On the denominator values and barycentric weights of rational interpolants

We improve upon the method of Zhu and Zhu [A method for directly finding the denominator values of rational interpolants, J. Comput. Appl. Math. 148 (2002) 341–348] for finding the denominator values of rational interpolants, reducing considerably the number of arithmetical operations required for their computation. In a second stage, we determine the points (if existent) which can be discarded from the rational interpolation problem. Furthermore, when the interpolant has a linear denominator, we obtain a formula for the barycentric weights which is simpler than the one found by Berrut and Mittelmann [Matrices for the direct determination of the barycentric weights of rational interpolation, J. Comput. Appl. Math. 78 (1997) 355–370]. Subsequently, we give a necessary and sufficient condition for the rational interpolant to have a pole.     

Cubic spline solution of singularly perturbed boundary value problems with significant first derivatives

We develop a numerical technique for a class of singularly perturbed two-point singular boundary value problems on an uniform mesh using polynomial cubic spline. The scheme derived in this paper is second-order accurate. The resulting linear system of equations has been solved by using a tri-diagonal solver. Numerical results are provided to illustrate the proposed method and to compared with the methods in [R.K. Mohanty, Urvashi Arora, A family of non-uniform mesh tension spline methods for singularly perturbed two-point singular boundary value problems with significant first derivatives, Appl. Math. Comput., 172 (2006) 531–544; M.K. Kadalbajoo, V.K. Aggarwal, Fitted mesh B-spline method for solving a class of singular singularly perturbed boundary value problems, Int. J. Comput. Math. 82 (2005) 67–76].     

A new family of iterative methods for solving system of nonlinear algebric equations

Homotopy perturbation method (HPM) is applied to construct a new iterative method for solving system of nonlinear algebric equations. Comparison of the result obtained by the present method with that obtained by revised Adomian decomposition method [Hossein Jafari, Varsha Daftardar-Gejji, Appl. Math. Comput. 175 (2006) 1–7] reveals that the accuracy and fast convergence of the new method.     

On mean-square stability properties of a new adaptive stochastic Runge–Kutta method

We analyze the mean-square (MS) stability properties of a newly introduced adaptive time-stepping stochastic Runge–Kutta method which relies on two local error estimators based on drift and diffusion terms of the equation [A. Foroush Bastani, S.M. Hosseini, A new adaptive Runge–Kutta method for stochastic differential equations, J. Comput. Appl. Math. 206 (2007) 631–644]. In the same spirit as [H. Lamba, T. Seaman, Mean-square stability properties of an adaptive time-stepping SDE solver, J. Comput. Appl. Math. 194 (2006) 245–254] and with applying our adaptive scheme to a standard linear multiplicative noise test problem, we show that the MS stability region of the adaptive method strictly contains that of the underlying stochastic differential equation. Some numerical experiments confirms the validity of the theoretical results.     

A reconsideration on convergence of three-step iterations for asymptotically nonexpansive mappings

We illustrate that the control conditions of the main convergence theorems of Yao and Noor [Convergence of three-step iterations for asymptotically nonexpansive mappings, Appl. Math. Comput. in press] are incorrect. We also provide new control conditions which are complementary to Nilsrakoo and Saejung’s results [W. Nilsrakoo, S. Saejung, A new three-step fixed point iteration scheme for asymptotically nonexpansive mappings, Appl. Math. Comput. 181 (2006) 1026–1034].     

Nonlinear optimization exclusion tests for finding all solutions of nonlinear equations

Exclusion tests are a well known tool in the area of interval analysis for finding the zeros of a function over a compact domain. Recently, K. Georg developed linear programming exclusion tests based on Taylor expansions. In this paper, we modify his approach by choosing another objective function and using nonlinear constraints to make the new algorithm converges faster than the algorithm in [K. Georg, A new exclusion test, J. Comput. Appl. Math. 152 (2003) 147–160]. In this way, we reduce the number of subinterval in each level. The computational complexity for the new tests are investigated. Also, numerical results and comparisons will be presented.     

Adomian decomposition method for solution of differential-algebraic equations

Solutions of differential algebraic equations is considered by Adomian decomposition method. In E. Babolian, M.M. Hosseini [Reducing index and spectral methods for differential-algebraic equations, J. Appl. Math. Comput. 140 (2003) 77] and M.M. Hosseini [An index reduction method for linear Hessenberg systems, J. Appl. Math. Comput., in press], an efficient technique to reduce index of semi-explicit differential algebraic equations has been presented. In this paper, Adomian decomposition method is applied to reduced index problems. The scheme is tested for some examples and the results demonstrate reliability and efficiency of the proposed methods.     

A nonmonotone trust-region method of conic model for unconstrained optimization

In this paper, we present a nonmonotone trust-region method of conic model for unconstrained optimization. The new method combines a new trust-region subproblem of conic model proposed in [Y. Ji, S.J. Qu, Y.J. Wang, H.M. Li, A conic trust-region method for optimization with nonlinear equality and inequality 4 constrains via active-set strategy, Appl. Math. Comput. 183 (2006) 217–231] with a nonmonotone technique for solving unconstrained optimization. The local and global convergence properties are proved under reasonable assumptions. Numerical experiments are conducted to compare this method with the method of [Y. Ji, S.J. Qu, Y.J. Wang, H.M. Li, A conic trust-region method for optimization with nonlinear equality and inequality 4 constrains via active-set strategy, Appl. Math. Comput. 183 (2006) 217–231].     

Asymptotic relations between the Hahn-type polynomials and Meixner–Pollaczek, Jacobi, Meixner and Krawtchouk polynomials

It has been shown in Ferreira et al. [Asymptotic relations in the Askey scheme for hypergeometric orthogonal polynomials, Adv. in Appl. Math. 31(1) (2003) 61–85], López and Temme [Approximations of orthogonal polynomials in terms of Hermite polynomials, Methods Appl. Anal. 6 (1999) 131–146; The Askey scheme for hypergeometric orthogonal polynomials viewed from asymptotic analysis, J. Comput. Appl. Math. 133 (2001) 623–633] that the three lower levels of the Askey table of hypergeometric orthogonal polynomials are connected by means of asymptotic relations. In Ferreira et al. [Limit relations between the Hahn polynomials and the Hermite, Laguerre and Charlier polynomials, submitted for publication] we have established new asymptotic connections between the fourth level and the two lower levels. In this paper, we continue with that program and obtain asymptotic expansions between the fourth level and the third level: we derive 16 asymptotic expansions of the Hahn, dual Hahn, continuous Hahn and continuous dual Hahn polynomials in terms of Meixner–Pollaczek, Jacobi, Meixner and Krawtchouk polynomials. From these expansions, we also derive three new limits between those polynomials. Some numerical experiments show the accuracy of the approximations and, in particular, the accuracy in the approximation of the zeros of those polynomials.     

A study of the solutions of the combined sine–cosine-Gordon equation

We have studied the solutions of the combined sine–cosine-Gordon Equation found by Wazwaz [A.M. Wazwaz, Travelling wave solutions for combined and double combined sine-cosine-Gordon equations by the variable separated ODE method, Appl. Math. Comput. 177 (2006) 755] using the variable separated ODE method. These solutions can be transformed into a new form. We have derived the relation between the phase of the combined sine–cosine-Gordon equation and the parameter in these solutions. Its applications in physical systems are also discussed.     

Mixed finite element analysis of a thermally nonlinear coupled problem

In Loula and Zhou [Comput Appl Math 20 (2001), 321–339], a thermally coupled nonlinear elliptic system modeling a large class of engineering problems was considered, and some mathematical and numerical analyses (C⁰ Lagrangian finite elements combined with a fixed point algorithm) were given. To continue our work, we propose in this article a mixed method for the potential equation and present the corresponding analyses and numerical implementations. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Padé iteration method for regularization

In this study we present iterative regularization methods using rational approximations, in particular, Padé approximants, which work well for ill-posed problems. We prove that the (k, j)-Padé method is a convergent and order optimal iterative regularization method in using the discrepancy principle of Morozov. Furthermore, we present a hybrid Padé method, compare it with other well-known methods and found that it is faster than the Landweber method. It is worth mentioning that this study is a completion of the paper [A. Kirsche, C. Böckmann, Rational approximations for ill-conditioned equation systems, Appl. Math. Comput. 171 (2005) 385–397] where this method was treated to solve ill-conditioned equation systems.     

Existence theorem and algorithm for a general implicit variational inequality in Banach space

By using the generalized f-projection operator, the existence theorem of solutions for the general implicit variational inequality GIVI(T-ξ,K) is proved without assuming the monotonicity of operators in reflexive and smooth Banach space. An iterative algorithm for approximating solution of the general implicit variational inequality is suggested also, and the convergence for this iterative scheme is shown. These theorems extend the corresponding results of Wu and Huang [K.Q. Wu, N.J. Huang, Comput. Math. Appl. 54 (2007) 399–406], Wu and Huang [K.Q. Wu, N.J. Huang, Bull. Austral. Math. Soc. 73 (2006) 307–317], Zeng and Yao [L.C. Zeng, J.C. Yao, J. Optimiz. Theory Appl. 132 (2) (2007) 321–337] and Li [J. Li, J. Math. Anal. Appl. 306 (2005) 55–71].     

Tensor complementarity problems: the GUS-property and an algorithm

In this paper, we study the global uniqueness and solvability (GUS-property) of tensor complementarity problems (TCPs) for some special structured tensors. The modulus equation for TCPs is also proposed, and based on this equation, we develop the corresponding nonsmooth Newton’s method, which extends the existing method given in the work of Zheng H and Li W. The modulus-based nonsmooth Newton’s method for solving linear complementarity problems. J Comput Appl Math. 2015;288: 116-126. Numerical examples are given to demonstrate the efficiency of the proposed algorithm.     

Convergence of generalized AOR iterative method for linear systems with strictly diagonally dominant matrices

In this paper, some improvements on Darvishi and Hessari [On convergence of the generalized AOR method for linear systems with diagonally dominant coefficient matrices, Appl. Math. Comput. 176 (2006) 128–133] are presented for bounds of the spectral radius of _lω,r$l_{\omega ,r}$, which is the iterative matrix of the generalized AOR (GAOR) method. Subsequently, some new sufficient conditions for convergence of GAOR method will be given, which improve some results of Darvishi and Hessari [On convergence of the generalized AOR method for linear systems with diagonally dominant coefficient matrices, Appl. Math. Comput. 176 (2006) 128–133].     

A survey of numerical techniques for solving singularly perturbed ordinary differential equations

This survey paper contains a surprisingly large amount of material and indeed can serve as an introduction to some of the ideas and methods of singular perturbation theory. Starting from Prandtl's work a large amount of work has been done in the area of singular perturbations. This paper limits its coverage to some standard singular perturbation models considered by various workers and the numerical methods developed by numerous researchers after 1984–2000. The work done in this area during the period 1905–1984 has already been surveyed by the first author of this paper, see [Appl. Math. Comput. 30 (1989) 223] for details. Due to the space constraints we have covered only singularly perturbed one-dimensional problems.     

A communication-less parallel algorithm for tridiagonal Toeplitz systems

Diagonally dominant tridiagonal Toeplitz systems of linear equations arise in many application areas and have been well studied in the past. Modern interest in numerical linear algebra is often focusing on solving classic problems in parallel. In McNally [Fast parallel algorithms for tri-diagonal symmetric Toeplitz systems, MCS Thesis, University of New Brunswick, Saint John, 1999], an m processor Split & Correct algorithm was presented for approximating the solution to a symmetric tridiagonal Toeplitz linear system of equations. Nemani [Perturbation methods for circulant-banded systems and their parallel implementation, Ph.D. Thesis, University of New Brunswick, Saint John, 2001] and McNally (2003) adapted the works of Rojo [A new method for solving symmetric circulant tri-diagonal system of linear equations, Comput. Math. Appl. 20 (1990) 61–67], Yan and Chung [A fast algorithm for solving special tri-diagonal systems, Computing 52 (1994) 203–211] and McNally et al. [A split-correct parallel algorithm for solving tri-diagonal symmetric Toeplitz systems, Internat. J. Comput. Math. 75 (2000) 303–313] to the non-symmetric case. In this paper we present relevant background from these methods and then introduce an m processor scalable communication-less approximation algorithm for solving a diagonally dominant tridiagonal Toeplitz system of linear equations.     

Recent advances in linear analysis of convergence for splittings for solving ODE problems

In the nineties, Van der Houwen et al. (see, e.g., [P.J. van der Houwen, B.P. Sommeijer, J.J. de Swart, Parallel predictor–corrector methods, J. Comput. Appl. Math. 66 (1996) 53–71; P.J. van der Houwen, J.J.B. de Swart, Triangularly implicit iteration methods for ODE-IVP solvers, SIAM J. Sci. Comput. 18 (1997) 41–55; P.J. van der Houwen, J.J.B. de Swart, Parallel linear system solvers for Runge–Kutta methods, Adv. Comput. Math. 7 (1–2) (1997) 157–181]) introduced a linear analysis of convergence for studying the properties of the iterative solution of the discrete problems generated by implicit methods for ODEs. This linear convergence analysis is here recalled and completed, in order to provide a useful quantitative tool for the analysis of splittings for solving such discrete problems. Indeed, this tool, in its complete form, has been actively used when developing the computational codes BiM and BiMD [L. Brugnano, C. Magherini, The BiM code for the numerical solution of ODEs, J. Comput. Appl. Math. 164–165 (2004) 145–158. Code available at: http://www.math.unifi.it/~brugnano/BiM/index.html; L. Brugnano, C. Magherini, F. Mugnai, Blended implicit methods for the numerical solution of DAE problems, J. Comput. Appl. Math. 189 (2006) 34–50]. Moreover, the framework is extended for the case of special second order problems. Examples of application, aimed to compare different iterative procedures, are also presented.     

An Algorithm for Solving Optimization Problems with One Linear Objective Function and Finitely Many Constraints of Fuzzy Relation Inequalities

An optimization model with one linear objective function and fuzzy relation equation constraints was presented by Fang and Li (1999) as well as an efficient solution procedure was designed by them for solving such a problem. A more general case of the problem, an optimization model with one linear objective function and finitely many constraints of fuzzy relation inequalities, is investigated in this paper. A new approach for solving this problem is proposed based on a necessary condition of optimality given in the paper. Compared with the known methods, the proposed algorithm shrinks the searching region and hence obtains an optimal solution fast. For some special cases, the proposed algorithm reaches an optimal solution very fast since there is only one minimum solution in the shrunk searching region. At the end of the paper, two numerical examples are given to illustrate this difference between the proposed algorithm and the known ones.     

Approximation theorems for total quasi-?-asymptotically nonexpansive mappings with applications

The purpose of this article is to prove some approximation theorems of common fixed points for countable families of total quasi-?-asymptotically nonexpansive mappings which contain several kinds of mappings as its special cases in Banach spaces. In order to get the approximation theorems, the hybrid algorithms are presented and are used to approximate the common fixed points. Using this result, we also discuss the problem of strong convergence concerning the maximal monotone operators in a Banach space. The results of this article extend and improve the results of Matsushita and Takahashi [S. Matsushita, W. Takahashi, A strong convergence theorem for relatively nonexpansive mappings in Banach spaces, J. Approx. Theor. 134 (2005) 257-266], Plubtieng and Ungchittrakool [S. Plubtieng, K. Ungchittrakool, Hybrid iterative methods for convex feasibility problems and fixed point problems of relatively nonexpansive mappings in Banach spaces, J. Approx. Theor. 149 (2007) 103-115], Li, Su [H. Y. Li, Y. F. Su, Strong convergence theorems by a new hybrid for equilibrium problems and variational inequality problems, Nonlinear Anal. 72(2) (2010) 847-855], Su, Xu and Zhang [Y.F. Su, H.K. Xu, X. Zhang, Strong convergence theorems for two countable families of weak relatively nonexpansive mappings and applications, Nonlinear Anal. 73 (2010) 3890-3960], Wang et al. [Z.M. Wang, Y.F. Su, D.X. Wang, Y.C. Dong, A modified Halpern-type iteration algorithm for a family of hemi-relative nonexpansive mappings and systems of equilibrium problems in Banach spaces, J. Comput. Appl. Math. 235 (2011) 2364-2371], Chang et al. [S.S. Chang, H.W. Joseph Lee, Chi Kin Chan, A new hybrid method for solving a generalized equilibrium problem solving a variational inequality problem and obtaining common fixed points in Banach spaces with applications, Nonlinear Anal. 73 (2010) 2260-2270], Chang et al. [S.S. Chang, C.K. Chan, H.W. Joseph Lee, Modified block iterative algorithm for quasi-?-asymptotically nonexpansive mappings and equilibrium problem in Banach spaces, Appl. Math. Comput. 217 (2011) 7520-7530], Ofoedu and Malonza [E.U. Ofoedu, D.M. Malonza, Hybrid approximation of solutions of nonlinear operator equations and application to equation of Hammerstein-type, Appl. Math. Comput. 217 (2011) 6019-6030] and Yao et al. [Y.H. Yao, Y.C. Liou, S.M. Kang, Strong convergence of an iterative algorithm on an infinite countable family of nonexpansive mappings, Appl. Math. Comput. 208 (2009) 211-218].