A Block–Stodola eigensolution technique for large algebraic systems with non-symmetrical matrices

A Block–Stodola eigensolution method is presented for large algebraic eigensystems of the form AU = λ BU where A is real but non-symmetric. The steps in this method parallel those of a previous technique for the case when both A and B were real and symmetric. The essence of the technique is simultaneous iteration using a group of trial vectors instead of only one vector as is the case in the classical Stodola–Vianello iteration method. The problem is then transformed into a subspace where a direct solution of the reduced algebraic eigenvalue problem is sought. The main advantage is the significant reduction of computational effort in extracting a subset of eigenvalues and corresponding eigenvectors. Theorems from linear algebra serve to underlie the basis of the present technique. Complex eigendata that emerge during iteration can be handled without doubling the size of the problem. Higher order eigenvalue problems are reducible to first order form for which this technique is applicable. The treatment of the quadratic eigenvalue problem illustrates the details of this extension.     

Trefftz-type boundary elements for the evaluation of symmetric coefficient matrices

The classical Trefftz-method can be generalized such that different types of finite elements and boundary elements are obtained. In a Trefftz-type approach we utilize functions which a priori satisfy the governing differential equations. In this paper the systematic construction of singular Trefftz-trial functions for elasticity problems is discussed. For convenience a list of solution representations and particular solutions is given which did not appear together elsewhere. The Trefftz-trial functions with singular expressions on the boundary are constructed such that the physical components (stresses, strains, displacements) remain finite in the solution domain and on the boundary. The unknown coefficients of the linearly independent Trefftz-trial functions for the physical components can be obtained by using a variational formulation. The symmetric coefficient matrix in the discussed procedure can be obtained from the evaluation of boundary integrals. As an application of the proposed boundary element algorithm, the symmetric stiffness matrices of subdomains (finite element domains) are calculated. For the numerical example the solution domain is decomposed into triangular subdomains so that a standard finite element program could be used to assemble the system of equations. The chosen example is meant as a simple test for the proposed algorithm and should not be understood as a proposal for a new triangular finite element. Using the proposed boundary element techniques, symmetric stiffness matrices for irregular shaped subdomains (finite elements) can be derived. However, in order to use the method in a finite element package for the coupling of irregular shaped subdomains some program modifications will be necessary.     

Simultaneous tridiagonalization of two symmetric matrices

We show how to simultaneously reduce a pair of symmetric matrices to tridiagonal form by congruence transformations. No assumptions are made on the non‐singularity or definiteness of the two matrices. The reduction follows a strategy similar to the one used for the tridiagonalization of a single symmetric matrix via Householder reflectors. Two algorithms are proposed, one using non‐orthogonal rank‐one modifications of the identity matrix and the other, more costly but more stable, using a combination of Householder reflectors and non‐orthogonal rank‐one modifications of the identity matrix with minimal condition numbers. Each of these tridiagonalization processes requires O(n³) arithmetic operations and respects the symmetry of the problem. We illustrate and compare the two algorithms with some numerical experiments. Copyright © 2003 John Wiley & Sons, Ltd.     

Ordering symmetric sparse matrices for small profile and wavefront

The ordering of large sparse symmetric matrices for small profile and wavefront or for small bandwidth is important for the efficiency of frontal and variable‐band solvers. In this paper, we look at the computation of pseudoperipheral nodes and compare the effectiveness of using an algorithm based on level‐set structures with using the spectral method as the basis of the Reverse Cuthill–McKee algorithm for bandwidth reduction. We also consider a number of ways of improving the performance and efficiency of Sloan's algorithm for profile and wavefront reduction, including the use of different weights, the use of supervariables, and implementing the priority queue as a binary heap. We also examine the use of the spectral ordering in combination with Sloan's algorithm. The design of software to implement the reverse Cuthill–McKee algorithm and a modified Sloan's algorithm is discussed. Extensive numerical experiments that justify our choice of algorithm are reported on. Copyright © 1999 John Wiley & Sons, Ltd.     

Subspace iteration accelrated by using Chebyshev polynomials for eigenvalue problems with symmetric matrices

Bathe's algorithm of subspace iteration for the solution of the eigenvalue problem with symmetric matrices is improved by incorporating an acceleration technique using Chebyshev polynomials. This method of acceleration is particularly effective for this kind of iteration. The rate of convergence of the iteration scheme presented is considerably improved when compared with the original one, and satisfactory rates of convergence can be obtained for a wider range of eigenvalues.     

A method for solving high-order real symmetric eigenvalue problems

A new method for solving structural dynamics problems has been proposed by the author in a separate paper.¹ This method of solution produces a high order real symmetric eigenvalue problem of the form ( A ? λ B ? λ² C ? λ³ D …) U = 0. An algorithm for solving such an eigenvalue problem using simultaneous iteration is presented in this paper. Methods of accelerating the convergence and reducing the amount of computation are also described. A numerical example is given in which the algorithm is used to calculate the eigenvalues and eigenvectors of a framed structure.     

A subspace iteration algorithm for the eigensolution of large structures subject to non‐classical viscous damping

This paper describes an efficient and numerically stable algorithm for accurately computing the solutions of the quadratic eigenproblem associated to non‐proportionally viscously damped structures characterized by symmetric matrices. Combining the simultaneous inverse iteration with a generalized Rayleigh–Ritz analysis, the proposed procedure is well suited for extracting the subset of the lowest natural frequencies, the corresponding subcritical damping ratios and the mode shapes of large dissipative systems with non‐classical viscous damping. The iterative process exploits the specific nature of non‐proportionally damped structures, takes full advantage of the banded configuration of the structural matrices involved in the eigenproblem, avoids the computation of the left eigenvectors and circumvents the use of complex algebra owing to a unitary transformation strategy. An academic test case and an industrial numerical example are presented to highlight the effectiveness of the algorithm. Copyright © 2000 John Wiley & Sons, Ltd.     

PSO和Cholesky分解的KELM的基因表达数据分类

核极限学习机(KELM)可使低维空间中线性不可分的数据变得线性可分,增加了ELM算法的鲁棒性,但KELM算法的输入权值参数采用随机初始化,容易导致算法不稳定.为此,本研究提出用粒子群优化算法对KELM中的权值初始参数进行优化、设定,以得到优化的分类器PSO-KELM.由于该算法输出权值求解采用传统的矩阵求逆运算,导致计算复杂,因此再对KELM的输出权值采用Cholesky分解进行优化.经一些标准基因数据集的实验表明,提出的PSO-KELM算法与已有的ELM、KELM、PSO-ELM相比分类精度更高,适用于基因表达数据分类.     

左移2进制gcd算法的改进

求两个整数的最大公因子(gcd)是密码学中重要的算法.左移gcd算法是对右移gcd算法在执行效率方面的改进.提出了一个改进的左移2进制gcd算法.分析和实验均表明,改进算法比原算法具有更高的效率.     

A block solver for large,unsymmetric, sparse,banded matrices with symmetric profiles

A block equation solver for the solution of large, sparse, banded unsymmetric system of linear equations is presented in this paper. The method employs Crout variation of Gauss elimination technique for the solution. The solver ensures the efficient use of the available memory by doing block factorization and storage. It uses a skyline storage scheme which will avoid unnecessary operations on zero elements above the skyline which has found widespread use in banded symmetric solvers. A FORTRAN code with ample comments is provided. © 1997 John Wiley & Sons, Ltd.     

A dissection solver with kernel detection for symmetric finite element matrices on shared memory computers

A direct solver for symmetric sparse matrices from finite element problems is presented. The solver is supposed to work as a local solver of domain decomposition methods for hybrid parallelization on cluster systems of multi‐core CPUs, and then it is required to run on shared memory computers and to have an ability of kernel detection. Symmetric pivoting with a given threshold factorizes a matrix with a decomposition introduced by a nested bisection and selects suspicious null pivots from the threshold. The Schur complement constructed from the suspicious null pivots is examined by a factorization with 1 × 1 and 2 × 2 pivoting and by a robust kernel detection algorithm based on measurement of residuals with orthogonal projections onto supposed image spaces. A static data structure from the nested bisection and a block sub‐structure for Schur complements at all bisection levels can use level 3 BLAS routines efficiently. Asynchronous task execution for each block can reduce idle time of processors drastically, and as a result, the solver has high parallel efficiency. Competitive performance of the developed solver to Intel Pardiso on shared memory computers is shown by numerical experiments. Copyright © 2014 John Wiley & Sons, Ltd.     

A geometry-based algorithm for cloning real grains

We introduce a computational algorithm to “clone” the grain morphologies of a sample of real grains that have been digitalized. This cloning algorithm allows us to generate an arbitrary number of cloned grains that satisfy the same distributions of morphological features displayed by their parents and can be included into a numerical Discrete Element Method simulation. This study is carried out in three steps. First, distributions of morphological parameters such as aspect ratio, roundness, principal geometric directions, and spherical radius, called the morphological DNA, are extracted from the parents. Second, the geometric stochastic cloning (GSC) algorithm, relying purely on statistical distributions of the aforementioned parameters, is explained, detailed, and used to generate a pool of clones from its parents’ morphological DNA. Third, morphological DNA is extracted from the pool of clones and compared to the one obtained from a similar pool of parents, and the distribution of volume-surface ratio is used to perform quality control. Then, from these results, the error (mutation) in the GSC process is analyzed and used to discuss the algorithm’s drawbacks, knobs (parameters) tuning, as well as potential improvements.     

A simultaneous iteration algorithm for symmetric eigenvalue problems

A FORTRAN IV algorithm is presented for determining sets of dominant eigenvalues and corresponding eigenvectors of symmetric matrices. It is also extended to the solution of the equations of natural vibration of a structure for which symmetric stiffness and mass matrices are available. The matrices are stored and processed in variable bandwidth form, thus enabling advantage to be gained from sparseness in the equations. Some of the procedures may also be used to solve symmetric positive definite equations such as those arising from the static analysis of structures loaded within the elastic range.     

A subspace iteration method for the eigensolution of large undamped gyroscopic systems

This paper presents an efficient and numerically stable algorithm for computing accurately the eigenvalues of smallest moduli and the associated eigenvectors of the quadratic eigenproblem which arises in the analysis of spinning systems. Derived from the well-known standard subspace iteration method, this solution procedure takes full advantage of the banded configuration of the structural matrices, and of the specific nature of gyroscopic systems. A numerical example is presented to show the efficiency and accuracy of the algorithm.     

基于遗传算法的LQR算法中权矩阵的优化分析

LQR控制算法是目前结构主动控制设计分析时最广泛采用的方法。在LQR算法中,权矩阵Q和R的选取直接影响着结构的动力反应和控制力。目前如何确定最优形式和大小的Q和R以获得全局最优控制力仍然是个难题。采用遗传算法,在一定的目标函数下对LQR算法中权矩阵Q和R进行优化,使其控制效果能够满足结构性能要求,并对优化结果下的动力反应进行了分析比较和结果的验证。     

On the decomposition of symmetric tensors into traceless symmetric tensors

The procedure of the decomposition of symmetric tensors defined in E_n into traceless symmetric tensors is given. Then this procedure is applied to symmetric tensors of 2-7 order.     

Two bracketing theorems characterizing the eigensolution for the h-version of the finite element method

This paper demonstrates that the classical inclusion principle is in general not valid for the h-version of the finite element method. Whereas the inclusion principle is valid for second-order systems discretized by the h-version of the finite element method, provided linear interpolation functions are used as admissible functions, the principle is not valid for fourth-order systems. To characterize the computed eigenvalues for fourth-order systems discretized by the h-version of the finite element method, this paper formulates two bracketing theorems.     

Further research on the performance of consistent mass matrices using BEM for symmetric/nonsymmetric formulations

This paper deals with the performance of consistent mass matrices for the 2-D scalar wave propagation problem using the Boundary Element Method (BEM), and proposes a new global functional set of base functions capable of avoiding domain integrations, suitable for symmetric and nonsymmetric formulations. The method can be applied to arbitrary shaped two-dimensional domains divided into triangular, rectangular and arbitrary shaped quadrilateral linear or curvilinear (e.g. circular) internal cells. The theory is sustained by numerical results for a rectangular and a circular acoustical cavity under Neumann and Dirichlet boundary conditions.     

Block diagonalization of Laplacian matrices of symmetric graphs via group theory

In this article, group theory is employed for block diagonalization of Laplacian matrices of symmetric graphs. The inter‐relation between group diagonalization methods and algebraic‐graph methods developed in recent years are established. Efficient methods are presented for calculating the eigenvalues and eigenvectors of matrices having canonical patterns. This is achieved by using concepts from group theory, linear algebra, and graph theory. These methods, which can be viewed as extensions to the previously developed approaches, are illustrated by applying to the eigensolution of the Laplacian matrices of symmetric graphs. The methods of this paper can be applied to combinatorial optimization problems such as nodal and element ordering and graph partitioning by calculating the second eigenvalue for the Laplacian matrices of the models and the formation of their Fiedler vectors. Considering the graphs as the topological models of skeletal structures, the present methods become applicable to the calculation of the buckling loads and the natural frequencies and natural modes of skeletal structures. Copyright © 2006 John Wiley & Sons, Ltd.