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QR algorithm with two‐sided Rayleigh quotient shifts 二维瑞利商移位QR算法
IF 4.3 3区 数学
Numerical Linear Algebra with Applications Pub Date : 2023-01-02 DOI: 10.1002/nla.2487
X. Chen, Hongguo Xu
{"title":"QR algorithm with two‐sided Rayleigh quotient shifts","authors":"X. Chen, Hongguo Xu","doi":"10.1002/nla.2487","DOIUrl":"https://doi.org/10.1002/nla.2487","url":null,"abstract":"We introduce the two‐sided Rayleigh quotient shift to the QR algorithm for non‐Hermitian matrices to achieve a cubic local convergence rate. For the singly shifted case, the two‐sided Rayleigh quotient iteration is incorporated into the QR iteration. A modified version of the method and its truncated version are developed to improve the efficiency. Based on the observation that the Francis double‐shift QR iteration is related to a 2D Grassmann–Rayleigh quotient iteration, A doubly shifted QR algorithm with the two‐sided 2D Grassmann–Rayleigh quotient double‐shift is proposed. A modified version of the method and its truncated version are also developed. Numerical examples are presented to show the convergence behavior of the proposed algorithms. Numerical examples also show that the truncated versions of the modified methods outperform their counterparts including the standard Rayleigh quotient single‐shift and the Francis double‐shift.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":null,"pages":null},"PeriodicalIF":4.3,"publicationDate":"2023-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41684014","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Multi‐view side information‐incorporated tensor completion 多视图侧信息合并张量补全
IF 4.3 3区 数学
Numerical Linear Algebra with Applications Pub Date : 2022-12-19 DOI: 10.1002/nla.2485
Yingjie Tian, Xiaotong Yu, Saiji Fu
{"title":"Multi‐view side information‐incorporated tensor completion","authors":"Yingjie Tian, Xiaotong Yu, Saiji Fu","doi":"10.1002/nla.2485","DOIUrl":"https://doi.org/10.1002/nla.2485","url":null,"abstract":"Tensor completion originates in numerous applications where data utilized are of high dimensions and gathered from multiple sources or views. Existing methods merely incorporate the structure information, ignoring the fact that ubiquitous side information may be beneficial to estimate the missing entries from a partially observed tensor. Inspired by this, we formulate a sparse and low‐rank tensor completion model named SLRMV. The ℓ0$$ {ell}_0 $$ ‐norm instead of its relaxation is used in the objective function to constrain the sparseness of noise. The CP decomposition is used to decompose the high‐quality tensor, based on which the combination of Schatten p$$ p $$ ‐norm on each latent factor matrix is employed to characterize the low‐rank tensor structure with high computation efficiency. Diverse similarity matrices for the same factor matrix are regarded as multi‐view side information for guiding the tensor completion task. Although SLRMV is a nonconvex and discontinuous problem, the optimality analysis in terms of Karush‐Kuhn‐Tucker (KKT) conditions is accordingly proposed, based on which a hard‐thresholding based alternating direction method of multipliers (HT‐ADMM) is designed. Extensive experiments remarkably demonstrate the efficiency of SLRMV in tensor completion.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":null,"pages":null},"PeriodicalIF":4.3,"publicationDate":"2022-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43167030","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Inexact inner–outer Golub–Kahan bidiagonalization method: A relaxation strategy 非精确内外Golub-Kahan双对角化方法:一种松弛策略
IF 4.3 3区 数学
Numerical Linear Algebra with Applications Pub Date : 2022-12-12 DOI: 10.1002/nla.2484
Vincent Darrigrand, Andrei Dumitrasc, Carola Kruse, Ulrich Rüde
{"title":"Inexact inner–outer Golub–Kahan bidiagonalization method: A relaxation strategy","authors":"Vincent Darrigrand, Andrei Dumitrasc, Carola Kruse, Ulrich Rüde","doi":"10.1002/nla.2484","DOIUrl":"https://doi.org/10.1002/nla.2484","url":null,"abstract":"We study an inexact inner–outer generalized Golub–Kahan algorithm for the solution of saddle-point problems with a two-times-two block structure. In each outer iteration, an inner system has to be solved which in theory has to be done exactly. Whenever the system is getting large, an inner exact solver is, however, no longer efficient or even feasible and iterative methods must be used. We focus this article on a numerical study showing the influence of the accuracy of an inner iterative solution on the accuracy of the solution of the block system. Emphasis is further given on reducing the computational cost, which is defined as the total number of inner iterations. We develop relaxation techniques intended to dynamically change the inner tolerance for each outer iteration to further minimize the total number of inner iterations. We illustrate our findings on a Stokes problem and validate them on a mixed formulation of the Poisson problem.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":null,"pages":null},"PeriodicalIF":4.3,"publicationDate":"2022-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138502932","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
A fast and accurate algorithm for solving linear systems associated with a class of negative matrix 一类负矩阵线性方程组的快速精确求解算法
IF 4.3 3区 数学
Numerical Linear Algebra with Applications Pub Date : 2022-12-08 DOI: 10.1002/nla.2483
Zhao Yang
{"title":"A fast and accurate algorithm for solving linear systems associated with a class of negative matrix","authors":"Zhao Yang","doi":"10.1002/nla.2483","DOIUrl":"https://doi.org/10.1002/nla.2483","url":null,"abstract":"A class of negative matrices including Vandermonde‐like matrices tends to be extremely ill‐conditioned, and linear systems associated with this class of matrices appear in the polynomial interpolation problems. In this article, we present a fast and accurate algorithm with O(n2)$$ Oleft({n}^2right) $$ complexity to solve the linear systems whose coefficient matrices belong to the class of negative matrix. We show that the inverse of any such matrix is generated in a subtraction‐free manner. Consequently, the solutions of linear systems associated with the class of negative matrix are accurately determined by parameterization matrices of coefficient matrices, and a pleasantly componentwise forward error is provided to illustrate that each component of the solution is computed to high accuracy. Numerical experiments are performed to confirm the claimed high accuracy.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":null,"pages":null},"PeriodicalIF":4.3,"publicationDate":"2022-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44783694","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
A class of improved conjugate gradient methods for nonconvex unconstrained optimization 非凸无约束优化的一类改进共轭梯度方法
IF 4.3 3区 数学
Numerical Linear Algebra with Applications Pub Date : 2022-12-08 DOI: 10.1002/nla.2482
Qingjie Hu, Hongrun Zhang, Zhijuan Zhou, Yu Chen
{"title":"A class of improved conjugate gradient methods for nonconvex unconstrained optimization","authors":"Qingjie Hu, Hongrun Zhang, Zhijuan Zhou, Yu Chen","doi":"10.1002/nla.2482","DOIUrl":"https://doi.org/10.1002/nla.2482","url":null,"abstract":"In this paper, based on a new class of conjugate gradient methods which are proposed by Rivaie, Dai and Omer et al. we propose a class of improved conjugate gradient methods for nonconvex unconstrained optimization. Different from the above methods, our methods possess the following properties: (i) the search direction always satisfies the sufficient descent condition independent of any line search; (ii) these approaches are globally convergent with the standard Wolfe line search or standard Armijo line search without any convexity assumption. Moreover, our numerical results also demonstrated the efficiencies of the proposed methods.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":null,"pages":null},"PeriodicalIF":4.3,"publicationDate":"2022-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42363692","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Issue Information 问题信息
IF 4.3 3区 数学
Numerical Linear Algebra with Applications Pub Date : 2022-12-01 DOI: 10.1002/nla.2449
{"title":"Issue Information","authors":"","doi":"10.1002/nla.2449","DOIUrl":"https://doi.org/10.1002/nla.2449","url":null,"abstract":"","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":null,"pages":null},"PeriodicalIF":4.3,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46794803","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A parameterized extended shift‐splitting preconditioner for nonsymmetric saddle point problems 非对称鞍点问题的参数化扩展移位分裂预条件
IF 4.3 3区 数学
Numerical Linear Algebra with Applications Pub Date : 2022-11-29 DOI: 10.1002/nla.2478
Seryas Vakili, G. Ebadi, C. Vuik
{"title":"A parameterized extended shift‐splitting preconditioner for nonsymmetric saddle point problems","authors":"Seryas Vakili, G. Ebadi, C. Vuik","doi":"10.1002/nla.2478","DOIUrl":"https://doi.org/10.1002/nla.2478","url":null,"abstract":"In this article, a parameterized extended shift‐splitting (PESS) method and its induced preconditioner are given for solving nonsingular and nonsymmetric saddle point problems with nonsymmetric positive definite (1,1) part. The convergence analysis of the PESS$$ PESS $$ iteration method is discussed. The distribution of eigenvalues of the preconditioned matrix is provided. A number of experiments are given to verify the efficiency of the PESS$$ PESS $$ method for solving nonsymmetric saddle‐point problems.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":null,"pages":null},"PeriodicalIF":4.3,"publicationDate":"2022-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47934281","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
On estimation of the optimal parameter of the modulus‐based matrix splitting algorithm for linear complementarity problems on second‐order cones 二阶锥上线性互补问题的基于模的矩阵分裂算法的最优参数估计
IF 4.3 3区 数学
Numerical Linear Algebra with Applications Pub Date : 2022-11-29 DOI: 10.1002/nla.2480
Zhizhi Li, Huai Zhang
{"title":"On estimation of the optimal parameter of the modulus‐based matrix splitting algorithm for linear complementarity problems on second‐order cones","authors":"Zhizhi Li, Huai Zhang","doi":"10.1002/nla.2480","DOIUrl":"https://doi.org/10.1002/nla.2480","url":null,"abstract":"There are many studies on the well‐known modulus‐based matrix splitting (MMS) algorithm for solving complementarity problems, but very few studies on its optimal parameter, which is of theoretical and practical importance. Therefore and here, by introducing a novel mapping to explicitly cast the implicit fixed point equation and thus obtain the iteration matrix involved, we first present the estimation approach of the optimal parameter of each step of the MMS algorithm for solving linear complementarity problems on the direct product of second‐order cones (SOCLCPs). It also works on single second‐order cone and the non‐negative orthant. On this basis, we further propose an iteration‐independent optimal parameter selection strategy for practical usage. Finally, the practicability and effectiveness of the new proposal are verified by comparing with the experimental optimal parameter and the diagonal part of system matrix. In addition, with the optimal parameter, the effectiveness of the MMS algorithm can indeed be greatly improved, even better than the state‐of‐the‐art solvers SCS and SuperSCS that solve the equivalent SOC programming.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":null,"pages":null},"PeriodicalIF":4.3,"publicationDate":"2022-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45285312","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Backward error analysis of specified eigenpairs for sparse matrix polynomials 稀疏矩阵多项式指定特征对的后向误差分析
IF 4.3 3区 数学
Numerical Linear Algebra with Applications Pub Date : 2022-11-21 DOI: 10.1002/nla.2476
Sk. Safique Ahmad, Prince Kanhya
{"title":"Backward error analysis of specified eigenpairs for sparse matrix polynomials","authors":"Sk. Safique Ahmad, Prince Kanhya","doi":"10.1002/nla.2476","DOIUrl":"https://doi.org/10.1002/nla.2476","url":null,"abstract":"This article studies the unstructured and structured backward error analysis of specified eigenpairs for matrix polynomials. The structures we discuss include T$$ T $$ ‐symmetric, T$$ T $$ ‐skew‐symmetric, Hermitian, skew Hermitian, T$$ T $$ ‐even, T$$ T $$ ‐odd, H$$ H $$ ‐even, H$$ H $$ ‐odd, T$$ T $$ ‐palindromic, T$$ T $$ ‐anti‐palindromic, H$$ H $$ ‐palindromic, and H$$ H $$ ‐anti‐palindromic matrix polynomials. Minimally structured perturbations are constructed with respect to Frobenius norm such that specified eigenpairs become exact eigenpairs of an appropriately perturbed matrix polynomial that also preserves sparsity. Further, we have used our results to solve various quadratic inverse eigenvalue problems that arise from real‐life applications.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":null,"pages":null},"PeriodicalIF":4.3,"publicationDate":"2022-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43936506","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Fast solution of three‐dimensional elliptic equations with randomly generated jumping coefficients by using tensor‐structured preconditioners 用张量结构预条件快速求解具有随机跳跃系数的三维椭圆方程
IF 4.3 3区 数学
Numerical Linear Algebra with Applications Pub Date : 2022-11-17 DOI: 10.1002/nla.2477
B. Khoromskij, V. Khoromskaia
{"title":"Fast solution of three‐dimensional elliptic equations with randomly generated jumping coefficients by using tensor‐structured preconditioners","authors":"B. Khoromskij, V. Khoromskaia","doi":"10.1002/nla.2477","DOIUrl":"https://doi.org/10.1002/nla.2477","url":null,"abstract":"In this paper, we propose and analyze the numerical algorithms for fast solution of periodic elliptic problems in random media in ℝd$$ {mathbb{R}}^d $$ , d=2,3$$ d=2,3 $$ . Both the two‐dimensional (2D) and three‐dimensional (3D) elliptic problems are considered for the jumping equation coefficients built as a checkerboard type configuration of bumps randomly distributed on a large L×L$$ Ltimes L $$ , or L×L×L$$ Ltimes Ltimes L $$ lattice, respectively. The finite element method discretization procedure on a 3D n×n×n$$ ntimes ntimes n $$ uniform tensor grid is described in detail, and the Kronecker tensor product approach is proposed for fast generation of the stiffness matrix. We introduce tensor techniques for the construction of the low Kronecker rank spectrally equivalent preconditioner in a periodic setting to be used in the framework of the preconditioned conjugate gradient iteration. The discrete 3D periodic Laplacian pseudo‐inverse is first diagonalized in the Fourier basis, and then the diagonal matrix is reshaped into a fully populated third‐order tensor of size n×n×n$$ ntimes ntimes n $$ . The latter is approximated by a low‐rank canonical tensor by using the multigrid Tucker‐to‐canonical tensor transform. As an example, we apply the presented solver in numerical analysis of stochastic homogenization method where the 3D elliptic equation should be solved many hundred times, and where for every random sampling of the equation coefficient one has to construct the new stiffness matrix and the right‐hand side. The computational characteristics of the presented solver in terms of a lattice parameter L$$ L $$ and the grid‐size, nd$$ {n}^d $$ , in both 2D and 3D cases are illustrated in numerical tests. Our solver can be used in various applications where the elliptic problem should be solved for a number of different coefficients for example, in many‐particle dynamics, protein docking problems or stochastic modeling.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":null,"pages":null},"PeriodicalIF":4.3,"publicationDate":"2022-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46110171","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
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