On sparse lu factorization procedures for the solution of parabolic differential equations in three space dimensions

The numerical solution of partial differential equations in 3 dimensions by finite difference methods leads to the problem of solving large order sparse structured linear systems.

In this paper, a factorization procedure in algorithmic form is derived yielding direct and iterative methods of solution of some interesting boundary value problems in physics and engineering.     

Computation of eigenvalues of a real matrix

This paper presents a computational procedure for finding eigenvalues of a real matrix based on Alternate Quadrant Interlocking Factorization, a parallel direct method developed by Rao in 1994 for the solution of the general linear system Ax=b. The computational procedure is similar to LR algorithm as studied by Rutishauser in 1958 for finding eigenvalues of a general matrix. After a series of transformations the eigenvalues are obtained from simple 2×2 matrices derived from the main and cross diagonals of the limit matrix. A sufficient condition for the convergence of the computational procedure is proved. Numerical examples are given to demonstrate the method.     

一种基于消去树的LDL分解方法及其在营销优化计算中的应用

基于分解因子消去树理论，提出一种新的LDL分解计算方法，它利用消去树结构，将LDL分解过程分为非零元素结构预测和数值计算两部分，从而有效避免多余的内存开销和数值运算．并给出其应用与One to One营销优化算法的实现方法．测试结果表明，该方法能以较少的乘法运算次数快速确定分解因子，从而大幅度提高One to One营销优化算法的性能．     

FETI‐DP,BDDC, and block Cholesky methods

The FETI‐DP and BDDC algorithms are reformulated using Block Cholesky factorizations, an approach which can provide a useful framework for the design of domain decomposition algorithms for solving symmetric positive definite linear system of equations. Instead of introducing Lagrange multipliers to enforce the coarse level, primal continuity constraints in these algorithms, a change of variables is used such that each primal constraint corresponds to an explicit degree of freedom. With the new formulation of these algorithms, a simplified proof is provided that the spectra of a pair of FETI‐DP and BDDC algorithms, with the same set of primal constraints, are essentially the same. Numerical experiments for a two‐dimensional Laplace's equation also confirm this result. Copyright © 2005 John Wiley & Sons, Ltd.     

对称矩阵的改进Cholesky分解在特征值问题中的应用

利用对称短阵的改进Cholesky分解，估计在给定区间(a，b)内对称短阵A特征值的个数,进而发展成一种求对称短阵特征值的近似算法。     

Parametrizing Quantum States and Channels

This work describes one parametrization of quantum states and channels and several of its possible applications. This parametrization works in any dimension and there is an explicit algorithm which produces it. Included in the list of applications are a simple characterization of pure states, an explicit formula for one additive entropic quantity which does not require knowledge of eigenvalues, and an algorithm which finds one Kraus operator representation for a quantum operation without recourse to eigenvalue and eigenvector calculations. PACS: 03.67a, 03.67-Hk, 03.67-Lx