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。
{"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. <br/>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}
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.
{"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. <br/> 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}
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.
{"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. <br/> 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}
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.
{"title":"Structure Preserving Quaternion Biconjugate Gradient Method","authors":"Tao Li, Qing-Wen Wang","doi":"10.1137/23m1547299","DOIUrl":"https://doi.org/10.1137/23m1547299","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 306-326, March 2024. <br/> 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.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"123 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139559967","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}
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.
{"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. <br/> 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}
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.
{"title":"Generic Eigenstructures of Hermitian Pencils","authors":"Fernando De Terán, Andrii Dmytryshyn, Froilán M. Dopico","doi":"10.1137/22m1523297","DOIUrl":"https://doi.org/10.1137/22m1523297","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 260-283, March 2024. <br/> 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.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"105 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139495078","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}
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.
{"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. <br/> 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}
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.
{"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. <br/> 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}
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.
{"title":"Projectively and Weakly Simultaneously Diagonalizable Matrices and their Applications","authors":"Wentao Ding, Jianze Li, Shuzhong Zhang","doi":"10.1137/22m1507656","DOIUrl":"https://doi.org/10.1137/22m1507656","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 167-202, March 2024. <br/> 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.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"24 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139482982","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}
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.
{"title":"Communication Avoiding Block Low-Rank Parallel Multifrontal Triangular Solve with Many Right-Hand Sides","authors":"Patrick Amestoy, Olivier Boiteau, Alfredo Buttari, Matthieu Gerest, Fabienne Jézéquel, Jean-Yves L’Excellent, Theo Mary","doi":"10.1137/23m1568600","DOIUrl":"https://doi.org/10.1137/23m1568600","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 148-166, March 2024. <br/> 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.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"23 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139462130","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}
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