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Nonsymmetric algebraic Riccati equations are a class of nonlinear matrix equations with wide applications. In this paper, numerical solution of nonsymmetric algebraic Riccati equations is studied. A generalized ALI iteration method is proposed to solve the equation, which is an extension of the ALI iteration method and the MALI iteration method. Theoretical analysis and numerical experiments show that the proposed method is feasible and is more effective than the ALI iteration method and the MALI iteration method. [ABSTRACT FROM AUTHOR]

In this paper, we trace back the genesis of Aitken's Δ2 process and Shanks' sequence transformation. These methods, which are extrapolation methods, are used for accelerating the convergence of sequences of scalars, vectors, matrices, and tensors. They had, and still have, many important applications in numerical analysis and in applied mathematics. They are related to continued fractions and Padé approximants. We go back to the roots of these methods and analyze the original contributions. New and detailed explanations on the building and properties of Shanks' transformation and its kernel are provided. We then review their historical algebraic and algorithmic developments. We also analyze how they were involved in the solution of systems of linear and nonlinear equations, in particular in the methods of Steffensen, Pulay, and Anderson. Testimonies by various actors of the domain are given. The paper can also serve as an introduction to this domain of numerical analysis. [ABSTRACT FROM AUTHOR]

Discretization of dynamical models defined through partial differential equations leads to large-scale systems. Time-depending condition involves an iterative integration of such kind of systems. In this paper, a novel technique based on overlapped domain decomposition, without preconditioner and scalable, is presented. Due to the domain decomposition, subproblems are solved in parallel without communications, cutting off the computation time and optimizing the computational cost. This direct method describes an optimized parallel strategy to solve the initial problem, providing the exact solution, up to rounding errors. Moreover, it takes into account both physical nature of the problem and deriving numerical properties of the system. It is highly recommended in case of band matrices and a long time interval because of an increasing gain in terms of performance and computational cost with the number of integrations. A deep analysis of the computational cost concludes the paper. [ABSTRACT FROM AUTHOR]

Hessenberg matrices frequently arise in many application areas and have been attracted much attention in recent years. In the current paper, we present two numerical algorithms of $$O(n^2)$$ for computing the determinant of an n-by- n general Hessenberg matrix. The algorithms are all suited for implementation using Computer Algebra Systems such as MATLAB and MAPLE. Some numerical examples are provided in order to demonstrate the performance of the proposed algorithms. [ABSTRACT FROM AUTHOR]

Linear and nonlinear regression analyses based on the least-squares method play a fundamental role in the evaluation of scientific data. A large number of valuable papers have dealt with various applications of the least-squares method in the chemical literature. They, however, usually contain tremendous formulas for computing the error estimates of the estimated parameters. This paper presents a simple numerical solution based on the well-known simplex method to this problem. Elaborate enzyme kinetic data published earlier have been chosen to test the simplex method extended with error estimation. The capability of the numerical method is demonstrated by the revision of the originally calculated error propagation. Our program might prove useful in handling either chemical or biological data. [ABSTRACT FROM AUTHOR]

The main concern of this paper is with the stable discretisation of linear partial differential equations of evolution with time-varying coefficients. We commence by demonstrating that an approximation of the first derivative by a skew-symmetric matrix is fundamental in ensuring stability for many differential equations of evolution. This motivates our detailed study of skew-symmetric differentiation matrices for univariate finite-difference methods. We prove that, in order to sustain a skew-symmetric differentiation matrix of order $$p\ge 2$$ , a grid must satisfy $$2p-3$$ polynomial conditions. Moreover, once it satisfies these conditions, it supports a banded skew-symmetric differentiation matrix of this order and of the bandwidth $$2p-1$$ , which can be derived in a constructive manner. Some applications require not just skew-symmetry, but also that the growth in the elements of the differentiation matrix is at most linear in the number of unknowns. This is always true for our tridiagonal matrices of order 2 but need not be true otherwise, a subject which we explore further. Another subject which we examine is the existence and practical construction of grids that support skew-symmetric differentiation matrices of a given order. We resolve this issue completely for order-two methods. We conclude the paper with a list of open problems and their discussion. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce a novel approach based on the operational matrix of orthonormal Bernoulli polynomial for the numerical solution of the class of singular second-order boundary value problems that arise in physiology. The main thrust of this approach is to decompose the domain of the problem into two subintervals. The singularity, which lies in the first subinterval, is removed via the application of an operational matrix procedure based on differentiating that is applied to surmount the singularity. Then, in the second subdomain, which is outside the vicinity of the singularity, the resulting problem is treated via employing the proposed basis. The performance of the numerical scheme is assessed and tested on specific test problems. The oxygen diffusion problem in spherical cells and a nonlinear heat-conduction model of the human head are discussed as illustrative examples. The numerical outcomes indicate that the method yields highly accurate results and is computationally more efficient than the existing ones. [ABSTRACT FROM AUTHOR]

The mixed principal eigenvalue of p -Laplacian (equivalently, the optimal constant of weighted Hardy inequality in L space) is studied in this paper. Several variational formulas for the eigenvalue are presented. As applications of the formulas, a criterion for the positivity of the eigenvalue is obtained. Furthermore, an approximating procedure and some explicit estimates are presented case by case. An example is included to illustrate the power of the results of the paper. [ABSTRACT FROM AUTHOR]

Solid wall boundary conditions have been an area of active research within the context of Smoothed Particle Hydrodynamics (SPH) for quite a while. Ferrand et al. (Int. J. Numer. Methods Fluids 71(4), 446-472, 2012) presented a novel approach using a renormalization factor in the SPH approximation. The computation of this factor depends on an integral along the boundary of the domain and in their original paper Ferrand et al. gave an analytical formulation for the 2-D case using the Wendland kernel. In this paper the formulation will be extended to 3-D, again providing analytical formulae. Due to the boundary being two dimensional a domain decomposition algorithm needs to be employed in order to obtain special integration domains. For these the analytical formulae will be presented when using the Wendland kernel. The algorithm presented within this paper is applied to several academic test-cases for which either analytical results or simulations with other methods are available. It will be shown that the present formulation produces accurate results and provides a significant improvement compared to approximative methods. [ABSTRACT FROM AUTHOR]

In this paper, we present a new spectral residual method for solving large-scale positive definite generalized eigenvalue problems. The proposed algorithm is equipped with a dwindling multidimensional nonmonotone filter, in which the envelope dwindles as the step length decreases. We have also employed a relaxed nonmonotone line search technique in the structure of the algorithm which allows it to enjoy the nonmonotonicity from scratch. Under some mild and standard assumptions, the global convergence property of the proposed algorithm is established. An implementation of the new algorithm on some test problems shows the efficiency and effectiveness of the proposed algorithm in practice. [ABSTRACT FROM AUTHOR]

A new fast direct linear equation solver for the boundary element method (BEM) is presented in this paper. The idea of the new fast direct solver stems from the concept of the hierarchical off-diagonal low-rank matrix. The hierarchical off-diagonal low-rank matrix can be decomposed into the multiplication of several diagonal block matrices. The inverse of the hierarchical off-diagonal low-rank matrix can be calculated efficiently with the Sherman-Morrison-Woodbury formula. In this paper, a more general and efficient approach to approximate the coefficient matrix of the BEM with the hierarchical off-diagonal low-rank matrix is proposed. Compared to the current fast direct solver based on the hierarchical off-diagonal low-rank matrix, the proposed method is suitable for solving general 3-D boundary element models. Several numerical examples of 3-D potential problems with the total number of unknowns up to above 200,000 are presented. The results show that the new fast direct solver can be applied to solve large 3-D BEM models accurately and with better efficiency compared with the conventional BEM. [ABSTRACT FROM AUTHOR]

The importance of unsupervised clustering and topic modeling is well recognized with ever-increasing volumes of text data available from numerous sources. Nonnegative matrix factorization (NMF) has proven to be a successful method for cluster and topic discovery in unlabeled data sets. In this paper, we propose a fast algorithm for computing NMF using a divide-and-conquer strategy, called DC-NMF. Given an input matrix where the columns represent data items, we build a binary tree structure of the data items using a recently-proposed efficient algorithm for computing rank-2 NMF, and then gather information from the tree to initialize the rank- k NMF, which needs only a few iterations to reach a desired solution. We also investigate various criteria for selecting the node to split when growing the tree. We demonstrate the scalability of our algorithm for computing general rank- k NMF as well as its effectiveness in clustering and topic modeling for large-scale text data sets, by comparing it to other frequently utilized state-of-the-art algorithms. The value of the proposed approach lies in the highly efficient and accurate method for initializing rank- k NMF and the scalability achieved from the divide-and-conquer approach of the algorithm and properties of rank-2 NMF. In summary, we present efficient tools for analyzing large-scale data sets, and techniques that can be generalized to many other data analytics problem domains along with an open-source software library called SmallK. [ABSTRACT FROM AUTHOR]

This paper investigates modified function projective synchronization (MFPS) for complex dynamical networks with mixed time-varying and hybrid asymmetric coupling delays, which is composed of state coupling, time-varying delay coupling and distributed time-varying delay coupling. In contrast to previous results, the coupling configuration matrix needs not be symmetric or irreducible. The MFPS of delayed complex dynamical networks is considered via either hybrid control or hybrid pinning control with nonlinear and adaptive linear feedback control, which contains error linear term, time-varying delay error linear term and distributed time-varying delay error linear term. Based on Lyapunov stability theory, adaptive control technique, the parameter update law and the technique of dealing with some integral terms, we will show that control may be used to manipulate the scaling functions matrix such that the drive system and response networks could be synchronized up to the desired scaling function matrix. Numerical examples are given to demonstrate the effectiveness of the proposed method. The results in this article generalize and improve the corresponding results of the recent works. [ABSTRACT FROM AUTHOR]

In this paper, we give new convergence results for the basic fixed point iteration and its two inversion-free variants for finding the maximal positive definite solution of the matrix equation $$X+A^{*}X^{-1}A+B^{*}X^{-1}B=Q$$ , proposed by Long et al. (Bull Braz Math Soc 39:371-386, 2008) and Vaezzadeh et al. (Adv Differ Equ 2013). The new results are illustrated by numerical examples. [ABSTRACT FROM AUTHOR]

Background, aim, and scope: Propagation of parametric uncertainty in life cycle inventory (LCI) models is usually performed based on probabilistic Monte Carlo techniques. However, alternative approaches using interval or fuzzy numbers have been proposed based on the argument that these provide a better reflection of epistemological uncertainties inherent in some process data. Recent progress has been made to integrate fuzzy arithmetic into matrix-based LCI using decomposition into α-cut intervals. However, the proposed technique implicitly assumes that the lower bounds of the technology matrix elements give the highest inventory results, and vice versa, without providing rigorous proof. Materials and methods: This paper provides formal proof of the validity of the assumptions made in that paper using a formula derived in 1950. It is shown that an increase in the numerical value of an element of the technology matrix A results in a decrease of the corresponding element of the inverted matrix A, provided that the latter is non-negative. Results: It thus follows that the assumption used in fuzzy uncertainty propagation using matrix-based LCI is valid when A does not contain negative elements. Discussion: In practice, this condition is satisfied by feasible life cycle systems whose component processes have positive scaling factors. However, when avoided processes are used in order to account for the presence of multifunctional processes, this condition will be violated. We then provide some guidelines to ensure that the necessary conditions for fuzzy propagation are met by an LCI model. Conclusions: The arguments presented here thus provide rigorous proof that the algorithm developed for fuzzy matrix-based LCI is valid under specified conditions, namely when the inverse of the technology matrix is non-negative. Recommendations and perspectives: This paper thus gives the conditions for which computationally efficient propagation of uncertainties in fuzzy LCI models is strictly valid. [ABSTRACT FROM AUTHOR]

This paper deals with quadratic equivalence, normal forms of observability, characteristic matrices and normal quadratic numbers for nonlinear Single-Input Single-Output systems. We investigated both cases: nonlinear systems linearly observable and nonlinear systems with one linear unobservable mode. Particularly, the effect of the normal quadratic numbers on the observer design is pointed out. Finally, a faster observability analysis is proposed using characteristic matrices and normal quadratic numbers. Throughout the paper, academic examples as well as bio-reactor example highlight our purpose. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study on the spectrum of Hermitizable tridiagonal matrices. As an illustration of the application of the author's recent results on Hermitizable matrices, an explicit criterion for discrete spectrum of the matrices is presented, with a slight and technical restriction. The problem is well known, but from the author's knowledge, it has been largely opened for quite a long time. It is important in various application, in quantum mechanics for instance. The main tool to solve the problem is the isospectral technique developed a few years ago. Two alternative constructions of the isospectral operator are presented; they are helpful in theoretical analysis and in numerical computations, respectively. Some illustrated examples are included. [ABSTRACT FROM AUTHOR]

Understanding the sustainable mining dynamics prevents economic uncertainties, operational errors and severe environmental degradation during mining. The mine dynamics vary depending on multiple factors that can affect overall efficiency. A computational model would help in assessing mining operations. This paper proposes a set of novel computational normed linear vector space models to determine various aspects of mining operations and the estimation of mineral associations in an ore complex by employing matrix algebra. The proposed computational models consider that, distribution of minerals and mine operations can be formulated based on multidimensional vector space, which is computable in nature. Analysis of range of influence interaction matrix (RIIM) basing on a 3-mineral model is carried out to assess the mining dynamics. In addition to RIIM, the cause–effect (C–E) interaction model is employed to assess the interaction intensity and dominance of minerals in an ore deposit in terms of availability as well as extraction. The C–E interrelationships between minerals are evaluated basing on the generic coding method. The transformations of structural domains resulting in multidimensional space are further assessed to envision C–E variations between Cu–Ni–Pb as a mineral complex. The projection analysis based on nonlinear regression and numerical evaluations of computational models provide an insight of the life time of a mine. The proposed computational approaches can be applied in combination on n-mineral models to aid in decision making during all stages of sustainable mining. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with a new algorithm for multichain finite state Markov decision processes which finds an average optimal policy through the decomposition of the state space into some communicating classes and a transient class. For each communicating class, a relatively optimal policy is found, which is used to find an optimal policy by applying the value iteration algorithm. Using a pattern matrix determining the behaviour pattern of the decision process, the decomposition of the state space is effectively done, so that the proposed algorithm simplifies the structured one given by the excellent Leizarowitz’s paper (Math Oper Res 28:553–586, 2003). Also, a numerical example is given to comprehend the algorithm. [ABSTRACT FROM AUTHOR]

Abstract  For a real square matrix A and an integer d ⩾ 0, let A (d) denote the matrix formed from A by rounding off all its coefficients to d decimal places. The main problem handled in this paper is the following: assuming that A (d) has some property, under what additional condition(s) can we be sure that the original matrix A possesses the same property? Three properties are investigated: nonsingularity, positive definiteness, and positive invertibility. In all three cases it is shown that there exists a real number α(d), computed solely from A (d) (not from A), such that the following alternative holds if d > α(d), then nonsingularity (positive definiteness, positive invertibility) of A (d) implies the same property for A if d < α(d) and A (d) is nonsingular (positive definite, positive invertible), then there exists a matrix A′ with A′(d) = A (d) which does not have the respective property. For nonsingularity and positive definiteness the formula for α(d) is the same and involves computation of the NP-hard norm ‖ ‖∞,1; for positive invertibility α(d) is given by an easily computable formula. [ABSTRACT FROM AUTHOR]

In this paper a new predictor-corrector noninterior method for LCP is presented, in which the predictor step is generated by the Levenberg-Marquadt method, which is new in the predictor-corrector type methods, and the corrector step is generated as in [3]. The method has the following merits: (i) any cluster point of the iteration sequence is a solution of the P0 LCP; (ii) if the generalized Jacobian is nonsingular at a solution point, then the whole sequence converges to the (unique) solution of the P0 LCP superlinearly; (iii) for the P0 LCP, if an accumulation point of the iteration sequence satisfies the strict complementary condition, then the whole sequence converges to this accumulation point superlinearly. Preliminary numerical experiments are reported to show the efficiency of the algorithm. [ABSTRACT FROM AUTHOR]

The paper reviews the development of the theory of weighted pseudoinversion. Weighted pseudoinverse matrices with singular weights are determined and investigated. Theorems of the existence and uniqueness of these matrices are provided. Weighted pseudoinverses with singular weights are related to weighted normal pseudosolutions. Weighted pseudoinverses with singular weights are represented in terms of coefficients of characteristic polynomials of symmetric and symmetrizable matrices. Their expansions into matrix power series and products and limit representations are obtained. Decompositions of weighed pseudoinverses are determined on the basis of the obtained weighed singular decomposition of matrices with singular weights. [ABSTRACT FROM AUTHOR]

In this paper, we establish some conditions for the existence and the representations for the common ( P, Q) generalized reflexive and anti-reflexive solutions of matrix equations $$AX=B$$ and $$XC=D$$ , where P and Q are two generalized reflection matrices. Moreover, in corresponding solution set of the equations, the explicit expression of the nearest matrix to a given matrix in the Frobenius norm has been presented. [ABSTRACT FROM AUTHOR]

In this paper, the unique solvability of the absolute value equation is further discussed. From the perspective of some special matrices and iteration forms, some new and useful results for the unique solvability of the absolute value equation are obtained. [ABSTRACT FROM AUTHOR]

We present a new model for two phase Darcy flows in fractured media, in which fractures are modelled as submanifolds of codimension one with respect to the surrounding domain (matrix). Fractures can act as drains or as barriers, since pressure discontinuities at the matrix-fracture interfaces are permitted. Additionally, a layer of damaged rock at the matrix-fracture interfaces is accounted for. The numerical analysis is carried out in the general framework of the Gradient Discretisation Method. Compactness techniques are used to establish convergence results for a wide range of possible numerical schemes; the existence of a solution for the two phase flow model is obtained as a byproduct of the convergence analysis. A series of numerical experiments conclude the paper, with a study of the influence of the damaged layer on the numerical solution. [ABSTRACT FROM AUTHOR]

In this paper, we present error bounds for the linear complementarity problem (LCP) with the system matrix being an MB-matrix based on its equivalent LCP, from which some error bounds for LCP with special system matrices are derived. The numerical results show the sharpness of the proposed bounds. [ABSTRACT FROM AUTHOR]

The medium is composed of an elliptic inclusion and many confocal elliptic layers. The crack is embedded in the elliptic inclusion. The remote loading is applied at the remote place of the matrix. Complex variable method and conformal mapping are used to study the mentioned problem. This paper provides a numerical solution for the mentioned crack problem. The continuity condition for the traction and displacement along the interface is reduced to a relation of two sets of Laurent series coefficients for the complex potentials defined in the interior or exterior to the interface. This formulation is called the matrix transfer method in this paper. From the following three conditions, the traction-free condition along creak face, the continuity condition for the traction and displacement along the interfaces and the remote loading condition, the problem is finally solved. Servable numerical examples are provided. For the exterior finite matrix case, the relevant solution is also provided. [ABSTRACT FROM AUTHOR]

Let TH be the space of ( T + H)-matrices of order n. The paper considers the following question: Which unitary matrices U satisfy the condition ∀A ∈ TH → U* AU ∈ TH? A criterion for verifying whether a given matrix U has this property is proposed. Bibliography: 4 titles. [ABSTRACT FROM AUTHOR]

In this paper, we consider the system of linear equations $A\tilde {x}=\tilde {b}$, where $A\in \mathbb {R}^{n \times n}$ is a crisp H-matrix and $\tilde {b}$ is a fuzzy n-vector. We then investigate the existence and uniqueness of a fuzzy solution to this system. The results can also be used for the class of M-matrices and strictly diagonally dominant matrices. Finally, some numerical examples are given to illustrate the presented theoretical results. [ABSTRACT FROM AUTHOR]

A matrix with zero diagonal is called a Euclidean distance matrix when the matrix values are measurements of distances between points in a Euclidean space. Because of data errors such a matrix may not be exactly Euclidean and it is desirable in many applications to find the best Euclidean matrix which approximates the non-Euclidean matrix. In this paper the problem is formulated as a smooth unconstrained minimization problem, for which rapid convergence can be obtained. Comparative numerical results are reported. [ABSTRACT FROM AUTHOR]

Evaluating the level of inconsistency of pairwise comparisons is often a crucial step in multi criteria decision analysis. Several inconsistency indices have been proposed in the literature to estimate the deviation of expert’s judgments from a situation of full consistency. This paper surveys and analyzes ten indices from the numerical point of view. Specifically, we investigate degrees of agreement between them to check how similar they are. Results show a wide range of behaviors, ranging from very strong to very weak degrees of agreement. [ABSTRACT FROM AUTHOR]

In this paper, we investigate spectral expansion for the asymptotically spectral differential operators generated in L2m (−∞,∞) by ordinary differential expressions of arbitrary order with periodic matrix-valued coefficients. [ABSTRACT FROM AUTHOR]

The paper derives and investigates the Jacobi methods for the generalized eigenvalue problem Ax = λBx, where A is a symmetric and B is a symmetric positive definite matrix. The methods first “normalize” B to have the unit diagonal and then maintain that property during the iterative process. The global convergence is proved for all such methods. That result is obtained for the large class of generalized serial strategies from Hari and Begović Kovač (Trans. Numer. Anal. (ETNA) 47, 107-147, 2017). Preliminary numerical tests confirm a high relative accuracy of some of those methods, provided that both matrices are positive definite and the spectral condition numbers of ΔAAΔAand ΔBBΔBare small, for some nonsingular diagonal matrices ΔAand ΔB. [ABSTRACT FROM AUTHOR]

This paper revisits, in a multi-thread context, the so-called multi-parameter or block conjugate gradient (B-CG) methods, first proposed as sequential algorithms by O’Leary and Brezinski, for the solution of the linear system Ax=b, for an n-dimensional symmetric positive definite matrix A. Instead of the scalar parameters of the classical CG algorithm, which minimizes a scalar functional at each iteration, multiple descent and conjugate directions are updated simultaneously. Implementation involves the use of multiple threads and the algorithm is referred to as cooperative CG (CCG) to emphasize that each thread now uses information that comes from the other threads. It is shown that for a sufficiently large matrix dimension n, the use of an optimal number of threads results in a worst case flop count of O(n7/3) in exact arithmetic. Numerical experiments on a multi-core, multi-thread computer, for synthetic and real matrices, illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

Completely positive (CP) tensors, which correspond to a generalization of CP matrices, allow to reformulate or approximate a general polynomial optimization problem (POP) with a conic optimization problem over the cone of CP tensors. Similarly, completely positive semidefinite (CPSD) tensors, which correspond to a generalization of positive semidefinite (PSD) matrices, can be used to approximate general POPs with a conic optimization problem over the cone of CPSD tensors. In this paper, we study CP and CPSD tensor relaxations for general POPs and compare them with the bounds obtained via a Lagrangian relaxation of the POPs. This shows that existing results in this direction for quadratic POPs extend to general POPs. Also, we provide some tractable approximation strategies for CP and CPSD tensor relaxations. These approximation strategies show that, with a similar computational effort, bounds obtained from them for general POPs can be tighter than bounds for these problems obtained by reformulating the POP as a quadratic POP, which subsequently can be approximated using CP and PSD matrices. To illustrate our results, we numerically compare the bounds obtained from these relaxation approaches on small scale fourth-order degree POPs. [ABSTRACT FROM AUTHOR]

An error bound for the linear complementarity problem (LCP) when the involved matrices are QN-matrices with positive diagonal entries is presented by Dai et al. (Error bounds for the linear complementarity problem of QN-matrices. Calcolo, 53:647-657, 2016), and there are some limitations to this bound because it involves a parameter. In this paper, for LCP with the involved matrix A being a QN-matrix with positive diagonal entries an alternative bound which depends only on the entries of A is given. Numerical examples are given to show that the new bound is better than that provided by Dai et al. in some cases. [ABSTRACT FROM AUTHOR]

We discuss a new approach to the problem of recovering signal from frame coefficients with erasures. It is known that, under the assumption that the erasure set of indices for a given frame satisfies the minimal redundancy condition, there exists a synthesizing dual frame which enables us to perfectly reconstruct the original signal without recovering the lost coefficients. In this paper we describe further properties of such dual frames compensating for erasures. [ABSTRACT FROM AUTHOR]

In this paper, fuzzy linear system as $$\widetilde{A} \widetilde{X}+ \widetilde{B} \widetilde{X}=\widetilde{Y}$$ in which $$\widetilde{A}, \widetilde{B}$$ are $$n \times n$$ fuzzy matrices and $$\widetilde{X}, \widetilde{Y}$$ are $$n \times 1$$ fuzzy vectors is studied. A new method to solve such systems based on interval linear system, interval inclusion linear system is proposed. Numerical examples are given to illustrate the ability of the proposed method. [ABSTRACT FROM AUTHOR]

Two-by-two block matrices of special form with square matrix blocks arise in important applications, such as in optimal control of partial differential equations and in high order time integration methods. Two solution methods involving very efficient preconditioned matrices, one based on a Schur complement reduction of the given system and one based on a transformation matrix with a perturbation of one of the given matrix blocks are presented. The first method involves an additional inner solution with the pivot matrix block but gives a very tight condition number bound when applied for a time integration method. The second method does not involve this matrix block but only inner solutions with a linear combination of the pivot block and the off-diagonal matrix blocks. Both the methods give small condition number bounds that hold uniformly in all parameters involved in the problem, i.e. are fully robust. The paper presents shorter proofs, extended and new results compared to earlier publications. [ABSTRACT FROM AUTHOR]

In this paper we investigate the onset of convection in a saturated porous medium where uniform suction into a horizontal and uniformly hot bounding surface induces a stationary thermal boundary layer. Particular attention is paid to how the well-known linear stability characteristics of this boundary layer are modified by the presence of local thermal nonequilibrium effects. The basic conduction state is determined and it is found that the boundary layer forms two distinct regions when the porosity is small or when the conductivity of the fluid is small compared with that of the solid. A linearised stability analysis is performed which results in an ordinary differential eigenvalue problem for the critical Darcy-Rayleigh number as a function of the wave number and the two nondimensional parameters, $$H$$ and $$\gamma $$ , which are associated with local thermal nonequilibrium. This eigenvalue problem is solved numerically by first approximating the equations by fourth order compact finite differences, and then the critical Rayleigh number is computed iteratively using the inverse power method and minimised over the wavenumber. The variation of the critical Rayleigh number and wavenumber with $$H$$ and $$\gamma $$ is presented. One of the unusual effects of local thermal nonequilibrium is that there exists a parameter regime within which the neutral curve is bimodal. [ABSTRACT FROM AUTHOR]

In this paper, we discuss some new numerical methods to solve a fully fuzzy linear system (FFLS) with triangular fuzzy numbers of the form $$ ( {m,\alpha ,\beta }) $$. Almost every existing method that intends to solve a FFLS confines the coefficient matrix and the solutions to be non-negative fuzzy numbers. The main intent of the proposed methods is to remove these restrictions and widen the scope of fuzzy linear systems in scientific applications. The methods are illustrated with the help of numerical examples and are conceptually easy to understand and apply in real life situations. [ABSTRACT FROM AUTHOR]

With the upper bound of Kullback-Leibler distance between a matrix variate Beta-distribution and a normal distribution, this paper gives the conditions under which a matrix-variate Beta-distribution will approach uniformly and asymptotically a normal distribution. [ABSTRACT FROM AUTHOR]

In this paper, we present a new alternating local Hermitian and skew-Hermitian Splitting preconditioner for solving saddle point problems. The spectral property of the preconditioned matrices is studies in detail. Theoretical results show all eigenvalues of the preconditioned matrices will generate two tight clusters, one is near (0, 0) and the other is near (2, 0) as the iteration parameter tends to zero from positive. Numerical experiments are given to validate the performances of the preconditioner. [ABSTRACT FROM AUTHOR]

In the last decade, the hierarchical matrix technique was introduced to deal with dense matrices in an efficient way. It provides a data-sparse format and allows an approximate matrix algebra of nearly optimal complexity. This paper is concerned with utilizing multiple processors to gain further speedup for the $${\mathcal {H}}$$ H -matrix algebra, namely matrix truncation, matrix–vector multiplication, matrix–matrix multiplication, and inversion. One of the most cost-effective solution for large-scale computation is distributed computing. Distribute-memory architectures provide an inexpensive way for an organization to obtain parallel capabilities as they are increasingly popular. In this paper, we introduce a new distribution scheme for $${\mathcal {H}}$$ H -matrices based on the corresponding index set. Numerical experiments applied to a BEM model will complement our complexity analysis. [ABSTRACT FROM AUTHOR]

In this paper we present a redesign of a linear algebra kernel of an interior point method to avoid the explicit use of problem matrices. The only access to the original problem data needed are the matrix-vector multiplications with the Hessian and Jacobian matrices. Such a redesign requires the use of suitably preconditioned iterative methods and imposes restrictions on the way the preconditioner is computed. A two-step approach is used to design a preconditioner. First, the Newton equation system is regularized to guarantee better numerical properties and then it is preconditioned. The preconditioner is implicit, that is, its computation requires only matrix-vector multiplications with the original problem data. The method is therefore well-suited to problems in which matrices are not explicitly available and/or are too large to be stored in computer memory. Numerical properties of the approach are studied including the analysis of the conditioning of the regularized system and that of the preconditioned regularized system. The method has been implemented and preliminary computational results for small problems limited to 1 million of variables and 10 million of nonzero elements demonstrate the feasibility of the approach. [ABSTRACT FROM AUTHOR]

We consider co-rotational wave maps from (3 + 1) Minkowski space into the three-sphere. This is an energy supercritical model which is known to exhibit finite time blow up via self-similar solutions. The ground state self-similar solution f is known in closed form and based on numerics, it is supposed to describe the generic blow up behavior of the system. In this paper we develop a rigorous linear perturbation theory around f. This is an indispensable prerequisite for the study of nonlinear stability of the self-similar blow up which is conducted in the companion paper (Donninger in Commun. Pure Appl. Math., 64(8), ). In particular, we prove that f is linearly stable if it is mode stable. Furthermore, concerning the mode stability problem, we prove new results that exclude the existence of unstable eigenvalues with large imaginary parts and also, with real parts larger than $${\frac{1}{2}}$$. The remaining compact region is well-studied numerically and all available results strongly suggest the nonexistence of unstable modes. [ABSTRACT FROM AUTHOR]

This paper shows that the stability or nonstability of switched systems does not depend on the eigenvalues of the matrices. The result is obtained by giving examples of no stable (resp. no unstable) switched linear systems consisting of stable (resp. unstable) matrices. Moreover, for the first time all kinds of eigenvalues are considered, as well as two general results for establishing the stability or nonstability of switched linear systems are presented. [ABSTRACT FROM AUTHOR]

The corotational method is an attractive approach to derive non-linear finite beam elements. In a number of papers, this method was employed to investigate the non-linear dynamic analysis of 2D beams. However, most of the approaches found in the literature adopted either a lumped mass matrix or linear local interpolations to derive the inertia terms (which gives the classical linear and constant Timoshenko mass matrix), although local cubic interpolations were used to derive the elastic force vector and the tangent stiffness matrix. In this paper, a new corotational formulation for dynamic nonlinear analysis is presented. Cubic interpolations are used to derive both the inertia and elastic terms. Numerical examples show that the proposed approach is more efficient than using lumped or Timoshenko mass matrices. [ABSTRACT FROM AUTHOR]

The single 2 dilation wavelet multipliers in one-dimensional case and single A-dilation (where A is any expansive matrix with integer entries and |det A| = 2) wavelet multipliers in twodimensional case were completely characterized by Wutam Consortium (1998) and Li Z., et al. (2010). But there exist no results on multivariate wavelet multipliers corresponding to integer expansive dilation matrix with the absolute value of determinant not 2 in L(ℝ). In this paper, we choose $2I_2 = \left( {\begin{array}{*{20}c} 2 & 0 \\ 0 & 2 \\ \end{array} } \right)$ as the dilation matrix and consider the 2 I-dilation multivariate wavelet Φ = { ψ, ψ, ψ}(which is called a dyadic bivariate wavelet) multipliers. Here we call a measurable function family f = { f, f, f} a dyadic bivariate wavelet multiplier if $\Psi _1 = \left\{ {\mathcal{F}^{ - 1} \left( {f_1 \widehat{\psi _1 }} \right),\mathcal{F}^{ - 1} \left( {f_2 \widehat{\psi _2 }} \right),\mathcal{F}^{ - 1} \left( {f_3 \widehat{\psi _3 }} \right)} \right\}$ is a dyadic bivariate wavelet for any dyadic bivariate wavelet Φ = { ψ, ψ, ψ}, where $\hat f$ and F denote the Fourier transform and the inverse transform of function f respectively. We study dyadic bivariate wavelet multipliers, and give some conditions for dyadic bivariate wavelet multipliers. We also give concrete forms of linear phases of dyadic MRA bivariate wavelets. [ABSTRACT FROM AUTHOR]

The paper suggests sufficient nonsingularity conditions for matrices in terms of certain determinantal relations of diagonal dominance type, which improve and generalize some known results. These conditions are used to describe new eigenvalue inclusion sets and to derive new two-sided bounds on the determinants of matrices satisfying them. Bibliography: 8 titles. [ABSTRACT FROM AUTHOR]