This paper introduces a novel approach for estimating the infinity norm of the inverse of monotone (inverse-positive) matrices and totally non-negative matrices. The proposed method is based on a partition of the index set into two complementary subsets. By considering partial row sums associated with this partition, the problem is reduced to the analysis of an auxiliary 2 × 2 matrix that captures the aggregated interaction between the subsets. The main result shows that if the constructed 2 × 2 matrix preserves the monotonicity property, then an explicit upper bound for the infinity norm of the inverse can be derived. This reduction principle enables greater flexibility in applications, as the partition can be adapted to the structure of the matrix, such as sparsity or block patterns. Furthermore, the new bounds generalize existing row-sum-based estimates, which can be viewed as the limit case corresponding to a trivial partition. The approach is extended to totally non-negative matrices via a standard sign transformation, preserving the structure of the obtained bounds. Numerical examples demonstrate that the proposed estimates can be significantly sharper than classical bounds, highlighting their practical relevance in stability analysis and numerical linear algebra.

Several applied linear algebra research areas, such as eigenvalue localizations, infinity norm estimations for matrix inverse, pseudospectra localizations, etc., are closely connected with special subclasses of nonsingular H-matrices. Many of them have already been investigated in details, but it seems that finding new such subclasses is far from completed. For example, in [22], interchanging the quantifiers in the definition of the Dashnic-Zusmanovich matrices, led to a new class, benefits of which was shown in [22], [13], [11]. In this paper we will apply similar idea to CKV class (also known in the literature under the name Σ-SDD class), and show several benefits from the obtained new class, which we will call CKV-type matrices.

An upper bound for the infinity norm for the inverse of Dashnic–Zusmanovich type matrices is given. It is proved that the upper bound is sharper than the well-known Varah's bound for strictly diagonally dominant matrices. By introducing a new subclass of P-matrices: Dashnic–Zusmanovich type B-matrices (DZ-type-B-matrices), and using the proposed infinity norm bound, an error bound is given for the linear complementarity problems of DZ-type-B-matrices. We also give a new pseudospectra localization to measure the distance to instability.

B lymphocytes, as a central part of adaptive immune responses, have the ability to fight against an almost unlimited numbers of pathogens. Impairment of B cell development, activation and differentiation to antibody secreting plasma cells can lead to malignancy, allergy, autoimmunity and immunodeficiency. However, the impact of environmental factors, such as hyperosmolality or osmotic stress caused by varying salt concentrations in different lymphoid organs, on these processes is not well-understood. Here, we report that B cells respond to osmotic stress in a biphasic manner. Initially, increased osmolality boosted B cell activation and differentiation as shown by an untimely downregulation of Pax5 as well as upregulation of CD138. However, in the second phase, we observed an increase in cell death and impaired plasmablast differentiation. Osmotic stress resulted in impaired class switch to IgG1, inhibition of phosphorylation of p38 mitogen-activated kinase and a delayed NFAT5 response. Overall, these findings demonstrate the importance of microenvironmental hyperosmolality and osmotic stress caused by NaCl for B cell activation and differentiation.

The aim of this paper is to obtain new lower bounds for the smallest singular value for some special subclasses of nonsingularH-matrices. This is done in two steps: first, unifying principle for deriving new upper bounds for the norm 1 of the inverse of an arbitrary nonsingular H-matrix is presented, and then, it is combined with some well-known upper bounds for the infinity norm of the inverse. The importance and efficiency of the results are illustrated by an example from ecological modelling, as well as on a type of large-scale matrices posessing a block structure, arising in boundary value problems.

In this paper, we consider the class of PH−matrices, a subclass of H−matrices and, using scaling characterization, we show that this class is closed under taking the Schur complement. We show that, under certain conditions, the Perron complement of a PH−matrix is a PH−matrix. We also present a way of constructing a scaling matrix for the given PH−matrix and we give eigenvalue localization for the Schur complement of a PH−matrix using only the entries of the original matrix. We illustrate this by numerical examples.

Motivated by the growing successful use of fractional differential equations in the modeling of different important phenomena, in this paper we derive tools for practical analysis of the robust asymptotic stability of a (incommensurate) fractional order linear system. First, the concept of fractional pseudospectra is introduced. Second, driven by the simplicity and usefulness of spectral localizations in the analysis of various matrix properties, we introduce adequate localization techniques using the ideas that come from diagonally dominant matrices, in order to localize the fractional pseudospectra. In such way, many theoretical and practical applications of pseudospectra (robust stability, transient behavior, nonnormal dynamics, etc.) in fractional order differential systems can be linked to the specificity of the matrix entries, allowing one to understand certain phenomena in practice better. Third, we consider the fractional distance to instability in ℓ∞, ℓ1 and ℓ2 norms, and determine efficient lower bounds. Finally, this novel approach is implemented on the realistic model of empirical food web to link the stability (that incorporates hereditary dynamics of living organisms) with the empirical data and their uncertainty limitations.

The convergence of modulus-based synchronous multisplitting accelerated overrelaxation iteration methods for linear complementarity problems is studied using the new technique by Zhang, Zhang and Ren. We show that this technique is particularly convenient in the a priori and a posteriori error analysis.

The theory of M- and H-matrices has become one of the basic tools in applied linear algebra and it has contributed to different areas of mathematical research and applications. Many results in numerical analysis, eigenvalue localization problems, analysis of iterative methods for solving systems of linear equations came from H-matrix theory. Also, many results in engineering rely on mathematical foundation that is, explicitly or implicitly, formulated in terms of H-matrices. In this talk, different matrix properties that guarantee nonsingularity of matrices and define different subclasses of H-matrices will be presented together with related results concerning Schur complement matrices, eigenvalue localization and bounds of the max-norm of the inverse matrix.

In this paper we present a new algorithm for the computation of the minimal Gersgorin set that can be considered an extension of the results from [5]. While the general approach to calculation of the boundary of the minimal Gersgorin set is kept, the core numerical calculation is changed. Namely, the problem is formulated in such a way that the eigenvalue computations are replaced by LU decompositions, allowing the algorithm to be used for larger matrices more efficiently. To illustrate the benefits, we compare both algorithms on several test matrices.

After a short review of the contemporary stability indicators recently used to assess behavior of the empirical food webs, we discuss the corresponding amplification and timescales of transient instability. Then, a novel robust measure that incorporates uncertainty level of empirical data and amplification-timescale frame in the stability analysis is introduced. As a result, more realistic notion of stability is achieved, and its usefulness is advocated. Finally, an efficient numerical algorithm for its computation is constructed to allow possible applications to high resolution food webs in the future. New stability indicator is computed for the soil food web and it is compared with the ones reported in the literature. (C) 2015 Elsevier B.V. All rights reserved.

In order to solve large sparse linear complementarity problems on parallel multiprocessor systems, modulus-based synchronous two-stage multisplitting iteration methods based on two-stage multisplittings of the system matrices were constructed and investigated by Bai and Zhang (Numer. Algoritm. 62, 59-77 2013). These iteration methods include the multisplitting relaxation methods such as Jacobi, Gauss-Seidel, SOR and AOR of the modulus type as special cases. In the same paper the convergence theory of these methods is developed, under the following assumptions: (i) the system matrix is an H +-matrix and (ii) one acceleration parameter is greater than the other. Here we show that the second assumption can be avoided, thus enabling us to obtain an improved convergence area. The result is obtained using the similar technique proposed by Cvetković and Kostić (Numer. Linear Algebra Appl. 21, 534-539 2014), and its usage is demonstrated by an example of the LCP.

In this paper we start from the known lower bounds for minimal singular value of the matrices possessing certain kind of the diagonal dominance property, and derive Euclidean norm estimates of the inverses of several new subclasses of the block H-matrices. The motivation comes from applications where the matrix in question has distinguished block structure, which can be exploited to obtain useful information. An example arising from ecological modeling illustrates the benefits of the presented approach.

Based on a short review of the different parameterization schemes for sub-grid scale surface fluxes in climate and other atmospheric models of different scales, the flux aggregation effect over a heterogeneous grid-box leading to the occurrence of Schmidt’s paradox is considered. To investigate this effect in the sub-grid scale parameterization, we have introduced a dynamical system approach, where the horizontal energy exchange is taken into account and is represented by a matrix of coupling parameters. Since it is, in general, very difficult to specify the quantities in that matrix, a sufficient condition for the asymptotic stability that can be applied for any coupling matrix is derived. Two theorems that consider the flux aggregation effect over a heterogeneous grid-box are proved. Finally, we have showed how, by their application, Schmidt’s paradox can be overcome. It is demonstrated through a numerical example of turbulent energy exchange over the grid-box including the part of the Prospect park, New York, USA.

It is well-known that for a given H-matrix A there exists a diagonal nonsingular matrix that scales A (by multiplying it from the right) to a strictly diagonally dominant (SDD) matrix. There are subclasses of H-matrices that can be fully characterised by the form of the corresponding diagonal scaling matrices. However, for some applications, it is not necessary to have such full characterisation. It is sufficient to find at least one scaling matrix that will do the job. The aim of this paper is to present a way of constructing a diagonal scaling matrix for one special subclass of H-matrices called Partition-Nekrasov matrices. As an application of this scaling approach, we obtain eigenvalue localisation for the corresponding Schur complement matrix, using only the entries of the original matrix.

In this paper we present a nonsingularity result which is a generalization of Nekrasov property by using two different permutations of the index set. The main motivation comes from the following observation: matrices that are Nekrasov matrices up to the same permutations of rows and columns, are nonsingular. But, testing all the permutations of the index set for the given matrix is too expensive. So, in some cases, our new nonsingularity criterion allows us to use the results already calculated in order to conclude that the given matrix is nonsingular. Also, we present new max-norm bounds for the inverse matrix and illustrate these results by numerical examples, comparing the results to some already known bounds for Nekrasov matrices.

Maximum norm bound of the inverse of a given matrix is an important issue in a wide range of applications. Motivated by this fact, we will extend the list of matrix classes for which upper bounds for max norms can be obtained. These classes are subclasses of block H-matrices, and they stand in a general position with corresponding point-wise classes. Efficiency of new results will be illustrated by numerical examples.

Modulus-based splitting, as well as multisplitting iteration methods, for linear complementarity problems are developed by Zhong-Zhi Bai. In related papers (see Bai, Z.-Z., Zhang, L.-L.: Modulus-Based Synchronous Multisplitting Iteration Methods for Linear Complementarity Problems. Numerical Linear Algebra with Applications 20 (2013) 425-439, and the references cited therein), the problem of convergence for two-parameter relaxation methods (accelerated overrelaxation-type methods) is analyzed under the assumption that one parameter is greater than the other. Here, we will show how we can avoid this assumption and, consequently, improve the convergence area. Copyright (C) 2013 John Wiley & Sons, Ltd.

From the application point of view, it is important to have a good upper bound for the maximum norm of the inverse of a given matrix A. In this paper we will give two simple and practical upper bounds for the maximum norm of the inverse of a Nekrasov matrix.