SIAM Journal on Matrix Analysis and Applications最新文献_第6页

A Fast Algorithm for Computing Macaulay Null Spaces of Bivariate Polynomial Systems 计算二元多项式系统麦考利无效空间的快速算法

IF 1.5 2区数学

SIAM Journal on Matrix Analysis and Applications Pub Date : 2024-01-24 DOI: 10.1137/23m1550414

Nithin Govindarajan, Raphaël Widdershoven, Shivkumar Chandrasekaran, Lieven De Lathauwer

{"title":"A Fast Algorithm for Computing Macaulay Null Spaces of Bivariate Polynomial Systems","authors":"Nithin Govindarajan, Raphaël Widdershoven, Shivkumar Chandrasekaran, Lieven De Lathauwer","doi":"10.1137/23m1550414","DOIUrl":"https://doi.org/10.1137/23m1550414","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 368-396, March 2024. Abstract.As a crucial first step towards finding the (approximate) common roots of a (possibly overdetermined) bivariate polynomial system of equations, the problem of determining an explicit numerical basis for the right null space of the system’s Macaulay matrix is considered. If [math] denotes the total degree of the bivariate polynomials of the system, the cost of computing a null space basis containing all system roots is [math] floating point operations through standard numerical algebra techniques (e.g., a singular value decomposition, rank-revealing QR-decomposition). We show that it is actually possible to design an algorithm that reduces the complexity to [math]. The proposed algorithm exploits the Toeplitz structures of the Macaulay matrix under a nongraded lexicographic ordering of its entries and uses the low displacement rank properties to efficiently convert it into a Cauchy-like matrix with the help of fast Fourier transforms. By modifying the classical Schur algorithm with total pivoting for Cauchy-like matrices, a compact representation of the right null space is eventually obtained from a rank-revealing LU-factorization. Details of the proposed method, including numerical experiments, are fully provided for the case wherein the polynomials are expressed in the monomial basis. Furthermore, it is shown that an analogous fast algorithm can also be formulated for polynomial systems expressed in the Chebyshev basis.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"65 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139559793","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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

An Efficient Algorithm for Integer Lattice Reduction 整数网格还原的高效算法

IF 1.5 2区数学

SIAM Journal on Matrix Analysis and Applications Pub Date : 2024-01-24 DOI: 10.1137/23m1557933

François Charton, Kristin Lauter, Cathy Li, Mark Tygert

{"title":"An Efficient Algorithm for Integer Lattice Reduction","authors":"François Charton, Kristin Lauter, Cathy Li, Mark Tygert","doi":"10.1137/23m1557933","DOIUrl":"https://doi.org/10.1137/23m1557933","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 353-367, March 2024. Abstract. A lattice of integers is the collection of all linear combinations of a set of vectors for which all entries of the vectors are integers and all coefficients in the linear combinations are also integers. Lattice reduction refers to the problem of finding a set of vectors in a given lattice such that the collection of all integer linear combinations of this subset is still the entire original lattice and so that the Euclidean norms of the subset are reduced. The present paper proposes simple, efficient iterations for lattice reduction which are guaranteed to reduce the Euclidean norms of the basis vectors (the vectors in the subset) monotonically during every iteration. Each iteration selects the basis vector for which projecting off (with integer coefficients) the components of the other basis vectors along the selected vector minimizes the Euclidean norms of the reduced basis vectors. Each iteration projects off the components along the selected basis vector and efficiently updates all information required for the next iteration to select its best basis vector and perform the associated projections.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"27 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139559971","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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

Constraint-Satisfying Krylov Solvers for Structure-Preserving DiscretiZations 保结构离散化的约束满足克雷洛夫求解器

IF 1.5 2区数学

SIAM Journal on Matrix Analysis and Applications Pub Date : 2024-01-23 DOI: 10.1137/22m1540624

James Jackaman, Scott MacLachlan

{"title":"Constraint-Satisfying Krylov Solvers for Structure-Preserving DiscretiZations","authors":"James Jackaman, Scott MacLachlan","doi":"10.1137/22m1540624","DOIUrl":"https://doi.org/10.1137/22m1540624","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 327-352, March 2024. Abstract. A key consideration in the development of numerical schemes for time-dependent partial differential equations (PDEs) is the ability to preserve certain properties of the continuum solution, such as associated conservation laws or other geometric structures of the solution. There is a long history of the development and analysis of such structure-preserving discretization schemes, including both proofs that standard schemes have structure-preserving properties and proposals for novel schemes that achieve both high-order accuracy and exact preservation of certain properties of the continuum differential equation. When coupled with implicit time-stepping methods, a major downside to these schemes is that their structure-preserving properties generally rely on an exact solution of the (possibly nonlinear) systems of equations defining each time step in the discrete scheme. For small systems, this is often possible (up to the accuracy of floating-point arithmetic), but it becomes impractical for the large linear systems that arise when considering typical discretization of space-time PDEs. In this paper, we propose a modification to the standard flexible generalized minimum residual iteration that enforces selected constraints on approximate numerical solutions. We demonstrate its application to both systems of conservation laws and dissipative systems.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"14 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139559792","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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

Structure Preserving Quaternion Biconjugate Gradient Method 结构保留四元双共轭梯度法

IF 1.5 2区数学

SIAM Journal on Matrix Analysis and Applications Pub Date : 2024-01-22 DOI: 10.1137/23m1547299

Tao Li, Qing-Wen Wang

引用次数: 0

Fast Non-Hermitian Toeplitz Eigenvalue Computations, Joining Matrixless Algorithms and FDE Approximation Matrices 快速非ermitian Toeplitz 特征值计算、无矩阵连接算法和 FDE 近似矩阵

IF 1.5 2区数学

SIAM Journal on Matrix Analysis and Applications Pub Date : 2024-01-19 DOI: 10.1137/22m1529920

Manuel Bogoya, Sergei M. Grudsky, Stefano Serra-Capizzano

{"title":"Fast Non-Hermitian Toeplitz Eigenvalue Computations, Joining Matrixless Algorithms and FDE Approximation Matrices","authors":"Manuel Bogoya, Sergei M. Grudsky, Stefano Serra-Capizzano","doi":"10.1137/22m1529920","DOIUrl":"https://doi.org/10.1137/22m1529920","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 284-305, March 2024. Abstract. The present work is devoted to the eigenvalue asymptotic expansion of the Toeplitz matrix [math], whose generating function [math] is complex-valued and has a power singularity at one point. As a consequence, [math] is non-Hermitian and we know that in this setting, the eigenvalue computation is a nontrivial task for large sizes. First we follow the work of Bogoya, Böttcher, Grudsky, and Maximenko and deduce a complete asymptotic expansion for the eigenvalues. In a second step, we apply matrixless algorithms, in the spirit of the work by Ekström, Furci, Garoni, Serra-Capizzano et al., for computing those eigenvalues. Since the inner and extreme eigenvalues have different asymptotic behaviors, we worked on them independently and combined the results to produce a high precision global numerical and matrixless algorithm. The numerical results are very precise, and the computational cost of the proposed algorithms is independent of the size of the considered matrices for each eigenvalue, which implies a linear cost when the entire spectrum is computed. From the viewpoint of real-world applications, we emphasize that the class under consideration includes the matrices stemming from the numerical approximation of fractional diffusion equations. In the final section a concise discussion on the matter and a few open problems are presented.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"213 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139506755","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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

Generic Eigenstructures of Hermitian Pencils 赫米特铅笔的通用特征结构

IF 1.5 2区数学

SIAM Journal on Matrix Analysis and Applications Pub Date : 2024-01-18 DOI: 10.1137/22m1523297

Fernando De Terán, Andrii Dmytryshyn, Froilán M. Dopico

引用次数: 0

The Joint Bidiagonalization of a Matrix Pair with Inaccurate Inner Iterations 矩阵对的联合对角线化与不精确的内部迭代

IF 1.5 2区数学

SIAM Journal on Matrix Analysis and Applications Pub Date : 2024-01-17 DOI: 10.1137/22m1541083

Haibo Li

{"title":"The Joint Bidiagonalization of a Matrix Pair with Inaccurate Inner Iterations","authors":"Haibo Li","doi":"10.1137/22m1541083","DOIUrl":"https://doi.org/10.1137/22m1541083","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 232-259, March 2024. Abstract. The joint bidiagonalization (JBD) process iteratively reduces a matrix pair [math] to two bidiagonal forms simultaneously, which can be used for computing a partial generalized singular value decomposition (GSVD) of [math]. The process has a nested inner-outer iteration structure, where the inner iteration usually cannot be computed exactly. In this paper, we study the inaccurately computed inner iterations of JBD by first investigating the influence of computational error of the inner iteration on the outer iteration, and then proposing a reorthogonalized JBD (rJBD) process to keep orthogonality of a part of Lanczos vectors. An error analysis of the rJBD is carried out to build up connections with Lanczos bidiagonalizations. The results are then used to investigate convergence and accuracy of the rJBD based GSVD computation. It is shown that the accuracy of computed GSVD components depends on the computing accuracy of inner iterations and the condition number of [math], while the convergence rate is not affected very much. For practical JBD based GSVD computations, our results can provide a guideline for choosing a proper computing accuracy of inner iterations in order to obtain approximate GSVD components with a desired accuracy. Numerical experiments are made to confirm our theoretical results.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"2 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139501437","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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

Deflation for the Off-Diagonal Block in Symmetric Saddle Point Systems 对称鞍点系统中对角线外块的放缩

IF 1.5 2区数学

SIAM Journal on Matrix Analysis and Applications Pub Date : 2024-01-17 DOI: 10.1137/22m1537266

Andrei Dumitrasc, Carola Kruse, Ulrich Rüde

{"title":"Deflation for the Off-Diagonal Block in Symmetric Saddle Point Systems","authors":"Andrei Dumitrasc, Carola Kruse, Ulrich Rüde","doi":"10.1137/22m1537266","DOIUrl":"https://doi.org/10.1137/22m1537266","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 203-231, March 2024. Abstract. Deflation techniques are typically used to shift isolated clusters of small eigenvalues in order to obtain a tighter distribution and a smaller condition number. Such changes induce a positive effect in the convergence behavior of Krylov subspace methods, which are among the most popular iterative solvers for large sparse linear systems. We develop a deflation strategy for symmetric saddle point matrices by taking advantage of their underlying block structure. The vectors used for deflation come from an elliptic singular value decomposition relying on the generalized Golub–Kahan bidiagonalization process. The block targeted by deflation is the off-diagonal one since it features a problematic singular value distribution for certain applications. One example is the Stokes flow in elongated channels, where the off-diagonal block has several small, isolated singular values, depending on the length of the channel. Applying deflation to specific parts of the saddle point system is important when using solvers such as CRAIG, which operates on individual blocks rather than the whole system. The theory is developed by extending the existing framework for deflating square matrices before applying a Krylov subspace method such as MINRES. Numerical experiments confirm the merits of our strategy and lead to interesting questions about using approximate vectors for deflation.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"43 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139495076","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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

Projectively and Weakly Simultaneously Diagonalizable Matrices and their Applications 投影和弱同时可对角矩阵及其应用

IF 1.5 2区数学

SIAM Journal on Matrix Analysis and Applications Pub Date : 2024-01-16 DOI: 10.1137/22m1507656

Wentao Ding, Jianze Li, Shuzhong Zhang

引用次数: 0

Communication Avoiding Block Low-Rank Parallel Multifrontal Triangular Solve with Many Right-Hand Sides 避免通信的块式低并行多前沿三角解法与多右边解法

IF 1.5 2区数学

SIAM Journal on Matrix Analysis and Applications Pub Date : 2024-01-12 DOI: 10.1137/23m1568600

Patrick Amestoy, Olivier Boiteau, Alfredo Buttari, Matthieu Gerest, Fabienne Jézéquel, Jean-Yves L’Excellent, Theo Mary

引用次数: 0