SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 478-503, March 2024. Abstract. We solve the problem of characterizing the existence of a polynomial matrix of fixed degree when its eigenstructure (or part of it) and some of its rows (columns) are prescribed. More specifically, we present a solution to the row (column) completion problem of a polynomial matrix of given degree under different prescribed invariants: the whole eigenstructure, all of it but the row (column) minimal indices, and the finite and/or infinite structures. Moreover, we characterize the existence of a polynomial matrix with prescribed degree and eigenstructure over an arbitrary field.
{"title":"Row or Column Completion of Polynomial Matrices of Given Degree","authors":"Agurtzane Amparan, Itziar Baragaña, Silvia Marcaida, Alicia Roca","doi":"10.1137/23m1564547","DOIUrl":"https://doi.org/10.1137/23m1564547","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 478-503, March 2024. <br/> Abstract. We solve the problem of characterizing the existence of a polynomial matrix of fixed degree when its eigenstructure (or part of it) and some of its rows (columns) are prescribed. More specifically, we present a solution to the row (column) completion problem of a polynomial matrix of given degree under different prescribed invariants: the whole eigenstructure, all of it but the row (column) minimal indices, and the finite and/or infinite structures. Moreover, we characterize the existence of a polynomial matrix with prescribed degree and eigenstructure over an arbitrary field.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139756466","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}
Hussam Al Daas, Grey Ballard, Laura Grigori, Suraj Kumar, Kathryn Rouse
SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 450-477, March 2024. Abstract. Multiple tensor-times-matrix (Multi-TTM) is a key computation in algorithms for computing and operating with the Tucker tensor decomposition, which is frequently used in multidimensional data analysis. We establish communication lower bounds that determine how much data movement is required (under mild conditions) to perform the Multi-TTM computation in parallel. The crux of the proof relies on analytically solving a constrained, nonlinear optimization problem. We also present a parallel algorithm to perform this computation that organizes the processors into a logical grid with twice as many modes as the input tensor. We show that, with correct choices of grid dimensions, the communication cost of the algorithm attains the lower bounds and is therefore communication optimal. Finally, we show that our algorithm can significantly reduce communication compared to the straightforward approach of expressing the computation as a sequence of tensor-times-matrix operations when the input and output tensors vary greatly in size.
{"title":"Communication Lower Bounds and Optimal Algorithms for Multiple Tensor-Times-Matrix Computation","authors":"Hussam Al Daas, Grey Ballard, Laura Grigori, Suraj Kumar, Kathryn Rouse","doi":"10.1137/22m1510443","DOIUrl":"https://doi.org/10.1137/22m1510443","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 450-477, March 2024. <br/> Abstract. Multiple tensor-times-matrix (Multi-TTM) is a key computation in algorithms for computing and operating with the Tucker tensor decomposition, which is frequently used in multidimensional data analysis. We establish communication lower bounds that determine how much data movement is required (under mild conditions) to perform the Multi-TTM computation in parallel. The crux of the proof relies on analytically solving a constrained, nonlinear optimization problem. We also present a parallel algorithm to perform this computation that organizes the processors into a logical grid with twice as many modes as the input tensor. We show that, with correct choices of grid dimensions, the communication cost of the algorithm attains the lower bounds and is therefore communication optimal. Finally, we show that our algorithm can significantly reduce communication compared to the straightforward approach of expressing the computation as a sequence of tensor-times-matrix operations when the input and output tensors vary greatly in size.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139756471","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 419-428, March 2024. Abstract. This is a further discussion of a previous work of the author on tensors with different rank and symmetric rank. We point out several obstructions towards extending a complex number example to the real number setting and discuss several further questions raised in the literature.
{"title":"More on Tensors with Different Rank and Symmetric Rank","authors":"Yaroslav Shitov","doi":"10.1137/23m1547159","DOIUrl":"https://doi.org/10.1137/23m1547159","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 419-428, March 2024. <br/> Abstract. This is a further discussion of a previous work of the author on tensors with different rank and symmetric rank. We point out several obstructions towards extending a complex number example to the real number setting and discuss several further questions raised in the literature.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139756314","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 429-449, March 2024. Abstract. We present a linear algebra formulation of backpropagation which allows the calculation of gradients by using a generically written “backslash” or Gaussian elimination on triangular systems of equations. Generally, the matrix elements are operators. This paper has three contributions: (i) it is of intellectual value to replace traditional treatments of automatic differentiation with a (left acting) operator theoretic, graph-based approach; (ii) operators can be readily placed in matrices in software in programming languages such as Julia as an implementation option; (iii) we introduce a novel notation, “transpose dot” operator “[math]” that allows for the reversal of operators. We further demonstrate the elegance of the operators approach in a suitable programming language consisting of generic linear algebra operators such as Julia [Bezanson et al., SIAM Rev., 59 (2017), pp. 65–98], and that it is possible to realize this abstraction in code. Our implementation shows how generic linear algebra can allow operators as elements of matrices. In contrast to “operator overloading,” where backslash would normally have to be rewritten to take advantage of operators, with “generic programming” there is no such need.
{"title":"Backpropagation through Back Substitution with a Backslash","authors":"Alan Edelman, Ekin Akyürek, Yuyang Wang","doi":"10.1137/22m1532871","DOIUrl":"https://doi.org/10.1137/22m1532871","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 429-449, March 2024. <br/> Abstract. We present a linear algebra formulation of backpropagation which allows the calculation of gradients by using a generically written “backslash” or Gaussian elimination on triangular systems of equations. Generally, the matrix elements are operators. This paper has three contributions: (i) it is of intellectual value to replace traditional treatments of automatic differentiation with a (left acting) operator theoretic, graph-based approach; (ii) operators can be readily placed in matrices in software in programming languages such as Julia as an implementation option; (iii) we introduce a novel notation, “transpose dot” operator “[math]” that allows for the reversal of operators. We further demonstrate the elegance of the operators approach in a suitable programming language consisting of generic linear algebra operators such as Julia [Bezanson et al., SIAM Rev., 59 (2017), pp. 65–98], and that it is possible to realize this abstraction in code. Our implementation shows how generic linear algebra can allow operators as elements of matrices. In contrast to “operator overloading,” where backslash would normally have to be rewritten to take advantage of operators, with “generic programming” there is no such need.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139756458","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}
Francesca Arrigo, Desmond J. Higham, Vanni Noferini, Ryan Wood
SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 397-418, March 2024. Abstract. We extend the notion of nonbacktracking walks from unweighted graphs to graphs whose edges have a nonnegative weight. Here the weight associated with a walk is taken to be the product over the weights along the individual edges. We give two ways to compute the associated generating function, and corresponding node centrality measures. One method works directly on the original graph and one uses a line graph construction followed by a projection. The first method is more efficient, but the second has the advantage of extending naturally to time-evolving graphs. Based on these generating functions, we define and study corresponding centrality measures. Illustrative computational results are also provided.
{"title":"Weighted Enumeration of Nonbacktracking Walks on Weighted Graphs","authors":"Francesca Arrigo, Desmond J. Higham, Vanni Noferini, Ryan Wood","doi":"10.1137/23m155219x","DOIUrl":"https://doi.org/10.1137/23m155219x","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 397-418, March 2024. <br/> Abstract. We extend the notion of nonbacktracking walks from unweighted graphs to graphs whose edges have a nonnegative weight. Here the weight associated with a walk is taken to be the product over the weights along the individual edges. We give two ways to compute the associated generating function, and corresponding node centrality measures. One method works directly on the original graph and one uses a line graph construction followed by a projection. The first method is more efficient, but the second has the advantage of extending naturally to time-evolving graphs. Based on these generating functions, we define and study corresponding centrality measures. Illustrative computational results are also provided.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139580909","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}
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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}
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