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In the recent years, there has been an intensive research of absolute value equations $$Ax-b=B|x|$$ . Various methods were developed, but less attention has been paid to approximating or bounding the solutions. We start filling this gap by proposing several outer approximations of the solution set. We present conditions for unsolvability and for existence of exponentially many solutions, too, and compare them with the known conditions. Eventually, we carried out numerical experiments to compare the methods with respect to computational time and quality of estimation. This helps in identifying the cases, in which the bounds are tight enough to determine the signs of the solution, and therefore also the solution itself. [ABSTRACT FROM AUTHOR]

We propose a new approach to computing a parametric solution (the so-called p-solution) to parametric interval linear systems. Solving such system is an important part of many scientific and engineering problems involving uncertainties. The parametric solution has many useful properties. It permits to compute an outer solution, an inner estimate of the interval hull solution, and intervals containing the lower and upper bounds of the interval hull solution. It can also be employed for solving various constrained optimisation problems related to the parametric interval linear system. The proposed approach improves both outer and inner bounds for the parametric solution set. In this respect, the new approach is competitive to most of the existing methods for solving parametric interval linear systems. Improved bounds on the parametric solution set guarantees improved bounds for the solutions of related optimisation problems. [ABSTRACT FROM AUTHOR]

This corrigendum provides the missing results of the paper in the title and also corrects the misprints in it. [ABSTRACT FROM AUTHOR]

An asynchronous version of the optimized Schwarz method for the solution of differential equations on a parallel computational environment is studied. In a one-way subdivision of the computational domain, with and without overlap, the method is shown to converge when the optimal artificial interface conditions are used. Convergence is also proved for the Laplacian operator under very mild conditions on the size of the subdomains, when approximate (non-optimal) interface conditions are utilized. Numerical results are presented on a large three-dimensional problem on modern parallel clusters and supercomputers illustrating the efficiency of the asynchronous approach. [ABSTRACT FROM AUTHOR]

It is established that the precise solutions on the minimum residual and matrix nearness problems of the quaternion matrix equation $${(AXB,DXE)=(C,F)}$$ for centrohermitian and skew-centrohermitian matrices, where X is an unknown quaternion matrix and A, B, C, D, E, and F are known quaternion matrices of suitable sizes. Moreover, an algorithm to get the solutions of the problems considered is provided, and a numerical example is also given to exemplify the results. [ABSTRACT FROM AUTHOR]

In this study, we propose a modified quasi-Chebyshev acceleration to the nonoverlopping multisplitting iteration method for solving the linear systems A x = b where A is a real symmetric positive definite matrix or an H-matrix. In the process of the parallel multisplitting method, the distributive tasks are parallelly computed by each processor, then a global modified acceleration is used to obtain the solution of the system A x = b for every τ steps, such that the efficiency of the computation can be improved. The convergence theory of the new algorithm is given under some reasonable conditions. Finally, numerical experiments show that the method is efficient and effective. [ABSTRACT FROM AUTHOR]

Several new criteria for identifying-tensors are obtained. As an application, some sufficient conditions of the positive definiteness for an even-order real symmetric tensor are given. Advantages of the obtained results are illustrated by numerical examples. [ABSTRACT FROM PUBLISHER]

An iterative method for the low-rank approximate solution of a class of generalized Lyapunov equations is studied. At each iteration, a standard Lyapunov equation is solved using Galerkin projection with an extended Krylov subspace method. This Lyapunov equation is solved inexactly, thus producing a nonstationary iteration. Several theoretical and computational issues are discussed so as to make the iteration efficient. Numerical experiments indicate that this method is competitive vis-à-vis the current state-of-the-art methods, both in terms of computational times and storage needs. [ABSTRACT FROM AUTHOR]

Letandbe two nonzero commutingand-quadratic matrices, respectively, wherewithand. The aim of this work is mainly to characterize all situations, where the linear combinationis a generalized quadratic matrix. The results established here cover many of the results in the literature related to idempotency, involutivity and tripotency of the linear combinations of idempotent and/or involutive matrices. [ABSTRACT FROM AUTHOR]

For the Hadamard productof a nonsingular-matrixand its inverse, some new inequalities for the minimum eigenvalue ofare obtained, which improve some related results. Finally, numerical example is given to show that the inequalities are better than some existing ones. [ABSTRACT FROM PUBLISHER]

Based on the Hermitian and skew-Hermitian splitting (HSS), we come up with a generalized HSS iteration method with a flexible shift-parameter for solving the non-Hermitian positive definite system of linear equations. This iteration method utilizes the optimization technique to obtain the optimal value of the flexible shift-parameter at iteration process. Both theory and experiment have shown that the new strategy is efficient. [ABSTRACT FROM AUTHOR]

Stimulated by odd-bipartite and even-bipartite hypergraphs, we define odd-bipartite (weakly odd-bipartie) and even-bipartite (weakly evenbipartite) tensors. It is verified that all even order odd-bipartite tensors are irreducible tensors, while all even-bipartite tensors are reducible no matter the parity of the order. Based on properties of odd-bipartite tensors, we study the relationship between the largest H-eigenvalue of a Z-tensor with nonnegative diagonal elements, and the largest H-eigenvalue of absolute tensor of that Ztensor. When the order is even and the Z-tensor is weakly irreducible, we prove that the largest H-eigenvalue of the Z-tensor and the largest H-eigenvalue of the absolute tensor of that Z-tensor are equal, if and only if the Z-tensor is weakly odd-bipartite. Examples show the authenticity of the conclusions. Then, we prove that a symmetric Z-tensor with nonnegative diagonal entries and the absolute tensor of the Z-tensor are diagonal similar, if and only if the Z-tensor has even order and it is weakly odd-bipartite. After that, it is proved that, when an even order symmetric Z-tensor with nonnegative diagonal entries is weakly irreducible, the equality of the spectrum of the Z-tensor and the spectrum of absolute tensor of that Z-tensor, can be characterized by the equality of their spectral radii. [ABSTRACT FROM AUTHOR]

Some new criteria for $\mathcal{H}$-tensors are obtained. As an application, some sufficient conditions of the positive definiteness for an even-order real symmetric tensor are given. The advantages of the results obtained are illustrated by numerical examples. [ABSTRACT FROM AUTHOR]

We compare three contrasting models of the perceived distance between root-position major and minor chords and test them against new empirical data. The models include a recent psychoacoustic model called spectral pitch-class distance, and two well-established music theoretical models –Tonnetzdistance and voice-leading distance. To allow a principled challenge, in the context of these data, of the assumptions behind each of the models, we compare them with a simple “benchmark” model that simply counts the number of common tones between chords. Spectral pitch-class distance andTonnetzhave the highest correlations with the experimental data and each other, and perform significantly better than the benchmark. The voice-leading model performs worse than the benchmark. We suggest that spectral pitch-class distance provides a psychoacoustic explanation for perceived triadic distance and its music theory representation, theTonnetz. The experimental data and the computational models are available in the Online Supplement (http://dx.doi.org/10.1080/17459737.2016.1152517). [ABSTRACT FROM AUTHOR]

Some sufficient conditions ensuring that the subdirect sum of Nekrasov matrices is in the class of Nekrasov matrices are given. In particular, we obtain that the-subdirect sumis a Nekrasov matrix, whenis a Nekrasov matrix andis a strictly diagonally dominant matrix. [ABSTRACT FROM AUTHOR]

Let A and B be nonsingular M-matrices. Some new convergent sequences of the lower bounds of the minimum eigenvalue $\tau(B\circ A^{-1})$ for the Hadamard product of B and $A^{-1}$ are given. Numerical examples are given to show that these sequences are better than some known results and could reach the true value of the minimum eigenvalue in some cases. [ABSTRACT FROM AUTHOR]

For the Hadamard product $A\circ A^{-1}$ of an M-matrix A and its inverse $A^{-1}$, some new inequalities for the minimum eigenvalue of $A\circ A^{-1}$ are derived. Numerical example is given to show that the inequalities are better than some known results. [ABSTRACT FROM AUTHOR]

We propose a generalization of the limited-memory Cholesky factorization of Lin and Moré (SIAM J. Sci. Comput. 21(1), 24-45, ) to the symmetric indefinite case with special interest in symmetric quasi-definite matrices. We use this incomplete factorization to precondition two formulations of linear systems arising from regularized interior-point methods for quadratic optimization. An advantage of the limited-memory approach is predictable memory requirements. We establish existence of incomplete factors when the input matrix is an H-matrix but our numerical results illustrate that the factorization succeeds more generally. An appropriate diagonal shift is applied whenever the input matrix is not quasi definite. As the memory parameter increases an efficiency measure of the preconditioner suggested by Scott and Tůma () improves. The combination of the 3×3 block formulation analyzed by Greif, Moulding, and Orban (SIAM J. Optim. 24(1), 49-83, ), the SYMAMD ordering, and a moderate memory parameter results in encouraging performance. [ABSTRACT FROM AUTHOR]

The well-known polar decomposition of the deformation gradient decomposes it into an orthogonal rotation and a symmetric stretch. We consider a Gram-Schmidt factorization of the deformation gradient, which decomposes it into a different orthogonal rotation with a right-triangular field that we call distortion. Properties of this distortion tensor are discussed, and a work-conjugate stress tensor is derived for this Lagrangian frame. The logarithm of distortion and its material derivative are then introduced, and their components are quantified, resulting in a new logarithmic measure for strain and its rate, distinct from Hencky strain (the logarithm of stretch) and its rate. An eigenprojection analysis and a first-order, differential, matrix equation solved using the technique of back substitution both produce the same matrix components describing the logarithm of distortion. Three homogeneous deformations illustrate similarities and differences between the logarithms of distortion and stretch. They are distinct measures of strain. The new logarithmic strain measure shows monotonic behavior under simple shear as opposed to the non-monotonic behavior of Hencky strain (see Fig. 2). [ABSTRACT FROM AUTHOR]

Given real nonzero coefficients a, A, b, B, a real parameter c and an inhomogeneity $${\varphi : \mathbb{R}\times\mathbb{R} \rightarrow \mathbb{R}}$$ , we present necessary and sufficient conditions for the existence and uniqueness of solutions of the equation $${f(ax+by+c) - Af(x) - Bf(y) = \varphi(x,y)}$$ . The solutions of this equation are given explicitly for each of the various cases emerging from the possibilities concerning the arithmetic nature of the four coefficients and the algebraic relationships between them. [ABSTRACT FROM AUTHOR]

We describe some Farkas-type necessary and sufficient conditions for solvability and feasibility of interval linear equations and inequalities, in a unified form. For the convenient choice in practice of theory and calculations, some different but equivalent forms of the Farkas-type conditions for interval linear systems are also discussed. [ABSTRACT FROM AUTHOR]

By applying the Kronecker product and vectorization operator, we extend the quasi-minimal residual variant of the biconjugate gradient stabilized (QMRCGSTAB) algorithm for solving the general coupled matrix equations and the periodic coupled matrix equations Several numerical examples are given to show the efficiency of the extended QMRCGSTAB algorithms. [ABSTRACT FROM AUTHOR]

A flexible version of the CMRH algorithm is presented that allows varying preconditioning at every step of the algorithm. A consequence of the flexibility of this new variant is that any iterative methods can be incorporated as a preconditioner in the inner steps. Theoretical results that relate the residual norm of the new algorithm and the flexible GMRES, the new algorithm with CMRH itself, are given. Numerical experiments are carried out to illustrate the effectiveness of the proposed algorithm in comparison with the standard CMRH algorithm, ILU-preconditioned CMRH variants and the flexible GMRES algorithm. [ABSTRACT FROM AUTHOR]

A graph matching is used to construct aggregation-based coarsening for an algebraic two-grid method. Effects of inexact coarse grid solve is analysed numerically for a highly discontinuous convection–diffusion coefficient matrix, and for problems from the Florida matrix market collection. The proposed strategy is found to be more robust compared to a classical algebraic multi-grid approach based on strength of connections. Basic properties of two-grid method are outlined. [ABSTRACT FROM AUTHOR]

An algorithm for computing the solution of indefinite least squares problems and of indefinite least squares problems with equality constrained is presented. Such problems arise when solving total least squares problems and in H-smoothing. The proposed algorithm relies only on stable orthogonal transformations reducing recursively the associated augmented matrix to proper block anti-triangular form. Some numerical results are reported showing the properties of the algorithm. [ABSTRACT FROM AUTHOR]

A new splitting iteration method is presented for the system of linear equations when the coefficient matrix is a non-Hermitian positive-definite matrix. The spectral radius, the optimal parameter, and some norm properties of the iteration matrix for the new method are discussed in detail. Based on these results, the new method is convergent under reasonable conditions for any non-Hermitian positive-definite linear system. Finally, the numerical examples show that the new method is more effective than the Hermitian and skew-Hermitian splitting iterative (or positive-definite and skew-Hermitian splitting iterative) method in central processing unit time. [ABSTRACT FROM PUBLISHER]

This study proposes an algorithm capable of working in parallel for solving variable statistics with large and sparse linear equations under given right hand side ranges. A comparative study to the direct linear programming method is conducted under a main central processor and up to four parallel processors. The studied results are reported computationally and discussed. Moreover, the approach can be adapted for the system under domain decompositions structure leading to a better efficiency experimentally in a case example. [ABSTRACT FROM AUTHOR]