共查询到20条相似文献,搜索用时 15 毫秒
1.
In this paper, we examine three algorithms in the ABS family and consider their storage requirements on sparse band systems. It is shown that, when using the implicit Cholesky algorithm on a band matrix with band width 2q+1, onlyq additional vectors are required. Indeed, for any matrix with upper band widthq, onlyq additional vectors are needed. More generally, ifa
kj
0,j>k, then thejth row ofH
i
is effectively nonzero ifj>i>k. The arithmetic operations involved in solving a band matrix by this method are dominated by (1/2)n
2
q. Special results are obtained forq-band tridiagonal matrices and cyclic band matrices.The implicit Cholesky algorithm may require pivoting if the matrixA does not possess positive-definite principal minors, so two further algorithms were considered that do not require this property. When using the implicit QR algorithm, a matrix with band widthq needs at most 2q additional vectors. Similar results forq-band tridiagonal matrices and cyclic band matrices are obtained.For the symmetric Huang algorithm, a matrix with band widthq requiresq–1 additional vectors. The storage required forq-band tridiagonal matrices and cyclic band matrices are again analyzed.This work was undertaken during the visit of Dr. J. Abaffy to Hatfield Polytechnic, sponsored by SERC Grant No. GR/E-07760. 相似文献
2.
Vamsi Kundeti Sanguthevar Rajasekaran 《Journal of Computational and Applied Mathematics》2010,235(3):756-764
Solving a sparse system of linear equations Ax=b is one of the most fundamental operations inside any circuit simulator. The equations/rows in the matrix A are often rearranged/permuted before factorization and applying direct or iterative methods to obtain the solution. Permuting the rows of the matrix A so that the entries with large absolute values lie on the diagonal has several advantages like better numerical stability for direct methods (e.g., Gaussian elimination) and faster convergence for indirect methods (such as the Jacobi method). Duff (2009) [3] has formulated this as a weighted bipartite matching problem (the MC64 algorithm). In this paper we improve the performance of the MC64 algorithm with a new labeling technique which improves the asymptotic complexity of updating dual variables from O(|V|+|E|) to O(|V|), where |V| is the order of the matrix A and |E| is the number of non-zeros. Experimental results from using the new algorithm, when benchmarked with both industry benchmarks and UFL sparse matrix collection, are very promising. Our algorithm is more than 60 times faster (than Duff’s algorithm) for sparse matrices with at least a million non-zeros. 相似文献
3.
V. P. Derevenskii 《Mathematical Notes》1999,66(1):51-61
Sufficient conditions for the solvability in quadratures of systems of matrix linear ordinary differential equations of first
order with one-sided multiplication by variable matrix coefficients are given in this paper. These conditions are stated in
terms of the theory of Lie algebras. We consider matrix equations of higher orders that are equivalent to such systems. An
illustrative example is considered.
Translated fromMatematicheskie Zametki, Vol. 66, No. 1, pp. 63–75, July, 1999. 相似文献
4.
I.L. El-Kalla 《Applied mathematics and computation》2010,217(8):3756-3763
In this paper, we demonstrate that an infinite number of successive integration by parts can be written in a closed form. This closed form can be used directly to prove that the analytic summation of Adomian series becomes identical to the closed form solution for some classes of differential and integral equations. 相似文献
5.
I. S. Krasil’shchik 《Acta Appl Math》1997,49(3):257-269
Three definitions for characteristics of linear differential operators in the category of modules over a commutative unitary algebra are given. These definitions are compared with each other and some basic fact concerning their properties are proved. It is shown that for algebras without zero divisors the characteristic ideal is involutive and is the support of the symbolic module. 相似文献
6.
In this paper, we introduced an accurate computational matrix method for solving systems of high order fractional differential equations. The proposed method is based on the derived relation between the Chebyshev coefficient matrix A of the truncated Chebyshev solution u(t) and the Chebyshev coefficient matrix A(ν) of the fractional derivative u(ν). The fractional derivatives are presented in terms of Caputo sense. The matrix method for the approximate solution for the systems of high order fractional differential equations (FDEs) in terms of Chebyshev collocation points is presented. The systems of FDEs and their conditions (initial or boundary) are transformed to matrix equations, which corresponds to system of algebraic equations with unknown Chebyshev coefficients. The remaining set of algebraic equations is solved numerically to yield the Chebyshev coefficients. Several numerical examples for real problems are provided to confirm the accuracy and effectiveness of the present method. 相似文献
7.
This paper presents an exponential matrix method for the solutions of systems of high‐order linear differential equations with variable coefficients. The problem is considered with the mixed conditions. On the basis of the method, the matrix forms of exponential functions and their derivatives are constructed, and then by substituting the collocation points into the matrix forms, the fundamental matrix equation is formed. This matrix equation corresponds to a system of linear algebraic equations. By solving this system, the unknown coefficients are determined and thus the approximate solutions are obtained. Also, an error estimation based on the residual functions is presented for the method. The approximate solutions are improved by using this error estimation. To demonstrate the efficiency of the method, some numerical examples are given and the comparisons are made with the results of other methods. Copyright © 2012 John Wiley & Sons, Ltd. 相似文献
8.
本文针对常微分方程数值方法稳定性问题,证明了一般方法的绝对稳定性定理,同时也指出了绝对稳定性条件的局限性.为了克服这种局限性,本文绘出了Jordan稳定性的概念,并建立了一个相应的判别定理. 相似文献
9.
矩阵微分方程的等度有界性 总被引:2,自引:0,他引:2
王林山 《纯粹数学与应用数学》1999,15(3):53-58,21
用矩阵李雅普诺夫函数研究了矩阵微分方程的等度有界性,给出了非自治矩阵微分方程等度有界性的几个判定定理,实例说明了主要定理的实用性 相似文献
10.
We prove the existence of a continuum of non-radial pairs (k,u) solutions to the following overposed problem div
in B
r
, u = 0 and
on ∂B
r
, where B
r
is the Euclidean ball centered at zero of radius r in
.
Dedicato a Marco e Andrea Provera. 相似文献
11.
12.
John Rizkallah 《代数通讯》2017,45(4):1785-1792
13.
A. V. Bayev 《Computational Mathematics and Mathematical Physics》2007,47(9):1452-1463
The problem of solving a system of linear algebraic equations is examined. An application of the Lagrange principle to the optimal recovery in this problem is described. New optimal methods that use available information about the errors in the data and a priori information about the solution are proposed for solving such systems. 相似文献
14.
15.
赵临龙 《数学的实践与认识》2014,(14)
对于常系数线性微分方程组:dx/dt=Ax(A是n阶实常数矩阵)通过特征根λ和对应的特征行向量K:K~T(A-λE)=0将微分方程组化为线性方程组:1°当有n个互异的特征根λ_1,λ_2,…,λ_n,对应的线性无关的特征行向量为K_1,K_2,…,K_n,若记K_i=(k_1,k_2,…,k_n)(i=1,2,…,n),则有方程组:(n∑i=1 k_ix_i)′=λ_j(n∑i=1 k_ix_I)(j=1,2,…,n);2°当有不同的特征根λ_1,λ_2,…,λ_m其重数分别为n_1,n_2,…,n_m,n_1+n_2+…+n_m=n,对应的线性无关的特征行向量为K_i=(k_1,K_2,…,k_n)(i=1,2,…,m),则有方程组:(n∑i=1 k_rx_r)′=λ_k(n∑i=1 k_rx_r)((A-λ_jE)x_(n_i)=0;i=1),(n∑i=1 k_rx_r)′=λ_j(n∑i=1k_rx_r)+c_(n_i)e~(λ_jt)((A-λ_kE)x_(i-1)=Ex_i,i=2,…,n_i). 相似文献
16.
Some examples of orthogonal matrix polynomials satisfying odd order differential equations 总被引:1,自引:1,他引:1
It is well known that if a finite order linear differential operator with polynomial coefficients has as eigenfunctions a sequence of orthogonal polynomials with respect to a positive measure (with support in the real line), then its order has to be even. This property no longer holds in the case of orthogonal matrix polynomials. The aim of this paper is to present examples of weight matrices such that the corresponding sequences of matrix orthogonal polynomials are eigenfunctions of certain linear differential operators of odd order. The weight matrices are of the form
W(t)=tαe-teAttBtB*eA*t,