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Abstract: This paper is concerned with the bounds of the Perron root of a nonnegative irreducible matrix . Two new methods utilizing the relationship between the Perron root of a nonnegative irreducible matrix and its generalized Perron complements are presented. The former method is efficient because it gives the bounds for only by calculating the row sums of the generalized Perron complement or even the row sums of submatrices and . And the latter gives the closest bounds (just in this paper) of . The results obtained by these methods largely improve the classical bounds. Numerical examples are given to illustrate the procedure and compare it with others, which shows that these methods are effective. [Copyright &y& Elsevier]

Abstract: This paper discusses the spectral properties of the nonsymmetric saddle point matrices of the form with A symmetric positive definite, B full rank, and C symmetric positive semidefinite. A new sufficient condition is obtained so that is diagonalizable with all its eigenvalues real and positive. This condition is weaker than that stated in the recent paper [J. Liesen, A note on the eigenvalues of saddle point matrices, Technical Report 10-2006, Institute of Mathematics, TU Berlin, 2006]. [Copyright &y& Elsevier]

Abstract: In this paper, high-resolution finite volume schemes are combined with an adaptive mesh technique inspired by multiresolution analysis to improve the computational efficiency for two-dimensional hyperbolic conservation laws. The method is conservative. Moreover, it is stable which is proven numerically in this paper. The computational grid is dynamically adapted so that higher spatial resolution is automatically allocated to regions where strong gradients are observed. Using this proposed scheme, we compute several two-dimensional model problems and a compressive rate ranging from about 5–10 is observed in all simulations. [Copyright &y& Elsevier]

Abstract: The paper is dealing with the problem of finding the densest packings of equal circles in the unit square. Recently, a global optimization method based exclusively on interval arithmetic calculations has been designed for this problem. With this method it became possible to solve the previously open problems of packing 28, 29, and 30 circles in the numerical sense: tight guaranteed enclosures were given for all the optimal solutions and for the optimum value. The present paper completes the optimality proofs for these cases by determining all the optimal solutions in the geometric sense. Namely, it is proved that the currently best-known packing structures result in optimal packings, and moreover, apart from symmetric configurations and the movement of well-identified free circles, these are the only optimal packings. The required statements are verified with mathematical rigor using interval arithmetic tools. [Copyright &y& Elsevier]

Abstract: A partition of unity method for the displacement obstacle problem of clamped Kirchhoff plates is considered in this paper. We derive optimal error estimates and present numerical results that illustrate the performance of the method. [Copyright &y& Elsevier]

Abstract: It is the purpose of this paper to derive two-step hybrid methods for , with oscillatory or periodic solutions, specially tuned to the behaviour of the solution, through the usage of the exponential fitting technique. The construction of two-step exponentially fitted hybrid methods is shown and their properties are discussed. Some numerical experiments confirming the theoretical expectations are provided. [Copyright &y& Elsevier]

Abstract: The use of explicit methods in the numerical treatment of differential equations of fractional order is an area not yet widely investigated. In this paper stability properties of some multistep methods of explicit type are investigated and new methods with larger intervals of stability are proposed. Some numerical experiments are presented in order to validate theoretical results. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the oscillation of Third-order difference equation with impulses. Some sufficient conditions for the oscillatory behavior of the solutions of Third-order impulsive difference equations are obtained. [Copyright &y& Elsevier]

Abstract: In this work we construct three novel families of approximating subdivision schemes that generate piecewise exponential polynomials and we show how to convert these into interpolating schemes of great interest in curve design for their ability to reproduce important analytical shapes and to provide highly smooth limit curves with a controllable tension. In particular, throughout this paper we will focus on the derivation of 6-point interpolating schemes that turn out to be unique in combining vital ingredients like -continuity, simplicity of definition, ease of implementation, user independency, tension control and ability to reproduce salient trigonometric and transcendental curves. [Copyright &y& Elsevier]

Abstract: In this paper, we propose some least-squares finite element procedures for linear and nonlinear parabolic equations based on first-order systems. By selecting the least-squares functional properly each proposed procedure can be split into two independent symmetric positive definite sub-procedures, one of which is for the primary unknown variable and the other is for the expanded flux unknown variable . Optimal order error estimates are developed. Finally we give some numerical examples which are in good agreement with the theoretical analysis. [Copyright &y& Elsevier]

Abstract: For the solution of elliptic problems, fractional step methods and in particular alternating directions (ADI) methods are iterative methods where fractional steps are sequential. Therefore, they only accept parallelization at low level. In [T. Lu, P. Neittaanmäki, X.C. Tai, A parallel splitting-up method for partial differential equations and its applications to Navier–Stokes equations, RAIRO Modél. Math. Anal. Numér. 26 (6) (1992) 673–708], Lu et al. proposed a method where the fractional steps can be performed in parallel. We can thus speak of parallel fractional step (PFS) methods and, in particular, simultaneous directions (SDI) methods. In this paper, we perform a detailed analysis of the convergence and optimization of PFS and SDI methods, complementing what was done in [T. Lu, P. Neittaanmäki, X.C. Tai, A parallel splitting-up method for partial differential equations and its applications to Navier–Stokes equations, RAIRO Modél. Math. Anal. Numér. 26 (6) (1992) 673–708]. We describe the behavior of the method and we specify the good choice of the parameters. We also study the efficiency of the parallelization. Some 2D, 3D and high-dimensional tests confirm our results. [Copyright &y& Elsevier]

Abstract: In this paper we consider constructing some higher-order modifications of Newton’s method for solving nonlinear equations which increase the order of convergence of existing iterative methods by one or two or three units. This construction can be applied to any iteration formula, and per iteration the resulting methods add only one additional function evaluation to increase the order. Some illustrative examples are provided and several numerical results are given to show the performance of the presented methods. [Copyright &y& Elsevier]

Abstract: In this paper, we study the superconvergence of the frictionless Signorini problem. When approximated by bilinear finite elements, by virtue of the information on the contact zone, we can derive a superconvergence rate of under a proper regularity assumption. Finally, a numerical test is given to verify our result. [Copyright &y& Elsevier]

Abstract: This paper proposes a new breakdown-free preconditioning technique, called SAINV-NS, of the AINV method of Benzi and Tuma for nonsymmetric positive definite matrices. The resulting preconditioner which is an incomplete factorization of the inverse of a nonsymmetric matrix will be used as an explicit right preconditioner for QMR, BiCGSTAB and GMRES(m) methods. The preconditoner is reliable (pivot breakdown can not occur) and effective at reducing the number of iterations. Some numerical experiments on test matrices are presented to show the efficiency of the new method and comparing to the AINV-A algorithm. [Copyright &y& Elsevier]

Abstract: In this paper, we analyze the spatial and spectral superconvergence properties of one-dimensional hyperbolic conservation law by a discontinuous Galerkin (DG) method. The analyses combine classical mathematical arguments with MATLAB experiments. Some properties of the DG schemes are discovered using discrete Fourier analyses: superconvergence of the numerical wave numbers, Radau structure of the X spatial error. [Copyright &y& Elsevier]

Abstract: This paper presents computable lower bounds of the penalty parameters for stable and convergent symmetric interior penalty Galerkin methods. In particular, we derive the explicit dependence of the coercivity constants with respect to the polynomial degree and the angles of the mesh elements. Numerical examples in all dimensions and for different polynomial degrees are presented. We show the numerical effects of loss of coercivity. [Copyright &y& Elsevier]

Abstract: The combined CFD-PBE (population balance models) are computationally intensive requiring efficient numerical methods for solving practical problems. In this paper, a high order method is presented based on the least-squares method (LSM) for the solution of a spatial-dependent population balance equation which includes advective processes. Numerical experiments are performed in order to study the behavior of the proposed method for one-dimensional cases using model problems with analytical solutions. [Copyright &y& Elsevier]

Abstract: In this paper we revisit the solution of ill-posed problems by preconditioned iterative methods from a Bayesian statistical inversion perspective. After a brief review of the most popular Krylov subspace iterative methods for the solution of linear discrete ill-posed problems and some basic statistics results, we analyze the statistical meaning of left and right preconditioners, as well as projected-restarted strategies. Computed examples illustrating the interplay between statistics and preconditioning are also presented. [Copyright &y& Elsevier]

Abstract: Graves-Morris (see [P.R. Graves-Morris, Vector valued interpolants I, Numer. Math. 42 (1983) 331–348 and P.R. Graves-Morris and C.D. Jenkins, Vector valued rational interpolants III, Constr. Approx. 2 (1986) 263–289]) defined the generalized inverse rational interpolants (GIRIs) in the form of with the divisibility condition , and proved the Uniqueness Theorem for GIRIs. However, this condition is found not necessary in some cases. In this paper, we remove this divisibility condition, define the extended generalized inverse rational interpolants (EGIRIs) and establish the Uniqueness Theorem for EGIRIs. One can see that the Uniqueness Theorem for GIRIs is the special case of the one for EGIRIs. [Copyright &y& Elsevier]

Abstract: In (simple) Pál-type interpolation the problem consists of finding a polynomial of degree that has prescribed values at, say, n pairwise distinct points and prescribed values for the rth derivative at (another) set of m distinct points. The problem is named regular if it has a unique solution for any set of data; this is equivalent to the homogeneous problem (all prescribed values are 0) having the trivial solution only. In this paper the effect of interchanging the value-nodes and the derivative-nodes on the regularity of the problem is studied for several sets of nodes. [Copyright &y& Elsevier]

Abstract: This paper is devoted to the study of the stability of the solutions of the first order implicit difference periodic equationWe prove that if there exist a pair of lower and upper solutions respectively of this problem together with a suitable condition on the function and it has at most two solutions in the sector , then one of such solutions is asymptotically stable. Moreover, it is proved the optimality of the given results, both in the conditions imposed on and in the number of the periodic solutions. [Copyright &y& Elsevier]

The authors found some errors in their above titled paper which should be corrected in this Corrigendum. [ABSTRACT FROM AUTHOR]