固体变温曲线拟合的新方法——特征值方法

本文提出了小温升条件下固体热传导系统变温过程拟合的新方法。结合机理研究和实验分析,辨识热系统特征值,进而把变温过程多元非线性拟合问题转化为多元线性回归问题。     

On relaxed nested factorization and combination preconditioning

The efficient solution of block tridiagonal linear systems arising from the discretization of convection–diffusion problem is considered in this paper. Starting with the classical nested factorization, we propose a relaxed nested factorization preconditioner. Then, several combination preconditioners are developed based on relaxed nested factorization and a tangential filtering preconditioner. Influence of the relaxation parameter is numerically studied, the results indicate that the optimal relaxation parameter should be close to but less than 1. The number of iteration counts exhibit an extremely sensitive behaviour. This phenomena resembles the behaviour of relaxed ILU preconditioner. For symmetric positive-definite coefficient matrix, we also show that the proposed combination preconditioner is convergent. Finally, numerous test cases are carried out with both additive and multiplicative combinations to verify the robustness of the proposed preconditioners.     

Efficiently preconditioned inexact Newton methods for large symmetric eigenvalue problems

In this paper, we propose an efficiently preconditioned Newton method for the computation of the leftmost eigenpairs of large and sparse symmetric positive definite matrices. A sequence of preconditioners based on the BFGS update formula is proposed, for the preconditioned conjugate gradient solution of the linearized Newton system to solve Au=q(u) u, q(u) being the Rayleigh quotient. We give theoretical evidence that the sequence of preconditioned Jacobians remains close to the identity matrix if the initial preconditioned Jacobian is so. Numerical results onto matrices arising from various realistic problems with size up to one million unknowns account for the efficiency of the proposed algorithm which reveals competitive with the Jacobi–Davidson method on all the test problems.     

Canonical form of para‐Hermitian pencils,generalized spectral factorization,and optimal control over frequency region

We generalize the J‐spectral factorization of para‐Hermitian proper rational matrix in the case when it has no constant inertia on the imaginary axis. This result and the presented numerical algorithm are based on a canonical form of para‐Hermitian matrix pencils. We apply the new spectral factorization to the optimal control of the proper plant by dynamic measurement feedback over a frequency region. Copyright © 2012 John Wiley & Sons, Ltd.     

An efficient method for computing leading eigenvalues and eigenvectors of large asymmetric matrices

An efficient method for computing a given number of leading eigenvalues (i.e., having largest real parts) and the corresponding eigenvectors of a large asymmetric matrixM is presented. The method consists of three main steps. The first is a filtering process in which the equationx = Mx is solved for an arbitrary initial conditionx(0) yielding:x(t)=e ^Mt x(0). The second step is the construction of (n+1) linearly independent vectorsv _m =M ^m x, 0mn orv _m =e ^mMt x ( being a short time interval). By construction, the vectorsv _m are combinations of only a small number of leading eigenvectors ofM. The third step consists of an analysis of the vectors {v _m } that yields the eigenvalues and eigenvectors. The proposed method has been successfully tested on several systems. Here we present results pertaining to the Orr-Sommerfeld equation. The method should be useful for many computations in which present methods are too slow or necessitate excessive memory. In particular, we believe it is well suited for hydrodynamic and mechanical stability investigations.     

一维结构系统重频特征问题的摄动传递矩阵法

基于传递矩阵法，给出一维结构系统重频振动特征问题的二阶摄动计算方法，该方法适用于一般的一维结构系统的实数和复数重频振动特征问题的摄动分析，并且可避免模态截断误差的影响。只要知道某阶重频的特征值和它的特征向量，用该方法便可准确地计算出其特征值和特征向量的一阶与二阶摄动。算例对弹性支承梁的复数重频振动特征问题进行摄动分析，摄动计算结果和精确计算结果吻合良好。     

网络拓扑的超能整循环图构造

循环图是一类重要的网络拓扑结构图,在并行计算和分布计算中发挥重要作用。图[G]的能量[E(G)]定义为图的特征值的绝对值之和。具有[n]个顶点的图[G]称为超能图如果图[G]的能量[E(G)>2n-2]。一个图称为循环图,若它是循环群上的Cayley图,即它的邻接矩阵是一个循环矩阵;整循环图是指循环图的特征值全为整数。借助Ramanujans和,利用Euler函数和Mobius函数,讨论了整循环图的超能性。利用Cartesian积图给出了一个构造超能整循环图的方法。     

Numerical Solutions to Critical Problem of Reflected Cylindrical Reactor

The multi-region critical problem can be transformed into an eigenvalue problem in the classical sense by using the method of Kuscer & Corngold and of Wing. This transformation is applied to derive a variational formulation for a reflected reactor.

An approximate critical value of the multiplying factor is determined by maximizing the Rayleigh quotient for radially and totally reflected cylindrical reactors. It is shown that this approximate critical value is an upper bound of the true critical value.

From the facts that the operator is self-adjoint and the eigenfunction is positive, an expression is derived for the upper and lower bounds of the true eigenvalue, by making use of the approximate distribution. The difference of the upper and lower bounds is an uncertainty of the presumption of the true critical value. It is found that we can compute the bounds to any required precision.

The narrow bounds are calculated for two radially and one totally reflected cylindrical reactors.     

Discrete Eigenvalues of Neutron Transport Equation with Quadratically Anisotropic Scattering

The general properties of discrete eigenvalues of the one-speed neutron transport equation with quadratically anisotropic scattering are studied analytically. We investigate the asymptotic behaviors of boundary curves of the complex discrete eigenvalues carefully and plot the distribution of the discrete eigenvalues in parameter space. Our predictions agree with numerical calculation     

A method for solving a stochastic eigenvalue problem applied to criticality

We present a novel approach to calculate stochastic eigenvalues of differential and integral equations using polynomial chaos theory. The method is applied to a criticality problem using the diffusion equation. This technique has the advantage of avoiding the non-linear terms in the conventional method of stochastic eigenvalue calculation but it does require an additional, ‘pseudo-time’, independent variable t.