查看更多>>摘要：In this expository paper, we discuss the properties and implementation of the numerical Jordan and Weyr canonical forms as computed by the algorithm of Kublanovskaya-RuheKagstrom. In contrast to the widespread negative opinion toward the numerical computation and implementation of the Jordan form, we emphasize its usefulness in the solution of some important problems of matrix analysis. The numerical Jordan form is defined by regularization of an ill-posed eigenvalue problem and, as opposed to the theoretical case, is not sensitive to small changes of the matrix elements. This makes possible its stable computation in the case of well-conditioned numerical structure. A short description of the algorithm of Kagstrom and Ruhe is presented and some examples are given to illustrate its performance. Also, an alternative numerical algorithm for finding the Jordan form proposed recently by Zeng and Li is briefly commented. Several misconceptions about the Jordan form are discussed and some ways to improve the existing numerical algorithms are briefly considered. It is asserted that reliable estimation of eigenvalue accuracy and sensitivity is impossible without knowing the Jordan structure of the matrix. As a case study, we present the computation of defective multiple eigenvalue sensitivity when the use of Jordan form is indispensable.(c) 2021 Elsevier Inc. All rights reserved.
查看更多>>摘要：We consider linear sparse recovery problems where additional structure regarding the support of the solution is known. The form of the structure considered is non-overlapping sets of indices that each contain part of the support. An algorithm based on iterative hard thresholding is proposed to solve this problem. The convergence and error of the method are analyzed with respect to mutual coherence. Numerical simulations are examined in the context of an inverse source problem, including modifications for off-grid recovery.(c) 2021 Elsevier Inc. All rights reserved.
查看更多>>摘要：We study a Riemannian metric on the cone of symmetric positive-definite matrices obtained from the Hessian of the power potential function (1 - det(X)(beta))/beta. We give explicit expressions for the geodesics and distance function, under suitable conditions. In the scalar case, the geodesic between two positive numbers coincides with a weighted power mean, while for matrices of size at least two it yields a notion of weighted power mean different from the ones given in the literature. As beta tends to zero, the power potential converges to the logarithmic potential, that yields a well-known metric associated with the matrix geometric mean; we show that the geodesic and the distance associated with the power potential converge to the weighted matrix geometric mean and the distance associated with the logarithmic potential, respectively. (c) 2021 Elsevier Inc. All rights reserved.
查看更多>>摘要：Scaffolds are certain tensors arising in the study of association schemes, and have been (implicitly) understood diagrammatically as digraphs with distinguished "root" nodes and with matrix edge weights, often taken from Bose-Mesner algebras. In this paper, we first present a slight modification of Martin's conjecture (2021) concerning a duality of scaffolds whose digraphs are embedded in a closed disk in the plane with root nodes all lying on the boundary circle, and then show that this modified conjecture holds true if we restrict ourselves to the class of translation association schemes, i.e., those association schemes that admit abelian regular automorphism groups.(c) 2021 Elsevier Inc. All rights reserved.
查看更多>>摘要：This article presents a randomized matrix-free method for approximating the trace of f (A), where A is a large symmetric matrix and f is a function analytic in a closed interval containing the eigenvalues of A. Our method uses a combination of stochastic trace estimation (i.e., Hutchinson's method), Chebyshev approximation, and multilevel Monte Carlo techniques. We establish general bounds on the approximation error of this method by extending an existing error bound for Hutchinson's method to multilevel trace estimators. Numerical experiments are conducted for common applications such as estimating the log-determinant, nuclear norm, and Estrada index. We find that using multilevel techniques can substantially reduce the variance of existing single-level estimators.(C) 2021 Elsevier Inc. All rights reserved.
查看更多>>摘要：Block Gram-Schmidt algorithms serve as essential kernels in many scientific computing applications, but for many commonly used variants, a rigorous treatment of their stability properties remains open. This work provides a comprehensive categorization of block Gram-Schmidt algorithms, particularly those used in Krylov subspace methods to build orthonormal bases one block vector at a time. Known stability results are assembled, and new results are summarized or conjectured for important communication-reducing variants. Additionally, new block versions of low-synchronization variants are derived, and their efficacy and stability are demonstrated for a wide range of challenging examples. Numerical examples are computed with a versatile MATLAB package hosted at https://github .com /katlund /BlockStab, and scripts for reproducing all results in the paper are provided. Block Gram-Schmidt implementations in popular software packages are discussed, along with a number of open problems. An appendix containing all algorithms type-set in a uniform fashion is provided.(c) 2021 Elsevier Inc. All rights reserved.
查看更多>>摘要：Kac-Moody algebras, g(A), are Lie algebras defined by generators and relations given by generalized Cartan matrices A. In this paper, we study the graded identities for Kac-Moody algebras when the matrix A is diagonal. More precisely, we provide a basis for the graded identities of g(A) equipped with its natural grading, the grading of Cartan type. These results are obtained over an arbitrary infinite field. We also compute the graded codimensions for these algebras and provide a basis for the vector space of the multihomogeneous polynomials of any given multidegree in the relatively free algebra. As the base field is infinite we have a vector space basis of the relatively free algebra. As a consequence of our results, we give an alternative proof of Theorem 17 in [15], and generalize it to characteristic two. Finally, we also describe a basis of the graded identities for the Heisenberg algebra with its natural grading, over any field. (C) 2021 Elsevier Inc. All rights reserved.
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