SIAM Journal on Matrix Analysis and Applications最新文献

{"title":"Efficient Vectors for Block Perturbed Consistent Matrices","authors":"Susana Furtado, Charles Johnson","doi":"10.1137/23m1580310","DOIUrl":"https://doi.org/10.1137/23m1580310","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 601-618, March 2024. <br/> Abstract. In prioritization schemes, based on pairwise comparisons, such as the analytical hierarchy process, it is important to extract a cardinal ranking vector from a reciprocal matrix that is unlikely to be consistent. It is natural to choose such a vector only from efficient ones. Recently, a method to generate inductively all efficient vectors for any reciprocal matrix has been discovered. Here we focus on the study of efficient vectors for a reciprocal matrix that is a block perturbation of a consistent matrix in the sense that it is obtained from a consistent matrix by modifying entries only in a proper principal submatrix. We determine an explicit class of efficient vectors for such matrices. Based on this, we give a description of all the efficient vectors in the 3-by-3 block perturbed case. In addition, we give sufficient conditions for the right Perron eigenvector of such matrices to be efficient and provide examples in which efficiency does not occur. Also, we consider a certain type of constant block perturbed consistent matrices, for which we may construct a class of efficient vectors, and demonstrate the efficiency of the Perron eigenvector. Appropriate examples are provided throughout.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"595 ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139756194","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}

{"title":"Perturbation and Inverse Problems of Stochastic Matrices","authors":"Joost Berkhout, Bernd Heidergott, Paul Van Dooren","doi":"10.1137/22m1489162","DOIUrl":"https://doi.org/10.1137/22m1489162","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 553-584, March 2024. <br/> Abstract. It is a classical task in perturbation analysis to find norm bounds on the effect of a perturbation [math] of a stochastic matrix [math] to its stationary distribution, i.e., to the unique normalized left Perron eigenvector. A common assumption is to consider [math] to be given and to find bounds on its impact, but in this paper, we rather focus on an inverse optimization problem called the target stationary distribution problem (TSDP). The starting point is a target stationary distribution, and we search for a perturbation [math] of the minimum norm such that [math] remains stochastic and has the desired target stationary distribution. It is shown that TSDP has relevant applications in the design of, for example, road networks, social networks, hyperlink networks, and queuing systems. The key to our approach is that we work with rank-1 perturbations. Building on those results for rank-1 perturbations, we provide heuristics for the TSDP that construct arbitrary rank perturbations as sums of appropriately constructed rank-1 perturbations.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"25 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139756461","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}

{"title":"Spectrum Maximizing Products Are Not Generically Unique","authors":"Jairo Bochi, Piotr Laskawiec","doi":"10.1137/23m1550621","DOIUrl":"https://doi.org/10.1137/23m1550621","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 585-600, March 2024. <br/> Abstract. It is widely believed that typical finite families of [math] matrices admit finite products that attain the joint spectral radius. This conjecture is supported by computational experiments and it naturally leads to the following question: are these spectrum maximizing products typically unique, up to cyclic permutations and powers? We answer this question negatively. As discovered by Horowitz around fifty years ago, there are products of matrices that always have the same spectral radius despite not being cyclic permutations of one another. We show that the simplest Horowitz products can be spectrum maximizing in a robust way; more precisely, we exhibit a small but nonempty open subset of pairs of [math] matrices [math] for which the products [math] and [math] are both spectrum maximizing.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"4 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139756204","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}

{"title":"Five-Precision GMRES-Based Iterative Refinement","authors":"Patrick Amestoy, Alfredo Buttari, Nicholas J. Higham, Jean-Yves L’Excellent, Theo Mary, Bastien Vieublé","doi":"10.1137/23m1549079","DOIUrl":"https://doi.org/10.1137/23m1549079","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 529-552, March 2024. <br/> Abstract. GMRES-based iterative refinement in three precisions (GMRES-IR3), proposed by Carson and Higham in 2018, uses a low precision LU factorization to accelerate the solution of a linear system without compromising numerical stability or robustness. GMRES-IR3 solves the update equation of iterative refinement using GMRES preconditioned by the LU factors, where all operations within GMRES are carried out in the working precision [math], except for the matrix–vector products and the application of the preconditioner, which require the use of extra precision [math]. The use of extra precision can be expensive, and is especially unattractive if it is not available in hardware; for this reason, existing implementations have not used extra precision, despite the absence of an error analysis for this approach. In this article, we propose to relax the requirements on the precisions used within GMRES, allowing the use of arbitrary precisions [math] for applying the preconditioned matrix–vector product and [math] for the rest of the operations. We obtain the five-precision GMRES-based iterative refinement (GMRES-IR5) algorithm which has the potential to solve relatively badly conditioned problems in less time and memory than GMRES-IR3. We develop a rounding error analysis that generalizes that of GMRES-IR3, obtaining conditions under which the forward and backward errors converge to their limiting values. Our analysis makes use of a new result on the backward stability of MGS-GMRES in two precisions. On hardware where three or more arithmetics are available, which is becoming very common, the number of possible combinations of precisions in GMRES-IR5 is extremely large. We provide an analysis of our theoretical results that identifies a relatively small subset of relevant combinations. By choosing from within this subset one can achieve different levels of tradeoff between cost and robustness, which allows for a finer choice of precisions depending on the problem difficulty and the available hardware. We carry out numerical experiments on random dense and SuiteSparse matrices to validate our theoretical analysis and discuss the complexity of GMRES-IR5.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"33 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139756455","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}

{"title":"A Unifying Framework for Higher Order Derivatives of Matrix Functions","authors":"Emanuel H. Rubensson","doi":"10.1137/23m1580589","DOIUrl":"https://doi.org/10.1137/23m1580589","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 504-528, March 2024. <br/> Abstract. We present a theory for general partial derivatives of matrix functions of the form [math], where [math] is a matrix path of several variables ([math]). Building on results by Mathias [SIAM J. Matrix Anal. Appl., 17 (1996), pp. 610–620] for the first order derivative, we develop a block upper triangular form for higher order partial derivatives. This block form is used to derive conditions for existence and a generalized Daleckiĭ–Kreĭn formula for higher order derivatives. We show that certain specializations of this formula lead to classical formulas of quantum perturbation theory. We show how our results are related to earlier results for higher order Fréchet derivatives. Block forms of complex step approximations are introduced, and we show how those are related to evaluation of derivatives through the upper triangular form. These relations are illustrated with numerical examples.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"77 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139756315","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}

{"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":"84 1","pages":""},"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}

{"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":"23 1","pages":""},"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}

{"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":"15 1","pages":""},"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}

{"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":"81 1","pages":""},"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}

{"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":"3 1","pages":""},"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}