求鳞状因子循环矩阵的逆阵及广义逆阵的快速算法

利用多项式快速算法，给出了求鳞状因子循环矩阵的逆阵、自反g-逆、群逆及Moore-Penrose逆的快速算法。该算法避免了一般快速算法中，要计算大量的三角函数等可能带来误差及影响效率的问题。该算法仅用到鳞状因子循环矩阵的第一行元素及对角阵D中的常数d1，d2，…，dn进行计算，在计算机上实现时只有舍入误差。特别地，在有理数域上用计算机求得的结果是精确的。     

非奇异H-矩阵的一组充分条件

非奇异H-矩阵在数值线性代数的理论与应用中起着重要作用,因此研究非奇异H-矩阵的判定条件有着非常重要的理论价值.本文根据广义严格α-链对角占优矩阵和广义严格α-对角占优矩阵的性质,通过引入迭代因子,给出了一组非奇异H-矩阵新的迭代判定条件.该判定条件推广和改进了相关已有结果,丰富和完善了非奇异H-矩阵的理论,最后用数值算例说明了其有效性.     

块对角占优块三对角方程组的块重叠分割无通信并行求解方法

基于并行计算的分治思想,对块三对角线性方程组的求解提出了一个块重叠分割无通信的高效可扩展并行算法(PBOPUC算法)。当系统严格块对角占优时,在机器精度内,得到与精确解等价的近似解。通过精度分析,得到子方程组的阶数与精度的关系,并用它来控制精度和并行效率。本文的算法已经在上海大学的高性能并行计算机"自强3000"上实现,结果说明,并行计算效率接近100%,加速比几乎是线性的。     

友循环矩阵求逆和广义逆的Euclid算法

利用多项式的Euclid算法给出了任意域上非奇异的友循环矩阵求逆矩阵的一个新算法,该算法同时推广到用于求任意域上奇异友循环矩阵的群逆和Moore-Penrose逆,最后给出了应用该算法的数值例子。     

求Hankel型线性方程组的一种算法

本文给出了矩阵为Hankel矩阵的充要条件,由此定义了一种新的矩阵-Hankel型矩阵,说明了Hankel矩阵是Hankel型矩阵的特殊情况.为了降低Hankel型线性方程组的计算量和减小这类算法的误差,利用Hankel型矩阵的位移性质,给出了求Hankel型线性方程组的一种算法.矩阵为Hankel矩阵时,该算法与Gohberg-Kailath-Koltracht算法相比计算量相当,但改进了精度;矩阵为一般Hankel型矩阵时,该算法与Cholesky分解算法相比计算量大为减少,极大改进了精度.     

α-对角占优矩阵的讨论及其应用

利用矩阵对角占优理论,讨论α-对角占优矩阵之间的蕴涵关系,并给出条件最弱的严格α<,1>-双对角占优矩阵的等价表征,作为应用得到非奇异H-矩阵新的判定准则,同时给出判定非奇异H-矩阵的算法,并通过数值结果表明本文判定方法的有效性和优越性.     

Vandermonde型方程组极小范数最小二乘解的快速算法

本文给出了求以n×m阶Vandermonde型矩阵为系数阵的线性方程组极小范数最小二乘解的快速算法.     

一类矩阵方程组的对称解及其最佳逼近

本文建立了求矩阵方程组AiXBi+GiXDi=Fi(i=1,2)对称解的迭代算法.使用该算法可以判断矩阵方程组是否有对称解.在有对称解时,能在有限步迭代后得到矩阵方程组的对称解;当选取特殊初始矩阵时,可得极小范数对称解.另外,在上述解集合中可得到给定矩阵的最佳逼近矩阵表达式.     

非奇异H矩阵的一组细分迭代判定条件

非奇异H矩阵是在科学和工程应用中经常遇到的一类特殊矩阵,研究其判定问题非常重要.本文根据α-链对角占优矩阵与非奇异H矩阵的关系,利用细分区间和构造迭代系数的思想,细分了矩阵的非对角占优行集合,给出了非奇异H矩阵的一组细分迭代判定条件,推广和改进了近期的一些结果.数值算例说明了该判定条件的有效性.     

绝对值方程的一种新的半光滑牛顿法(英文)

绝对值方程产生于求解区间线性方程组,其定义为Ax|x|=b,其中A为n阶实矩阵.线性互补问题可以转化为一个绝对值方程,因此许多重要的数学规划问题可以转化为绝对值方程.本文基于min-函数和FB-函数,提出了求解绝对值方程的半光滑牛顿算法.该算法在每一次迭代中只需要求解一个线性方程组.当区间矩阵[A I,A+I]正则时,该算法全局收敛且有限步收敛;即任意给定初始点x0∈Rn,算法在有限次迭代之后收敛于绝对值方程的解.数值实验表明了算法的有效性,特别是大规模问题求解中的适用性.     

多变量矩阵方程异类约束解的修正共轭梯度法

基于求解线性代数方程组的共轭梯度法,通过对相关矩阵和系数的修改,建立了一种求多矩阵变量矩阵方程异类约束解的修正共轭梯度法.该算法不要求等价线性代数方程组的系数矩阵具备正定性、可逆性或者列满秩性,因此算法总是可行的.利用该算法不仅可以判断矩阵方程的异类约束解是否存在,而且在有异类约束解,不考虑舍入误差时,可在有限步计算后求得矩阵方程的一组异类约束解;选取特殊初始矩阵时,可求得矩阵方程的极小范数异类约束解.另外,还可求得指定矩阵在异类约束解集合中的最佳逼近.算例验证了该算法的有效性.     

Block circulant matrices and applications in free vibration analysis of cyclically repetitive structures

In this paper, block circulant matrices and their properties are investigated. Basic concepts and the necessary theorems are presented and then their applications are discussed. It is shown that a circulant matrix can be considered as the sum of Kronecker products in which the first components have the commutativity property with respect to multiplication. The important fact is that the method for block diagonalization of these matrices is much simpler than the previously developed methods, and one does not need to find an additional matrix for orthogonalization. As it will be shown not only the matrices corresponding to domes in the form of Cartesian product, strong Cartesian product and direct product are circulant, but for other structures such as diamatic domes, pyramid domes, flat double layer grids, and some family of transmission towers these matrices are also block circulant.     

求置换因子循环矩阵的极小多项式和逆的算法

本文引入了任意域上置换因子循环矩阵,利用多项式环的理想的Gr(?)bner基的算法给出了任意域上置换因子循环矩阵的极小多项式和公共极小多项式的算法,同时给出了这类矩阵逆矩阵的两种算法最后,利用Schur补给出了任意域上具有置换因子循环矩阵块的分块矩阵逆的一个算法,在有理数域或模素数剩余类域上,这一算法可由代数系统软件CoCoA4．0实现。     

求解无限域动力刚度矩阵的双渐近算法

采用连分式算法可以有效地求解无限域动力刚度表示的比例边界有限元方程, 它具有收敛范围广、收敛速度快等优点. 该文在高频渐近连分式算法的基础上考虑了低频渐近, 发展了一种针对矢量波动方程的双渐近算法. 随着展开阶数的增加, 双渐近算法可以在全频域范围内快速逼近准确解. 引入了系数矩阵?X(i)来增强连分式算法的数值稳定性. 通过在高频极限、低频极限时满足动力刚度表示的比例边界有限元方程, 建立了递推关系以求得动力刚度矩阵. 通过二维半无限楔形体、三维均质弹性半空间数值算例表明, 双渐近算法比单渐近算法更稳定、优越.     

Circulant preconditioners for ill-conditioned boundary integral equations from potential equations

In this paper, we consider solving potential equations by the boundary integral equation approach. The equations so derived are Fredholm integral equations of the first kind and are known to be ill-conditioned. Their discretized matrices are dense and have condition numbers growing like O(n) where n is the matrix size. We propose to solve the equations by the preconditioned conjugate gradient method with circulant integral operators as preconditioners. These are convolution operators with periodic kernels and hence can be inverted efficiently by using fast Fourier transforms. We prove that the preconditioned systems are well conditioned, and hence the convergence rate of the method is linear. Numerical results for two types of regions are given to illustrate the fast convergence. © 1998 John Wiley & Sons, Ltd.     

A Matrix Decomposition MFS Algorithm for Biharmonic Problems in Annular Domains

The Method of Fundamental Solutions (MFS) is a boundary-type method for the solution of certain elliptic boundary value problems. In this work, we develop an efficient matrix decomposition MFS algorithm for the solution of biharmonic problems in annular domains. The circulant structure of the matrices involved in the MFS discretization is exploited by using Fast Fourier Transforms. The algorithm is tested numerically on several examples.     

Matrix decomposition RBF algorithm for solving 3D elliptic problems

In this study, we propose an efficient algorithm for the evaluation of the particular solutions of three-dimensional inhomogeneous elliptic partial differential equations using radial basis functions. The collocation points are placed on concentric spheres and thus the resulting global matrix possesses a block circulant structure. This structure is exploited to develop an efficient matrix decomposition algorithm for the solution of the resulting system. Further savings in the matrix decomposition algorithm are obtained by the use of fast Fourier transforms. The proposed algorithm is used, in conjunction with the method of fundamental solutions for the solution of three-dimensional inhomogeneous elliptic boundary value problems.     

High-resolution image reconstruction with multisensors

This article considers the problem of reconstructing a high-resolution image from multiple undersampled, shifted, degraded frames with subpixel displacement errors. This leads to a formulation involving a periodically shift-variant system model. The maximum a posteriori (MAP) estimation scheme is used subject to the assumption that the original high-resolution image is modeled by a stationary Markov-Gaussian random field. The resulting MAP formulation is expressed as a complex linear matrix equation, where the characterizing matrix involves the periodic block Toeplitz with Toeplitz block (BTTB) blur matrix and banded-BTTB inverse covariance matrix associated with the original image. By approximating the periodic-BTTB and the banded-BTTB matrices with, respectively, the periodic block circulant with circulant block (BCCB) and the banded-BCCB matrices, it is shown that the computation-intensive MAP formulation can be decomposed into a set of smaller matrix equations by using the two-dimensional discrete Fourier transform. Exact solutions are also considered through the use of the preconditioned conjugate gradient algorithm. Computer simulations are given to illustrate the procedure. © 1998 John Wiley & Sons, Inc. Int J Imaging Syst Technol, 9, 294–304, 1998     

Solution of arbitrarily dimensional matrix equation in computational electromagnetics by fast lifting wavelet‐like transform

A new wavelet matrix transform (WMT), operated by lifting wavelet‐like transform (LWLT), is applied to the solution of matrix equations in computational electromagnetics. The method can speedup the WMT without allocating auxiliary memory for transform matrices and can be implemented with the absence of the fast Fourier transform. Furthermore, to handle the matrix equation of arbitrarily dimension, a new in‐space preprocessing technique based on LWLT is constructed to eliminate the limitation in matrix dimension. Complexity analysis and numerical simulation show the superiority of the proposed algorithm in saving CPU time. Numerical simulations for scattering analysis of differently shaped objects are considered to validate the effectiveness of the proposed method. In particular, due to its generality, the proposed preprocessing technique can be extended to other engineering areas and therefore can pave a broad way for the application of the WMT. Copyright © 2009 John Wiley & Sons, Ltd.     

Solution of Algebraic Lyapunov Equation on Positive-Definite Hermitian Matrices by Using Extended Hamiltonian Algorithm

This communique is opted to study the approximate solution of the Algebraic Lyapunov equation on the manifold of positive-definite Hermitian matrices. We choose the geodesic distance between -AHX - XA and P as the cost function, and put forward the Extended Hamiltonian algorithm (EHA) and Natural gradient algorithm (NGA) for the solution. Finally, several numerical experiments give you an idea about the effectiveness of the proposed algorithms. We also show the comparison between these two algorithms EHA and NGA. Obtained results are provided and analyzed graphically. We also conclude that the extended Hamiltonian algorithm has better convergence speed than the natural gradient algorithm, whereas the trajectory of the solution matrix is optimal in case of Natural gradient algorithm (NGA) as compared to Extended Hamiltonian Algorithm (EHA). The aim of this paper is to show that the Extended Hamiltonian algorithm (EHA) has superior convergence properties as compared to Natural gradient algorithm (NGA). Upto the best of author’s knowledge, no approximate solution of the Algebraic Lyapunov equation on the manifold of positive-definite Hermitian matrices is found so far in the literature.