计算部分奇异值分解的隐式重新启动的双对角化Lanczos方法和精化的双对角化Lanczos方法

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.     

The Use of Refined Harmonic Shifts in the Implicitly Restarted Refined Harmonic Arnoldi Method

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.     

A harmonic restarted Arnoldi algorithm for calculating eigenvalues and determining multiplicity

A restarted Arnoldi algorithm is given that computes eigenvalues and eigenvectors. It is related to implicitly restarted Arnoldi, but has a simpler restarting approach. Harmonic and regular Rayleigh-Ritz versions are possible.For multiple eigenvalues, an approach is proposed that first computes eigenvalues with the new harmonic restarted Arnoldi algorithm, then uses random restarts to determine multiplicity. This avoids the need for a block method or for relying on roundoff error to produce the multiple copies.     

DQGMRES: a Direct Quasi-minimal Residual Algorithm Based on Incomplete Orthogonalization

We describe a Krylov subspace technique, based on incomplete orthogonalization of the Krylov vectors, which can be considered as a truncated version of GMRES. Unlike GMRES(m), the restarted version of GMRES, the new method does not require restarting. Like GMRES, it does not break down. Numerical experiments show that DQGMRES(k) often performs as well as the restarted GMRES using a subspace of dimension m=2k. In addition, the algorithm is flexible to variable preconditioning, i.e., it can accommodate variations in the preconditioner at every step. In particular, this feature allows the use of any iterative solver as a right-preconditioner for DQGMRES(k). This inner-outer iterative combination often results in a robust approach for solving indefinite non-Hermitian linear systems.     

求解PageRank问题的重启GMRES修正的多分裂迭代法

PageRank算法已经成为网络搜索引擎的核心技术。针对PageRank问题导出的线性方程组,首先将Krylov子空间方法中的重启GMRES(generalized minimal residual)方法与多分裂迭代(multi-splitting iteration,MSI)方法相结合,提出了一种重启GMRES修正的多分裂迭代法;然后,给出了该算法的详细计算流程和收敛性分析;最后,通过数值实验验证了该算法的有效性。     

计算最小奇异组的一个精化调和 Lanczos双对角化方法

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.     

Restarted Full Orthogonalization Method with Deflation for Shifted Linear Systems

In this paper, we study shifted restated full orthogonalization method with deflation for simultaneously solving a number of shifted systems of linear equations. Theoretical analysis shows that with the deflation technique, the new residual of shifted restarted FOM is still collinear with each other. Hence, the new approach can solve the shifted systems simultaneously based on the same Krylov subspace. Numerical experiments show that the deflation technique can significantly improve the convergence performance of shifted restarted FOM.     

An Implicitly Restarted Block Arnoldi Method in a Vector-Wise Fashion

In this paper, we develop an implicitly restarted block Arnoldi algorithm in a vector-wise fashion. The vector-wise construction greatly simplifies both the detection of necessary deflation and the actual deflation itself, so it is preferable to the block-wise construction. The numerical experiment shows that our algorithm is effective.     

添加近似误差的重新启动的simpler GMRES方法

本文给出了重新启动的LGMRES方法的一种代价更小的实现方式.这种做法基于消除以下减慢收敛速度的现象:重新启动的simpler GMRES的每次循环结束时得到的残向量经常交替方向,与重新启动的GMRES的情形类似.这种新的变形的方法的优点是它比重新启动的LGMRES所需要的计算量要少.大量的例子表明该方法计算速度更快.     

隐式重启精化全局Lanczos方法

The numerical methods for solving large symmetric eigenvalue problems are considered in this paper.Based on the global Lanczos process,a global Lanczos method for solving large symmetric eigenvalue problems is presented.In order to accelerate the convergence of the F-Ritz vectors,the refined global Lanczos method is developed.Combining the implicitly restarted strategy with the deflation technique,an implicitly restarted and refined global Lanczos method for computing some eigenvalues of large symmetric matrices is proposed.Numerical results show that the proposed methods are efficient.     

A Restarted Conjugate Gradient Method for Ill—posed Problems

This paper presents a restarted conjugate gradient iterative algorithm for solving ill-posed problems.The damped Morozov‘s discrepancy principle is used as a stopping rule,Numerical experiments are given to illustrate the efficiency of the method.     

计算大规模矩阵最大最小奇异值和奇异向量的两个精化Lanczos算法

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     

Heavy Ball Flexible GMRES Method for Nonsymmetric Linear Systems

Flexible GMRES (FGMRES) is a variant of preconditioned GMRES, which changes preconditioners at every Arnoldi step. GMRES often has to be restarted in order to save storage and reduce orthogonalization cost in the Arnoldi process. Like restarted GMRES, FGMRES may also have to be restarted for the same reason. A major disadvantage of restarting is the loss of convergence speed. In this paper, we present a heavy ball flexible GMRES method, aiming to recoup some of the loss in convergence speed in the restarted flexible GMRES while keep the benefit of limiting memory usage and controlling orthogonalization cost. Numerical tests often demonstrate superior performance of the proposed heavy ball FGMRES to the restarted FGMRES.     

一种灵活的混合GMRES算法

1　引　　言考虑线性方程组Ax =b (1 .1 )其中 A∈RN× N是非奇异的 .求解方程组 (1 .1 )的很多迭代方法都可归类于多项式法 ,即满足x(n) =x(0 ) +qn- 1 (A) r(0 ) ,degqn- 1 ≤ n -1这里 x(n) ,n≥ 0为第 n步迭代解 ,r(n) =b-Ax(n) 是对应的迭代残量 .等价地 ,r(n) =pn(A) r(0 ) ,degpn≤ n;pn(0 ) =1 (1 .2 )其中 pn(z) =1 -zqn- 1 (z)称为残量多项式 .或有r(n) -r(0 ) ∈ AKn(r(0 ) ,A)其中 Kn(v,A)≡span{ Aiv} n- 1 i=0 是对应于 v,A的 Krylov子空间 .对于非对称问题 ,可以用正交性条件r(n)⊥ AKn(r(0 ) ,A)来确定 (1 .2 )中的…     

An Arnoldi-Extrapolation algorithm for computing PageRank

The Arnoldi-type algorithm proposed by Golub and Greif [G. Golub, C. Greif, An Arnoldi-type algorithm for computing PageRank, BIT 46 (2006) 759-771] is a restarted Krylov subspace method for computing PageRank. However, this algorithm may not be efficient when the damping factor is high and the dimension of the search subspace is small. In this paper, we first develop an extrapolation method based on Ritz values. We then consider how to periodically knit this extrapolation method together with the Arnoldi-type algorithm. The resulting algorithm is the Arnoldi-Extrapolation algorithm. The convergence of the new algorithm is analyzed. Numerical experiments demonstrate the numerical behavior of this algorithm.     

Block Krylov–Schur method for large symmetric eigenvalue problems

Stewart’s Krylov–Schur algorithm offers two advantages over Sorensen’s implicitly restarted Arnoldi (IRA) algorithm. The first is ease of deflation of converged Ritz vectors, the second is the avoidance of the potential forward instability of the QR algorithm. In this paper we develop a block version of the Krylov–Schur algorithm for symmetric eigenproblems. Details of this block algorithm are discussed, including how to handle rank deficient cases and how to use varying block sizes. Numerical results on the efficiency of the block Krylov–Schur method are reported.     

IMPROVING EIGENVECTORS IN ARNOLDI'S METHOD

1. IntroductionArnoldi's method [1, 12] is used for computing.,a few selected eigenpairs of largeunsymmetric matrices. It hajs been investigated since the 1980s; see, e-g., [3--15].It is well known that the m--step Arnoldi processt as described in detail in Section 2,generates an orthonormal basis {yi}7=1 of the Krylov subspace Km(vi, A) spanned byvil Avi,... 5 Am--'v,. Here yi is an initial unit norm vector. The projected matrix ofA onto Km(vi, A) is represented by an m x m upper Hessenb…     

On the squared Smith method for large‐scale Stein equations

A squared Smith type algorithm for solving large‐scale discrete‐time Stein equations is developed. The algorithm uses restarted Krylov spaces to compute approximations of the squared Smith iterations in low‐rank factored form. Fast convergence results when very few iterations of the alternating direction implicit method are applied to the Stein equation beforehand. The convergence of the algorithm is discussed and its performance is demonstrated by several test examples. Copyright © 2013 John Wiley & Sons, Ltd.     

On-line scheduling on a batch machine to minimize makespan with limited restarts

We study the on-line scheduling on an unbounded batch machine to minimize makespan. In this model, jobs arrive over time and batches are allowed limited restarts. Any batch that contains a job which has already been restarted once cannot be restarted any more. We provide a best possible on-line algorithm for the problem with a competitive ratio .     

A new framework for implicit restarting of the Krylov–Schur algorithm

This paper introduces a new framework for implicit restarting of the Krylov–Schur algorithm. It is shown that restarting with arbitrary polynomial filter is possible by reassigning some of the eigenvalues of the Rayleigh quotient through a rank‐one correction, implemented using only the elementary transformations (translation and similarity) of the Krylov decomposition. This framework includes the implicitly restarted Arnoldi (IRA) algorithm and the Krylov–Schur algorithm with implicit harmonic restart as special cases. Further, it reveals that the IRA algorithm can be turned into an eigenvalue assignment method. Copyright © 2014 John Wiley & Sons, Ltd.