Enhanced algebraic substructuring for symmetric generalized eigenvalue problems

This article proposes a new substructuring algorithm to approximate the algebraically smallest eigenvalues and corresponding eigenvectors of a symmetric positive-definite matrix pencil $(A,M)$. The proposed approach partitions the graph associated with $(A,M)$ into a number of algebraic substructures and builds a Rayleigh–Ritz projection subspace by combining spectral information associated with the interior and interface variables of the algebraic domain. The subspace associated with interior variables is built by computing substructural eigenvectors and truncated Neumann series expansions of resolvent matrices. The subspace associated with interface variables is built by computing eigenvectors and associated leading derivatives of linearized spectral Schur complements. The proposed algorithm can take advantage of multilevel partitionings when the size of the pencil. Experiments performed on problems stemming from discretizations of model problems showcase the efficiency of the proposed algorithm and verify that adding eigenvector derivatives can enhance the overall accuracy of the approximate eigenpairs, especially those associated with eigenvalues located near the origin.     

On eigenvalue calculations for radiative transfer models that include polarization effects

Several representations of the dispersion matrix (z) basic to analytical solutions for a theory of radiative transfer that includes the effects of polarization are reported, and a method for computing the zeros of det (z) is discussed. Numerical results are given for several specific models.

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Zusammenfassung Verschiedene Darstellungen der Dispersionsmatrix (z), welche grundlegende Bedeutung für die analytischen Lösungen der Theorie der Strahlungsübertragung mit Polarisation hat, werden angegeben. Eine Methode zur Berechnung der Nullstellen von det (z) wird diskutiert. Es werden numerische Ergebnisse für verschiedene Modelle angegeben.

     

A block Chebyshev-Davidson method for linear response eigenvalue problems

We present a Chebyshev-Davidson method to compute a few smallest positive eigenvalues and corresponding eigenvectors of linear response eigenvalue problems. The method is applicable to more general linear response eigenvalue problems where some purely imaginary eigenvalues may exist. For the Chebyshev filter, a tight upper bound is obtained by a computable bound estimator that is provably correct under a reasonable condition. When the condition fails, the estimated upper bound may not be a true one. To overcome that, we develop an adaptive strategy for updating the estimated upper bound to guarantee the effectiveness of our new Chebyshev-Davidson method. We also obtain an estimate of the rate of convergence for the Ritz values by our algorithm. Finally, we present numerical results to demonstrate the performance of the proposed Chebyshev-Davidson method.     

Minimization principles and computation for the generalized linear response eigenvalue problem

The minimization principle and Cauchy-like interlacing inequalities for the generalized linear response eigenvalue problem are presented. Based on these theoretical results, the best approximations through structure-preserving subspace projection and a locally optimal block conjugate gradient-like algorithm for simultaneously computing the first few smallest eigenvalues with the positive sign are proposed. Numerical results are presented to illustrate essential convergence behaviors of the proposed algorithm.     

Algebraic Bethe Ansatz for SO(N)-invariant transfer matrices

A matrix version of an algebraic Bethe Ansatz is proposed for R-matrices of a special structure. It is shown that SO(N)-invariant R-matrices in spinor representation possess this special structure. Eigenvalues of transfer matrices corresponding to these R-matrices are computed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 169, pp. 122–140, 1988.     

用逼近方法建立代数无关性(Ⅱ)

设ｆ（ｘ）是以ａｎｘ为元素的连分式,其中系数ａｎ是正代数数．应用［１］中的判别法则,在某些仅与ａｎ有关的条件下证明了对任何绝对值互异的非零实代数数ａ１,…,ａｓ值ｆ（ａ１）代数无关．还对某些实超越数建立了的代数无关性．     

Algebraic Independence by Approximation Method (II)

Let f (x) be a continued fraction with elements a _n x, where coefficients a _n are positive algebraic numbers. Using the criterion of [l] for any nonzero real algebraic numbers α₁,...,α_s with distinct absolute values the algebraic independence of the values f(α₁), ..., f(α_s) is proved under certain assumption concerning only with a _n . For some transcendental numbers ξ the algebraic independence of values f(ξ^j)(j∈ℤ) is also established. Received March 27, 1998, Accepted September 28, 1998     

An inverse eigenvalue method for frequency isolation in spring–mass systems

The action of external vibrating forces on mechanical structures can cause severe damages when resonance occurs. The removal of natural frequencies of the structure from resonance bands is therefore of great importance. This problem is called frequency isolation problem and is the subject of this paper. A new inverse eigenvalue method is proposed and applied to spring–mass systems, which have generated much interest in the literature as prototypes of vibrating structures. The novelty of the method lies in using the zeros of the frequency response function at the last mass as control variables in an optimization problem to minimize the impact of redesign. Numerically accurate algorithms for computing the sensitivity with respect to the control variables are presented, which form the basis of an efficient multidimensional search strategy to solve the frequency isolation problem. Copyright © 2001 by John Wiley & Sons, Ltd.     

Algebraic and essentially algebraic composition operators onC(X)

Summary Given a compact Hausdorff spaceX, we may associate with every continuous mapa: X X a composition operatorC _a onC(X) by the rule(C _a f)(x) = f(a(x)). We describe all self-mapsa for whichC _a is an algebraic operator or an essentially algebraic operator (i.e. an operator algebraic modulo compact operators), determine the characteristic polynomialp _a (z) and the essentially characteristic polynomialq _a (z) in these cases and show how the connectivity ofX may be characterized in terms of the quotientsp _a (z)/q _a (z). Research supported by the Alfried Krupp Förderpreis für junge Hochschullehrer.     

Algebraic Theories of Brackets and Related (Co)Homologies

A general theory of the Frölicher–Nijenhuis and Schouten–Nijenhuis brackets in the category of modules over a commutative algebra is described. Some related structures and (co)homology invariants are discussed, as well as applications to geometry.     

Polynomial (chaos) approximation of maximum eigenvalue functions

Numerical Algorithms - In this note, the additive block diagonal preconditioner (Bai et al., Numer. Algorithms 62, 655–675 2013) and the block triangular preconditioner (Pearson and Wathen,...     

On an Algebraic Extension of A(E)

An algebraic extension of the algebra A(E), where E is a compactum in with nonempty connected interior, leads to a Banach algebra B of functions that are holomorphic on some analytic set K^{o 2} with boundary bK and continuous up to bK. The singular points of the spectrum of B and their defects are investigated. For the case in which B is a uniform algebra, the depth of B in the algebra C(bK) is estimated. In particular, conditions under which B is maximal on bK are obtained.     

某些无穷乘积的代数无关性(Ⅱ)

本文证明了某些具有代数系数的无穷乘积和幂级数定义的函数在某些代数数和超越数上值的代数无关性.     

Optimization techniques for Available Transfer Capability (ATC) and market calculations

The recent movement towards an open, competitive market environmentintroduced new optimization problems such as market clearingmechanism, bidding decision and Available Transfer Capability(ATC) calculation. These optimization problems are characterizedby the complexity of power systems and the uncertainties inthe electricity market. Accurate evaluation of the transfercapability of a transmission system is required to maximizethe utilization of the existing transmission systems in a competitivemarket environment. The transfer capability of the transmissionnetworks can be limited by various system constraints such asthermal, voltage and stability limits. The ability to incorporatesuch limits into the optimization problem is a challenge inthe ATC calculation from an engineering point of view. In thecompetitive market environment, a power supplier needs to findan optimal strategy that maximizes its own profits under variousuncertainties such as electricity prices and load. On the otherhand, an efficient market clearing mechanism is needed to increasethe social welfare, i.e. the sum of the consumers’ andproducers’ surplus. The need to maximize the social welfaresubject to system operational constraints is also a major challengefrom a societal point of view. This paper presents new optimizationtechniques motivated by the competitive electricity market environment.Numerical simulation results are presented to demonstrate theperformance of the proposed optimization techniques.     

Grid Methods in Computational Real Algebraic (and Semialgebraic) Geometry

In recent years, a family of numerical algorithms to solve problems in real algebraic and semialgebraic geometry has been slowly growing. Unlike their counterparts in symbolic computation they are numerically stable. But their complexity analysis, based on the condition of the data, is radically different from the usual complexity analysis in symbolic computation as these numerical algorithms may run forever on a thin set of ill-posed inputs.     

优化路问题的代数方法—论动态规划（Ⅱ）

本文用同一思路求解多阶段有向图中三种优化路问题:最优路、N阶最优路及多指标Pareto优化路问题,它们都服从嘉量原理,都用同一个代数公式表达它们的嘉量,并可在同一种表格中进行计算,只是所在半域不同,以本文的方法讨论动态规划中一些离散决定型典型应用问题,其提法、建模思路以及求解过程都有可观的扩大与改善。