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1.
矩阵对角化方法的再探讨   总被引:1,自引:0,他引:1  
向大晶 《数学通报》2000,(10):37-38
引言文 [1 ]— [3]对矩阵对角化方法的简化问题进行了讨论 ,给出了简便易行的判定和求法 .区别于传统的方法 ,文 [1 ]和 [2 ]把问题归结为矩阵的乘法运算 ,文 [3]则在特殊情形下把问题归结为求特征值与特征向量同步求解 .后者收到了判定和求解一体化的效果 .这种同步操作的思想已在文 [4]和 [5]中见到 ,但均未做到一步成功 .本文对此作进一步探讨 ,一方面改进了 [4]和 [5]的方法 ,使同步求解一步到位 ;另一方面较容易地得到矩阵对角化的十分简单的判定方法 ,以致于判定和求解都是从最终的λ—矩阵中“读”出来的 .其主要依据是以下两个定…  相似文献   

2.
文[1]研究了WKI特征值问题 φ_x=Uφ,U=,及相应的发展方程.本文利用[2]中的方法研究如下推广的WKI特征值问题  相似文献   

3.
孙继广 《计算数学》1991,13(4):369-381
特征值问题(1.1)的敏度分析,主要是指研究特征值λ(p)对于矩阵所含变数p_1,…,p_N的偏导数;这一研究,在结构动力优化等应用中,具有重要意义(见[2]、[4]、[5]、[14]).对于单特征值,以及对于矩阵A(p)与B(p)只含1个变数(即N=1)  相似文献   

4.
文[3]提出了求解大型对称矩阵特征值问题的DL(Davidson—Lanczos)方法。文[1]种[2]对[3]作了改进,分别提出了块DL方法和DL-Chebyshev方法。但当要求的特征值密集而不要求的特征值分离较好时,DL—Chebyshev方法的有效性和可靠性会下降。块DL方法虽克服了上述缺点,但计算量较大,收敛速度仍不理想。为此,本文提出并研究了块DL—Chebyshev方法。  相似文献   

5.
文[1、4]和[2、3]分别研究了求解实对称特征值问题的Davidson方法和Davidson Lanczos(简称DL)方法。本文推广了DL方法,给出并研究了块DL方法。  相似文献   

6.
读了《数学通报》一九九○年第三期《用正交变换化实二次型的标准形方法研究》(以下简称[1])一文之后,颇受启发。笔者这里就该文所举的例子提供一种更为简便的求正交特征向量的方法。这种方法不需要对矩阵进行初等变换,而只需要采用简单的算术运算。下面先用[1]中的例子来说明这种方法。例1 已知λ=1为[1]中矩阵  相似文献   

7.
本文研究了Finsler流形上的距离函数的Laplacian.利用指标引理和文献[4]中主要方法,获得了Ricci曲率有函数下界的Laplacian比较定理,改进了文献[6]和文献[7]的相关结果.  相似文献   

8.
设R(C)为实(复)数域,H~(n×n)为n×n的Hermitian矩阵的集合。当A(∈C~(n×n))的特征值皆为实数时,如不特殊说明,约定A的特征值满足λ_1(A)≥…≥λ_n(A)。文[1]有如下不等式, 令A=B=[(?)],知(1)式一般不成立,(1)式是[1]将[2]的关于奇异值不等式  相似文献   

9.
研究了定义在[0,1]上的Sturm-Liouville问题的特征值对势函数的连续依赖性.应用比较定理和定义区间单调性证明了:当部分区间[x0,1]上的势函数趋于无穷大时,[0,1]区间上的特征值渐进趋近于[0,x0]区间上的某个特征值.推广了一些作者对Sturm-Liouville问题研究的相应结果,并为其相应问题的研究提供了一个新的视角.  相似文献   

10.
一般地,扩散方程的系数q(x)与p(x)是由两组谱或者一组谱及其标准常数唯一确定的.运用Hochstadt与Lieberman的方法证明了:(a)如果给定区间[π/2,π]上的p(x)及区阿[0,π]上的q(x),则扩散方程的一组谱可唯一确定另一半区间[0,π/2]上系数p(x);(b)如果给定区间[π/2,π]上的g(x)及区间[0,π]上的p(x),则扩散方程的一组谱可唯一确定另一半区间[0,π/2]上系数q(x).  相似文献   

11.
The truncated version of the generalized minimal residual method (GMRES), the incomplete generalized minimal residual method (IGMRES), is studied. It is based on an incomplete orthogonalization of the Krylov vectors in question, and gives an approximate or quasi-minimum residual solution over the Krylov subspace. A convergence analysis of this method is given, showing that in the non-restarted version IGMRES can behave like GMRES once the basis vectors of Krylov subspace generated by the incomplete orthogonalization are strongly linearly independent. Meanwhile, some relationships between the residual norms for IOM and IGMRES are established. Numerical experiments are reported to show convergence behavior of IGMRES and of its restarted version IGMRES(m). Project supported by the China State Key Basic Researches, the National Natural Science Foundation of China (Grant No. 19571014), the Doctoral Program (97014113), the Foundation of Returning Scholars of China and the Natural Science Foundation of Liaoning Province.  相似文献   

12.
贾仲孝  张萍 《计算数学》2003,25(3):293-304
1.引言 在科学工程计算中经常需要计算大规模矩阵的少数最大或最小的奇异值及其所对应的奇异子空间。例如图像处理中要计算矩阵端部奇异值之比作为图像的分辨率,诸如此类的问题还存在于最小二乘问题、控制理论、量子化学中等等。然而大多实际问题中的矩阵是大型稀疏矩阵,且需要的是矩阵的部分奇异对。如果计算A的完全奇异值分解(SVD),则运算量和存储量极大,甚至不可能。因此必须寻求其它有效可靠的算法。 假设A的SVD为  相似文献   

13.
We combine Lanczos algorithm with the thought of the refined projection method using QR factorization and propose the refined biothogonalization Lanczos method for computing the desired eigenvalues of large unsymmetric matrix. With low cost of work space and flops the algorithm cures the disease that the Ritz vectors may not converge when the Ritz values converge usingthe Lanczos method. Numerical experiments show our algorithm is considerably more stable and efficient than its counterpart.  相似文献   

14.
This paper proposes a new shift scheme, called refined harmonic shifts, for use in the implicitly restarted refined harmonic Arnoldi method. Numerical experiments show that the implicitly restarted refined harmonic Arnoldi algorithm with refined harmonic shifts is better than the implicitly restarted harmonic Arnoldi algorithm with Morgan's harmonic shifts and the refined harmonic shifts are as efficient as Jia's refined shifts.  相似文献   

15.
不完全信息动态二维价格博弈模型及其分析   总被引:2,自引:0,他引:2  
单位生产成本为不完全信息条件下,本文首先讨论了两个企业关于具有一定替代性的两种产品价格的动态二维博弈模型,并求得其精练Bayes均衡.然后分析了当两种产品不存在替代关系时,企业对这两种产品价格进行动态博弈的精练Bayes均衡相当于对这两种产品单独进行博弈的精练Bayes均衡的简单组合.  相似文献   

16.
The incomplete orthogonalization method (IOM) proposed by Saad for computing a few eigenpairs of large nonsymmetric matrices is generalized into a block incomplete orthogonalization method (BIOM). It is studied how the departure from symmetry A – A H affects the conditioning of the block basis vectors generated by BIOM, and some relationships are established between the approximate eigenpairs obtained by BIOM and Ritz pairs. It is proved that BIOM behaves much like generalized block Lanczos methods if the basis vectors of the block Krylov subspace generated by it are strongly linearly independent. However, it is shown that BIOM may generate a nearly linearly dependent basis for a general nonsymmetric matrix. Numerical experiments illustrate the convergence behavior of BIOM.This work was supported in part by the Graduiertenkolleg at the University of Bielefeld, Germany.  相似文献   

17.
在很多实际应用中需要计算大规模矩阵的若干个最小奇异组.调和投影方法是计算内部特征对的常用方法,其原理可用于求解大规模奇异值分解问题.本文证明了,当投影空间足够好时,该方法得到的近似奇异值收敛,但近似奇异向量可能收敛很慢甚至不收敛.根据第二作者近年来提出的精化投影方法的原理,本文提出一种精化的调和Lanczos双对角化方法,证明了它的收敛性.然后将该方法与Sorensen提出的隐式重新启动技术相结合,开发出隐式重新启动的调和Lanczos双对角化算法(IRHLB)和隐式重新启动的精化调和Lanczos双对角化算法(IRRHLB).位移的合理选取是算法成功的关键之一,本文对精化算法提出了一种新的位移策略,称之为"精化调和位移".理论分析表明,精化调和位移比IRHLB中所用的调和位移要好,且可以廉价可靠地计算出来.数值实验表明,IRRHLB比IRHLB要显著优越,而且比目前常用的隐式重新启动的Lanczos双对角化方法(IRLB)和精化算法IRRLB更有效.  相似文献   

18.
According to the refined projection principle advocated by Jia[8], we improve the residual iteration method of quadratic eigenvalue problems and propose a refined residual iteration method. We study the restarting issue of the method and develop a practical algorithm. Preliminary numerical examples illustrate the efficiency of the method.  相似文献   

19.
The singular value decomposition problem is mathematically equivalent to the eigenproblem of an argumented matrix. Golub et al. give a bidiagonalization Lanczos method for computing a number of largest or smallest singular values and corresponding singular vertors, but the method may encounter some convergence problems. In this paper we analyse the convergence of the method and show why it may fail to converge. To correct this possible nonconvergence, we propose a refined bidiagonalization Lanczos method and apply the implicitly restarting technique to it, and we then present an implicitly restarted bidiagonalization Lanczos algorithm(IRBL) and an implicitly restarted refined bidiagonalization Lanczos algorithm (IRRBL). A new implicitly restarting scheme and a reliable and efficient algorithm for computing refined shifts are developed for this special structure eigenproblem.Theoretical analysis and numerical experiments show that IRRBL performs much better than IRBL.  相似文献   

20.
This article presents a new algorithm for obtaining a block diagonalization of Hankel matrices by means of truncated polynomial divisions, such that every block is a lower Hankel matrix. In fact, the algorithm generates a block LU-factorization of the matrix. Two applications of this algorithm are also presented. By the one hand, this algorithm yields an algebraic proof of Frobenius’ Theorem, which gives the signature of a real regular Hankel matrix by using the signs of its principal leading minors. On the other hand, the close relationship between Hankel matrices and linearly recurrent sequences leads to a comparison with the Berlekamp–Massey algorithm.  相似文献   

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