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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 Q2 MATHEMATICS, APPLIED Pub Date : 2024-01-24 DOI: 10.1137/23m1550414
Nithin Govindarajan, Raphaël Widdershoven, Shivkumar Chandrasekaran, Lieven De Lathauwer
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.
SIAM 期刊《矩阵分析与应用》第 45 卷第 1 期第 368-396 页,2024 年 3 月。摘要.作为寻找(可能过度确定的)二元多项式方程组的(近似)公共根的关键第一步,考虑了为方程组的麦考利矩阵的右空空间确定明确数值基础的问题。如果[math]表示系统的二元多项式的总阶数,那么通过标准的数值代数技术(如奇异值分解、秩揭示 QR 分解)计算包含系统所有根的空空间基的成本为[math]浮点运算。我们的研究表明,实际上可以设计一种算法,将复杂度降低到 [math]。所提出的算法利用了麦考利矩阵在其条目非分级词法排序下的托普利兹结构,并利用低位移秩的特性,借助快速傅立叶变换将其高效地转换为类考奇矩阵。通过修改经典的库尔算法,对类考奇矩阵进行总枢转,最终通过秩揭示 LU 因子化获得右空空间的紧凑表示。针对多项式用单项式基表示的情况,全面介绍了所提方法的细节,包括数值实验。此外,研究还表明,对于用切比雪夫基表示的多项式系统,也可以制定类似的快速算法。
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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 Q2 MATHEMATICS, APPLIED Pub Date : 2024-01-24 DOI: 10.1137/23m1557933
François Charton, Kristin Lauter, Cathy Li, Mark Tygert
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.
SIAM 矩阵分析与应用期刊》,第 45 卷,第 1 期,第 353-367 页,2024 年 3 月。 摘要整数网格是一组向量的所有线性组合的集合,其中向量的所有条目都是整数,线性组合中的所有系数也都是整数。网格还原指的是在给定网格中找到一个向量集,使这个子集的所有整数线性组合集合仍然是整个原始网格,并使子集的欧几里得规范减小。本文提出了简单、高效的网格还原迭代法,保证每次迭代都能单调地降低基向量(子集中的向量)的欧氏规范。每次迭代都会选择一个基向量,在这个基向量上,沿所选向量投影掉(整数系数)其他基向量的分量,可以使还原后基向量的欧氏常态最小化。每次迭代都会沿着选定的基向量投影出分量,并有效地更新下一次迭代所需的所有信息,以选择最佳基向量并执行相关的投影。
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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 Q2 MATHEMATICS, APPLIED Pub Date : 2024-01-23 DOI: 10.1137/22m1540624
James Jackaman, Scott MacLachlan
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.
SIAM 矩阵分析与应用期刊》,第 45 卷,第 1 期,第 327-352 页,2024 年 3 月。 摘要。开发时变偏微分方程(PDEs)数值方案的一个关键考虑因素是能否保留连续解的某些性质,如相关守恒定律或解的其他几何结构。此类结构保留离散化方案的开发和分析由来已久,包括证明标准方案具有结构保留特性,以及提出既能实现高阶精度又能精确保留连续微分方程某些特性的新型方案。当这些方案与隐式时间步进方法相结合时,其主要缺点是它们的结构保持特性通常依赖于离散方案中定义每个时间步进的(可能是非线性的)方程系统的精确解。对于小系统,这通常是可能的(达到浮点运算的精度),但对于考虑典型时空 PDE 离散化时出现的大型线性系统,这就变得不切实际了。在本文中,我们提出了对标准灵活广义最小残差迭代法的修改,对近似数值解强制执行选定的约束条件。我们演示了它在守恒定律系统和耗散系统中的应用。
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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 Q2 MATHEMATICS, APPLIED Pub Date : 2024-01-22 DOI: 10.1137/23m1547299
Tao Li, Qing-Wen Wang
SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 306-326, March 2024.
Abstract. This paper considers a novel structure-preserving method for solving non-Hermitian quaternion linear systems arising from color image deblurred problems. From the quaternion Lanczos biorthogonalization procedure that preserves the quaternion tridiagonal form at each iteration, we derive the quaternion biconjugate gradient method for solving the linear systems and then establish the convergence analysis of the proposed algorithm. Finally, we provide some numerical examples to illustrate the feasibility and validity of our method in comparison with the QGMRES, especially in terms of computing time.
SIAM 矩阵分析与应用期刊》,第 45 卷,第 1 期,第 306-326 页,2024 年 3 月。 摘要本文研究了一种新颖的结构保留方法,用于求解彩色图像去模糊问题中产生的非赫米四元线性系统。从每次迭代都保留四元数三边形的四元数 Lanczos 双正交化过程出发,我们推导出求解线性系统的四元数双共轭梯度法,然后建立了所提算法的收敛性分析。最后,我们提供了一些数值示例来说明我们的方法与 QGMRES 相比的可行性和有效性,特别是在计算时间方面。
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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 Q2 MATHEMATICS, APPLIED Pub Date : 2024-01-19 DOI: 10.1137/22m1529920
Manuel Bogoya, Sergei M. Grudsky, Stefano Serra-Capizzano
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.
SIAM 矩阵分析与应用期刊》,第 45 卷第 1 期,第 284-305 页,2024 年 3 月。 摘要。本研究致力于托普利兹矩阵[math]的特征值渐近展开,该矩阵的生成函数[math]是复值矩阵,在一点处有幂奇异性。因此,[math] 是非赫米特矩阵,我们知道,在这种情况下,对于大尺寸矩阵,特征值计算并非易事。首先,我们效仿博戈亚、伯彻、格鲁德斯基和马克西门科的工作,推导出特征值的完整渐近展开。第二步,我们根据 Ekström、Furci、Garoni、Serra-Capizzano 等人的研究成果,采用无矩阵算法计算这些特征值。由于内特征值和极值特征值具有不同的渐近行为,我们对它们进行了独立研究,并将结果结合起来,产生了一种高精度的全局数值和无矩阵算法。数值结果非常精确,而且所提算法的计算成本与每个特征值的矩阵大小无关,这意味着计算整个频谱时的成本是线性的。从实际应用的角度来看,我们强调所考虑的类别包括源于分数扩散方程数值近似的矩阵。最后,我们将对这一问题进行简要讨论,并提出一些有待解决的问题。
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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 Q2 MATHEMATICS, APPLIED Pub Date : 2024-01-18 DOI: 10.1137/22m1523297
Fernando De Terán, Andrii Dmytryshyn, Froilán M. Dopico
SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 260-283, March 2024.
Abstract. We obtain the generic complete eigenstructures of complex Hermitian [math] matrix pencils with rank at most [math] (with [math]). To do this, we prove that the set of such pencils is the union of a finite number of bundle closures, where each bundle is the set of complex Hermitian [math] pencils with the same complete eigenstructure (up to the specific values of the distinct finite eigenvalues). We also obtain the explicit number of such bundles and their codimension. The cases [math], corresponding to general Hermitian pencils, and [math] exhibit surprising differences, since for [math] the generic complete eigenstructures can contain only real eigenvalues, while for [math] they can contain real and nonreal eigenvalues. Moreover, we will see that the sign characteristic of the real eigenvalues plays a relevant role for determining the generic eigenstructures.
SIAM 矩阵分析与应用期刊》,第 45 卷,第 1 期,第 260-283 页,2024 年 3 月。 摘要。我们得到了秩最多为[math](含[math])的复赫米特[math]矩阵铅笔的一般完整特征结构。为此,我们证明这类铅笔的集合是有限数量的束闭包的联合,其中每个束是具有相同完整特征结构(直到不同有限特征值的特定值)的复赫米特[数学]铅笔的集合。我们还得到了此类束的显式数量及其标度。对应于一般赫尔墨斯铅笔的[math]和[math]两种情况表现出惊人的差异,因为对于[math],一般的完整特征结构只能包含实特征值,而对于[math],它们可以包含实和非实特征值。此外,我们还将看到,实特征值的符号特征对确定通用特征结构起着重要作用。
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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 Q2 MATHEMATICS, APPLIED Pub Date : 2024-01-17 DOI: 10.1137/22m1541083
Haibo Li
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.
SIAM 矩阵分析与应用期刊》,第 45 卷,第 1 期,第 232-259 页,2024 年 3 月。 摘要。联合对角线化(JBD)过程同时将一对矩阵[math]迭代还原为两个对角线形式,可用于计算[math]的部分广义奇异值分解(GSVD)。该过程具有嵌套的内-外迭代结构,其中内迭代通常无法精确计算。本文通过研究内迭代计算误差对外迭代的影响来研究 JBD 内迭代计算不准确的问题,然后提出一种重新正交化的 JBD(rJBD)过程,以保持部分 Lanczos 向量的正交性。对 rJBD 进行了误差分析,以建立与 Lanczos 对角线化的联系。然后利用分析结果研究基于 rJBD 的 GSVD 计算的收敛性和准确性。结果表明,计算出的 GSVD 分量的精度取决于内部迭代的计算精度和 [math] 的条件数,而收敛速度则不会受到太大影响。对于基于 JBD 的实际 GSVD 计算,我们的结果可以为选择合适的内迭代计算精度提供指导,从而获得具有理想精度的近似 GSVD 分量。数值实验证实了我们的理论结果。
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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 Q2 MATHEMATICS, APPLIED Pub Date : 2024-01-17 DOI: 10.1137/22m1537266
Andrei Dumitrasc, Carola Kruse, Ulrich Rüde
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.
SIAM 矩阵分析与应用期刊》,第 45 卷,第 1 期,第 203-231 页,2024 年 3 月。 摘要放缩技术通常用于移动孤立的小特征值簇,以获得更紧密的分布和更小的条件数。这种变化会对 Krylov 子空间方法的收敛行为产生积极影响,而 Krylov 子空间方法是大型稀疏线性系统最常用的迭代求解器之一。我们利用对称鞍点矩阵的底层块结构,开发了一种对称鞍点矩阵的放缩策略。用于放缩的向量来自椭圆奇异值分解,依赖于广义 Golub-Kahan 对角线化过程。放缩的目标块是离对角线块,因为它在某些应用中具有奇异值分布问题。其中一个例子是细长通道中的斯托克斯流,根据通道的长度,非对角线块有几个孤立的小奇异值。在使用 CRAIG 等求解器时,对鞍点系统的特定部分进行放缩非常重要,因为 CRAIG 等求解器对单个块而不是整个系统进行求解。该理论是通过扩展现有框架,在应用诸如 MINRES 等克雷洛夫子空间方法之前对正方形矩阵进行放缩而发展起来的。数值实验证实了我们策略的优点,并引出了关于使用近似向量进行放缩的有趣问题。
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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 Q2 MATHEMATICS, APPLIED Pub Date : 2024-01-16 DOI: 10.1137/22m1507656
Wentao Ding, Jianze Li, Shuzhong Zhang
SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 167-202, March 2024.
Abstract. Characterizing simultaneously diagonalizable (SD) matrices has been receiving considerable attention in recent decades due to its wide applications and its role in matrix analysis. However, the notion of SD matrices is arguably still restrictive for wider applications. In this paper, we consider two error measures related to the simultaneous diagonalization of matrices and propose several new variants of SD thereof; in particular, TWSD, TWSD-B, [math]-SD (SDO), DWSD, and [math]-SD (SDO). Those are all weaker forms of SD. We derive various sufficient and/or necessary conditions of them under different assumptions and show the relationships between these new notions. Finally, we discuss the applications of these new notions in, e.g., quadratically constrained quadratic programming and independent component analysis.
SIAM 矩阵分析与应用期刊》,第 45 卷,第 1 期,第 167-202 页,2024 年 3 月。 摘要。由于同时可对角化(SD)矩阵的广泛应用及其在矩阵分析中的作用,近几十年来,SD 矩阵的特征描述一直受到广泛关注。然而,可以说 SD 矩阵的概念对于更广泛的应用仍有限制。在本文中,我们考虑了与矩阵同时对角相关的两种误差度量,并提出了 SD 的几种新变体,特别是 TWSD、TWSD-B、[math]-SD (SDO)、DWSD 和 [math]-SD (SDO)。这些都是较弱形式的 SD。我们在不同的假设条件下推导出了它们的各种充分条件和/或必要条件,并展示了这些新概念之间的关系。最后,我们讨论了这些新概念在二次约束二次编程和独立分量分析等方面的应用。
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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 Q2 MATHEMATICS, APPLIED 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
SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 148-166, March 2024.
Abstract. Block low-rank (BLR) compression can significantly reduce the memory and time costs of parallel sparse direct solvers. In this paper, we investigate the performance of the BLR triangular solve phase, which we observe to be underwhelming when dealing with many right-hand sides (RHS). We explain that this is because the bottleneck of the triangular solve is not in accessing the BLR LU factors, but rather in accessing the RHS, which are uncompressed. Motivated by this finding, we propose several new hybrid variants, which combine the right-looking and left-looking communication patterns to minimize the number of accesses to the RHS. We confirm via a theoretical analysis that these new variants can significantly reduce the total communication volume. We assess the impact of this reduction on the time performance on a range of real-life applications using the MUMPS solver, obtaining up to 20% time reduction.
SIAM 矩阵分析与应用期刊》,第 45 卷,第 1 期,第 148-166 页,2024 年 3 月。 摘要块低秩 (BLR) 压缩可以显著降低并行稀疏直接求解器的内存和时间成本。在本文中,我们研究了 BLR 三角求解阶段的性能,我们观察到在处理许多右边(RHS)时,BLR 三角求解阶段的性能不尽如人意。我们解释说,这是因为三角求解的瓶颈不在于访问 BLR LU 因子,而在于访问未压缩的 RHS。受这一发现的启发,我们提出了几种新的混合变体,它们结合了右视和左视通信模式,最大限度地减少了访问 RHS 的次数。我们通过理论分析证实,这些新变体可以显著减少总通信量。我们使用 MUMPS 求解器评估了这种减少对一系列实际应用的时间性能的影响,结果发现时间最多可减少 20%。
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引用次数: 0
期刊
SIAM Journal on Matrix Analysis and Applications
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