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This paper is concerned with the inverse problem of constructing a symmetric nonnegative matrix from realizable spectrum. We reformulate the inverse problem as an underdetermined nonlinear matrix equation over a Riemannian product manifold. To solve it, we develop a Riemannian underdetermined inexact Newton dogleg method for solving a general underdetermined nonlinear equation defined between Riemannian manifolds and Euclidean spaces. The global and quadratic convergence of the proposed method is established under some mild assumptions. Then, we solve the inverse problem by applying the proposed method to its equivalent nonlinear matrix equation and a preconditioner for the perturbed normal Riemannian Newton equation is also constructed. Numerical tests show the efficiency of the proposed method for solving the inverse problem. [ABSTRACT FROM AUTHOR]

Modeling complex systems that consist of different types of objects leads to multilayer networks, where nodes in the different layers represent different kinds of objects. Nodes are connected by edges, which have positive weights. A multilayer network is associated with a supra-adjacency matrix. This paper investigates the sensitivity of the communicability in a multilayer network to perturbations of the network by studying the sensitivity of the Perron root of the supra-adjacency matrix. Our analysis sheds light on which edge weights to make larger to increase the communicability of the network, and which edge weights can be made smaller or set to zero without affecting the communicability significantly. [ABSTRACT FROM AUTHOR]

In this work, we consider symmetric positive definite pencils depending on two parameters. That is, we are concerned with the generalized eigenvalue problem A (x) − λ B (x) v = 0 , where A and B are symmetric matrix valued functions in ℝ n × n , smoothly depending on parameters x ∈ Ω ⊂ ℝ 2 ; furthermore, B is also positive definite. In general, the eigenvalues of this multiparameter problem will not be smooth, the lack of smoothness resulting from eigenvalues being equal at some parameter values (conical intersections). Our main goal is precisely that of locating parameter values where eigenvalues are equal. We first give general theoretical results for the present generalized eigenvalue problem, and then introduce and implement numerical methods apt at detecting conical intersections. Finally, we perform a numerical study of the statistical properties of coalescing eigenvalues for pencils where A and B are either full or banded, for several bandwidths. [ABSTRACT FROM AUTHOR]

A randomized quaternion singular value decomposition algorithm based on block Krylov iteration (RQSVD-BKI) is presented to solve the low-rank quaternion matrix approximation problem. The upper bounds of deterministic approximation error and expected approximation error for the RQSVD-BKI algorithm are also given. It is shown by numerical experiments that the running time of the RQSVD-BKI algorithm is smaller than that of the quaternion singular value decomposition, and the relative errors of the RQSVD-BKI algorithm are smaller than those of the randomized quaternion singular value decomposition algorithm in Liu et al. (SIAM J. Sci. Comput., 44(2): A870-A900 (2022)) in some cases. In order to further illustrate the feasibility and effectiveness of the RQSVD-BKI algorithm, we use it to deal with the problem of color image inpainting. [ABSTRACT FROM AUTHOR]

The Cauchy radius of a scalar polynomial is an upper bound on the magnitude of its zeros, and it is optimal among all bounds depending only on the moduli of the coefficients. It has the disadvantage of being implicit because it requires the solution of a nonlinear equation. In this note, simple and explicit upper bounds are derived that are useful approximations to the Cauchy radius. The general approach to obtain the aforementioned bounds is to embed scalar polynomials into the larger framework of their generalization to matrix polynomials and then use bounds on the eigenvalues of the latter. [ABSTRACT FROM AUTHOR]

A quasi-Toeplitz (QT) matrix is a semi-infinite matrix of the form A = T (a) + E where T(a) is the Toeplitz matrix with entries (T (a)) i , j = a j - i , for a j - i ∈ C , i , j ≥ 1 , while E is a matrix representing a compact operator in ℓ 2 . The matrix A is finitely representable if a k = 0 for k < - m and for k > n , given m , n > 0 , and if E has a finite number of nonzero entries. The problem of numerically computing eigenpairs of a finitely representable QT matrix is investigated, i.e., pairs (λ , v) such that A v = λ v , with λ ∈ C , v = (v j) j ∈ Z + , v ≠ 0 , and ∑ j = 1 ∞ | v j | 2 < ∞ . It is shown that the problem is reduced to a finite nonlinear eigenvalue problem of the kind W U (λ) β = 0 , where W is a constant matrix and U depends on λ and can be given in terms of either a Vandermonde matrix or a companion matrix. Algorithms relying on Newton's method applied to the equation det W U (λ) = 0 are analyzed. Numerical experiments show the effectiveness of this approach. The algorithms have been included in the CQT-Toolbox [Numer. Algorithms 81 (2019), no. 2, 741–769]. [ABSTRACT FROM AUTHOR]

A new class of tensors called K S -tensors, which is a subset of non-singular P -tensors and generalization of H + -tensors, is proposed. It is proved that the system of K S -tensor equations always has a unique positive solution for any positive right-hand side by proposing a positive increasing map. Two approaches based on dynamical system are presented to find the unique positive solution. The theoretical analysis results show that the convergence of the proposed models is guaranteed, and numerical examples further illustrate that the given models are feasible and effective in finding the positive solution of K S -tensor equations. [ABSTRACT FROM AUTHOR]

The joint bidiagonalization (JBD) process is a useful algorithm for the computation of the generalized singular value decomposition (GSVD) of a matrix pair. However, it always suffers from rounding errors, which causes the Lanczos vectors to lose their mutual orthogonality. In order to maintain some level of orthogonality, we present a semiorthogonalization strategy. Our rounding error analysis shows that the JBD process with the semiorthogonalization strategy can ensure that the convergence of the computed quantities is not affected by rounding errors and the final accuracy is high enough. Based on the semiorthogonalization strategy, we develop the joint bidiagonalization process with partial reorthogonalization (JBDPRO). In the JBDPRO algorithm, reorthogonalizations occur only when necessary, which saves a big amount of reorthogonalization work, compared with the full reorthogonalization strategy. Numerical experiments illustrate our theory and algorithm. [ABSTRACT FROM AUTHOR]