Laplace算子的特征值估计

设D为n维Euclid空间Rn的一个有界区域,且0＜λ1≤λ2≤…≤λk≤…是l阶Laplace算子的Dirichlet问题{(-△)lu=λu, 在D中,u=(e)u/(e)n=…=(e)l-1u/(e)nl-1=0,在(e)D上的特征值.得到了该问题用其前k个特征值来估计第(k+1)个特征值λk+1的不等式k∑i=1(λk+1-λi)≤1/n(4l(n+2l-2)]1/2{k∑i=1(λk+1-λi)1/2λil-1/lk∑i=1(λk+1-λi)1/2λi1/l}1/2,此不等式不依赖于区域D.对l≥3,上述不等式比所有已知的结果都要好.陈庆民与杨洪苍考虑了l=2的情形.我们的结果是他们结果的自然推广.当l=1时,我们的不等式蕴含杨洪苍不等式的弱形式.文中还给出了陈和杨的一个断言的直接证明.     

具有幂零奇点的七次Hamilton系统Abel积分的零点个数估计

本文研究了具有幂零奇点的七次Hamilton系统的Abel积分的零点个数问题.利用Picard-Fuchs方程法,得到了Abel积分I(h)=∮_(Γh)g(x,y)dx-f(x,y)dy在(0,1/4)上零点个数B(n≤3[(n-1)/4]),其中Γ_h是H(x,y)=x~4+y~4-x~8=h,h∈(0,1/4),所定义的卵形线f(x,y)=∑(1≤4i+4j+1≤n)aijx~(4i+1)y~4j)和g(x,y)=∑(1≤4i+4j+1≤n)bijx~4iy~(4j+1)是x和y的次数不超过n的多项式.     

L—自反Fuzzy矩阵的幂等性及正则性

本文利用对Fuzzy矩阵分块的方法,讨论了L-自反Fuzzy矩阵的幂等性及正则性。并利用标准基的性质证明了自反的,非奇异的Fuzzy矩阵的任一广义逆是自反的。本文总设(L,∧,∨)是完备的分配格并简记为L,其最大元最小元分别为1,0,L~(m×n)表示L上全体m×n矩阵的集合。有关记号参见[1]。得到的主要结果是: 命题1 设A∈L~(n×n),A=A~2且若某aii=0(1≤i≤n)则(1)A的第i行和其余各行相关;(2)A的第i列和其余各列相关;(3)若记A(i|i_~2为划去A的第i行,第i列所得     

An extremal property of positive operators and its spectral implications

The Perron-Frobenius theory of a positive operator T (defined on an ordered space E) is developed from an extremal characterization of its largest eigenvalue. This characterization has manifest intrinsic interest. Additionally, it is used to give a particularly revealing derivation of the basic results concerning the existence, multiplicity, and distribution of the eigenvalues of T of maximum modulus. A significant feature of this derivation is that the customary assumptions that the space E be complete and/or that its positive cone have a nonvoid interior are often unnecessary or can be replaced by weaker hypotheses more amenable to practical applications (see 1, 3). The extremal characterization proof of the distribution properties of the eigenvalues of maximum modulus is new in the infinite dimensional case. Also, several new results on the extent of applicability of the extremal characterization are given     

Estimate for the Gradient of the Solution to the Neumann Problem for (p,q)-Nonlinear Equations

We consider the Neumann boundary value problem for the class of (p,q)-nonlinear elliptic equations. The numbers p and q, 2 ⩽ p < q, characterize the power growth with respect to the gradient of eigenvalues of the leading matrix of the equation. An a priori estimate for the maximum of the modulus of the gradient of the solution is obtained in a neighborhood of the boundary of the domain for some interval of p and q. Bibliography: 5 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 31, 2005, pp. 47–57.     

A reflection on the implicitly restarted Arnoldi method for computing eigenvalues near a vertical line

In this article, we will study the link between a method for computing eigenvalues closest to the imaginary axis and the implicitly restarted Arnoldi method. The extension to computing eigenvalues closest to a vertical line is straightforward, by incorporating a shift. Without loss of generality we will restrict ourselves here to computing eigenvalues closest to the imaginary axis.In a recent publication, Meerbergen and Spence discussed a new approach for detecting purely imaginary eigenvalues corresponding to Hopf bifurcations, which is of interest for the stability of dynamical systems. The novel method is based on inverse iteration (inverse power method) applied on a Lyapunov-like eigenvalue problem. To reduce the computational overhead significantly a projection was added.This method can also be used for computing eigenvalues of a matrix pencil near a vertical line in the complex plane. We will prove in this paper that the combination of inverse iteration with the projection step is equivalent to Sorensen’s implicitly restarted Arnoldi method utilizing well-chosen shifts.     

On determinants of nonprincipal submatrices of hermitian matrices

It is our purpose to compute the maximum value of the modulus of the determinant of an m×m nonprincipal submatrix of an n×n hermitian (or real symmetric) matrix A, in terms of m, the eigenvalues of A, and the cardinality k of the set of common row and column indices of this submatrix.     

Convergence rate analysis of discrete-time Markovian jump systems

This paper is concerned with convergence rate analysis of a discrete-time Markovian jump linear system. First, two eigenvalue sets of some operator associated with the Markovian jump linear system are defined to characterize its stability. It is shown that the fastest and slowest convergence rate of the Markovian jump system can be determined by the eigenvalues having minimal modulus and maximal modulus, respectively. Finally, a linear matrix inequality based approach is established to design controllers such that the closed-loop system has a guaranteed convergence rate. A numerical example is carried out to illustrate the effectiveness of the proposed approach.     

Numerical Solution of an Elliptic Boundary-value Problem in the Complex Variable

A general, non-self-adjoint, complex, elliptic boundary-valueproblem is approximated by a system of complex, finite-differenceequations and solved by SOR techniques. Since transformationsof the complex difference equations to real equations lead tosystems which lack diagonal dominance, the difference equationsare solved iteratively as a complex system. A brief accountof complex SOR theory is provided. This includes block as wellas point methods and, for the complex point Jacobi matrix, thedistribution of the eigenvalues and the proof that there isone eigenvalue pair with maximum modulus are given. Importantconsiderations in the application of complex methods—accuracy,factors influencing convergence and the automatic determinationof convergence factors—are examined.     

Asymptotic Correction of More Sturm–Liouville Eigenvalue Estimates

The asymptotic correction technique of Paine, de Hoog and Anderssen can dramatically improve the accuracy of finite difference or finite element eigenvalues at negligible extra cost if closed form expressions are available for the errors in a simpler related problem. This paper gives closed form expressions for the errors in the eigenvalues of certain Sturm–Liouville problems obtained by various methods, thereby increasing the range of problems for which asymptotic correction can achieve maximum efficiency. It also investigates implementation of the method for more general problems.     

A bound for the determinant of the sum of two normal matrices

Hadamard's determinant theorem is used to obtain an upper bound for the modulus of the determinant of the sum of two normal matrices in terms of their eigenvalues. This bound is compared with another given by M. E. Miranda.     

A Note on the Calculation of Step-Lengths in Interior-Point Methods for Semidefinite Programming

In each iteration of an interior-point method for semidefinite programming, the maximum step-length that can be taken by the iterate while maintaining the positive semidefiniteness constraint needs to be estimated. In this note, we show how the maximum step-length can be estimated via the Lanczos iteration, a standard iterative method for estimating the extremal eigenvalues of a matrix. We also give a posteriori error bounds for the estimate. Numerical results on the performance of the proposed method against two commonly used methods for calculating step-lengths (backtracking via Cholesky factorizations and exact eigenvalues computations) are included.