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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]

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]

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]

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]

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 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 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]

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]

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]

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]

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]

The paper describes the congruent centralizer of the matrix $$ {\Delta}_n=\left(\begin{array}{llll}\hfill & \hfill & \hfill & 1\hfill \\ {}\hfill & \hfill & \dots \hfill & i\hfill \\ {}\hfill & 1\hfill & \dots \hfill & \hfill \\ {}1\hfill & i\hfill & \hfill & \hfill \end{array}\right) $$ , representing one of three blocks in the Horn-Sergeichuk canonical form, i.e., the set of matrices X such that X Δ X = Δ. [ABSTRACT FROM AUTHOR]

In this paper we pose some questions about superderivations on $${\mathbb {Z}}_{2}$$ -graded rings. Then we consider the quaternion rings and upper triangular matrix rings with special $${\mathbb {Z}}_{2}$$ -gradings and we check the answer to these questions about them. [ABSTRACT FROM AUTHOR]

The signature matrix structural analysis method developed by Pryce provides more structural information than the commonly used Pantelides method and applies to differential-algebraic equations (DAEs) of arbitrary order. It is useful to consider how existing methods using the Pantelides algorithm can benefit from such structural analysis. The dummy derivative method is a technique commonly used to solve DAEs that can benefit from such exploitation of underlying DAE structures and information found in the Signature Matrix method. This paper gives a technique to find structurally necessary dummy derivatives and how to use different block triangular forms effectively when performing the dummy derivative method and then provides a brief complexity analysis of the proposed approach. We finish by outlining an approach that can simplify the task of dummy pivoting. [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 design two numerical methods for solving some matrix feasibility problems, which arise in the quantum information science. By making use of the structured properties of linear constraints and the minimization theorem of symmetric matrix on manifold, the projection formulas of a matrix onto the feasible sets are given, and then the relaxed alternating projection algorithm and alternating projection algorithm on manifolds are designed to solve these problems. Numerical examples show that the new methods are feasible and effective. [ABSTRACT FROM AUTHOR]

In this paper, new error bounds for the linear complementarity problem are obtained when the involved matrix is a weakly chained diagonally dominant B-matrix. The proposed error bounds are better than some existing results. The advantages of the results obtained are illustrated by numerical examples. [ABSTRACT FROM AUTHOR]

The classes of right and left γ-generating matrices were introduced by D. Z. Arov in the 1980s. These matrices play an important role in the description of solutions of the indeterminate Nehari problem. In the present paper, the classes of the so-called generalized right and left γ-generating matrices, being resolvent matrices of the Nehari-Takagi problem, are introduced. For matrices of these classes, some factorization theorems are proved, and the connection between the class of generalized γ-generating matrices and the class of generalized j -inner matrix valued functions is found. Subclasses of singular, regular, and strongly regular generalized -generating matrices are introduced and studied. [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 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]

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]

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 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]

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]

The present paper presents a self-equilibrated boundary element formulation for elastostatic analysis of Reissner’s plate bending problems. In this formulation, rigid body movements are introduced to the generalized displacement fundamental solution, resulting in modified boundary element matrices with inherent equilibrium satisfaction. The procedure is quite general and easy to implement into existing boundary element codes. In the end of the paper, numerical examples are presented, illustrating the benefits of the proposed formulation. [ABSTRACT FROM AUTHOR]

Abstract  In this paper, we consider the numerical discretization of elliptic eigenvalue problems by Finite Element Methods and its solution by a multigrid method. From the general theory of finite element and multigrid methods, it is well known that the asymptotic convergence rates become visible only if the mesh width h is sufficiently small, h ≤ h 0. We investigate the dependence of the maximal mesh width h 0 on various problem parameters such as the size of the eigenvalue and its isolation distance. In a recent paper (Sauter in Finite elements for elliptic eigenvalue problems in the preasymptotic regime. Technical Report. Math. Inst., Univ. Zürich, 2007), the dependence of h 0 on these and other parameters has been investigated theoretically. The main focus of this paper is to perform systematic experimental studies to validate the sharpness of the theoretical estimates and to get more insights in the convergence of the eigenfunctions and -values in the preasymptotic regime. [ABSTRACT FROM AUTHOR]

Abstract  A 0-1 matrix has the Consecutive Ones Property (C1P) if there is a permutation of its columns that leaves the 1's consecutive in each row. The Consecutive Ones Submatrix (C1S) problem is, given a 0-1 matrix A, to find the largest number of columns of A that form a submatrix with the C1P property. Such a problem finds application in physical mapping with hybridization data in genome sequencing. Let (a, b)-matrices be the 0-1 matrices in which there are at most a 1's in each column and at most b 1's in each row. This paper proves that the C1S problem remains NP-hard for (i) (2, 3)-matrices and (ii) (3, 2)-matrices. This solves an open problem posed in a recent paper of Hajiaghayi and Ganjali. We further prove that the C1S problem is polynomial-time 0.8-approximatable for (2, 3)-matrices in which no two columns are identical and 0.5-approximatable for (2, ∞)-matrices in general. We also show that the C1S problem is polynomial-time 0.5-approximatable for (3, 2)-matrices. However, there exists an ε > 0 such that approximating the C1S problem for (∞, 2)-matrices within a factor of nε (where n is the number of columns of the input matrix) is NP-hard. [ABSTRACT FROM AUTHOR]

This paper extends the numerical method, originally developed by Onate and his colleagues for the static analysis of plates, to enable modal (vibration) and bifurcative (buckling) analyses of plates to be undertaken. The procedure uses a rotation-free triangular element, and is based on combining the finite element method and the finite volume method so as to produce a triangular element with only three displacement degrees of freedom. Elastic stiffness, stability (geometric stiffness) and mass matrices are developed using the mixed formulation, and these are implemented in routine computer coding in order to demonstrate the validity and efficacy of the solution technique when compared with results reported by independent investigators. Natural frequencies and local buckling coefficients are given for plates with interior line supports, and these are used to illustrate the prowess of the numerical technique for the analysis of a wide class of plate buckling and vibration problems, whose analysis by non-discretisation techniques has found favour recently in deference to general finite element modelling which tends to introduce some inaccuracies. The element developed in this paper, however, is both accurate and has the attractive meshing characteristics of the finite element method for analysing more arbitrary geometric structural topologies. [ABSTRACT FROM AUTHOR]

This paper deals with a finite element numerical quadrature method. It is applied for a class of second-order self-adjoint elliptic operators defined on a bounded domain in the plane. Isoparametric finite element transformations and triangular Lagrange finite elements are used. We establish the rate of convergence for approximate eigenvalues and eigenfunctions of second-order elliptic eigenvalue problems, obtained by a numerical quadrature finite element approximation. Thus the relationship between possible quadrature formulas and the optimal and almost optimal precision of the method is established. The emphasis of the paper is on the error analysis of the approximate eigenpairs. Numerical results confirming the theory are presented. [ABSTRACT FROM AUTHOR]

The paper presents a class of numerical methods to compute the stationary distribution of Markov chains (MCs) with large and structured state spaces. A popular way of dealing with large state spaces in Markovian modeling and analysis is to employ Kronecker-based representations for the generator matrix and to exploit this matrix structure in numerical analysis methods. This paper presents various multilevel (ML) methods for a broad class of MCs with a hierarchcial Kronecker structure of the generator matrix. The particular ML methods are inspired by multigrid and aggregation-disaggregation techniques, and differ among each other by the type of multigrid cycle, the type of smoother, and the order of component aggregation they use. Numerical experiments demonstrate that so far ML methods with successive over-relaxation as smoother provide the most effective solvers for considerably large Markov chains modeled as HMMs with multiple macrostates. [ABSTRACT FROM AUTHOR]

In this paper we present efficient symbolic techniques for probabilistic model checking. These have been implemented in PRISM, a tool for the analysis of probabilistic models such as discrete-time Markov chains, continuous-time Markov chains and Markov decision processes using specifications in the probabilistic temporal logics PCTL and CSL. Motivated by the success of model checkers such as SMV which use BDDs (binary decision diagrams), we have developed an implementation of PCTL and CSL model checking based on MTBDDs (multi-terminal BDDs) and BDDs. Existing work in this direction has been hindered by the generally poor performance of MTBDD-based numerical computation, which is often substantially slower than explicit methods using sparse matrices. The focus of this paper is a novel hybrid technique which combines aspects of symbolic and explicit approaches to overcome these performance problems. For typical examples, we achieve a dramatic improvement over the purely symbolic approach. In addition, thanks to the compact model representation using MTBDDs, we can verify systems an order of magnitude larger than with sparse matrices, while almost matching or even beating them for speed. [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]

The work presented in this paper discusses the transmission properties of elastic waves through multilayers of spheres which are periodically arranged in a homogeneous host material. The multilayers of spheres can have different kinds of planar defects. These defects are formed by removing one layer of spheres or by changing the radiuses or the material of scatterers in some layers. First, the reflection and transmission matrices of a single layer of spheres are obtained by the multiple scattering approaches and then the reflection and transmission matrices of multilayers of spheres are derived based on the polymerization method. Numerical examples are presented for 15 layers of spheres with one central planar defect layer or two symmetrically arranged planar defect layers. The influences of these planar defects on the frequency-dependent transmission curves are discussed. It is observed that the band gap can be widened evidently by introducing the planar defect, and the defect states appear in the band gap consequently. So the elaborately arranged planar defects are important in designing specific acoustic filters. [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]

As is well known, the diagonal-Schur complements of strictly diagonally dominant matrices are strictly diagonally dominant. In this paper, we verify the block diagonal-Schur complements of I-block strictly doubly diagonally dominant matrices are I-block strictly doubly diagonally dominant matrices, the same is true for II-block strictly doubly diagonally dominant matrices. The theoretical analysis is supported by numerical experiments. [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]

In this paper, we find matrix representation of a class of sixth order Sturm-Liouville problem (SLP) with separated, self-adjoint boundary conditions and we show that such SLP have finite spectrum. Alsofor a given matrix eigenvalue problem HX = λV X, where H is a block tridiagonal matrix and V is a block diagonal matrix, we find a sixth order boundary value problem of Atkinson type that is equivalent to matrix eigenvalue problem. [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, an adaptive sizing BFGS method for unconstrained optimization is proposed, whose scaling factor is automatically chosen by the signature of some terms of the approximate model's curvature. The scaling factor is always chosen less than or equal to one as required by the convergence property of the self-scaling BFGS method of Al-Baali (Comput Optim Appl 9:191-203, ), while the choosing strategy can ensure the sufficient positive definiteness of the updating matrices. Under mild conditions, the global convergence properties of Al-Baali on convex functions are proved. Numerical results on some test problems show the proposed method is competitive with its counterparts provided that the scaling factor is not too small. [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]

In this paper we present a convergence analysis of a positivity-preserving fitted finite volume element method (FVEM) for a generalized Black-Scholes equation transformed on finite interval, degenerating on both boundary points. We first formulate the FVEM as a Petrov-Galerkin finite element method using a spatial discretization, previously proposed by the author. The Gärding coercivity of the corresponding discrete bilinear form is established. We obtain stability and error bounds for the solution of the fully-discrete system. Analysis of the impact of the finite domain transformation on the numerical solution of the original problem is given. [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]

In this paper, we investigate variants of the well-known Golub and Welsch algorithm for computing nodes and weights of Gaussian quadrature rules for symmetric weights w in intervals (− a, a) (not necessarily bounded). The purpose is to reduce the complexity of the Jacobi eigenvalue problem stemming from Wilf's theorem and show the effectiveness of Matlab implementations of our variants for reducing the computer times compared to some other methods. Numerical examples on three test problems show the benefits of these variants. [ABSTRACT FROM AUTHOR]

Recently Miyajima presented algorithms to compute componentwise verified error bounds for the solution of full-rank least squares problems and underdetermined linear systems. In this paper we derive simpler and improved componentwise error bounds which are based on equalities for the error of a given approximate solution. Equalities are not improvable, and the expressions are formulated in a way that direct evaluation yields componentwise and rigorous estimates of good quality. The computed bounds are correct in a mathematical sense covering all sources of errors, in particular rounding errors. Numerical results show a gain in accuracy compared to previous results. [ABSTRACT FROM AUTHOR]

This paper deals with designing H filters of reduced order for two dimensional (2-D) continuous systems described by Roesser models, with uncertain state space matrices. These filters are characterized in terms of linear matrix inequalities (LMI), to minimize a bound on the H noise attenuation, by using homogeneous polynomially parameter-dependent matrices of arbitrary degree. The methodology is also particularized for full order and zero order (static) filters, where more simple LMI conditions are derived. Numerical examples are presented to illustrate the proposed methodology. [ABSTRACT FROM AUTHOR]

The global FOM and GMRES algorithms are among the effective methods to solve Sylvester matrix equations. In this paper, we study these algorithms in the case that the coefficient matrices are real symmetric (real symmetric positive definite) and extract two CG-type algorithms for solving generalized Sylvester matrix equations. The proposed methods are iterative projection methods onto matrix Krylov subspaces. Numerical examples are presented. [ABSTRACT FROM AUTHOR]