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In this paper we present some techniques for constructing high-order iterative methods in order to approximate the zeros of a non-linear equation f(x)=0, starting from a well-known family of cubic iterative processes. The first technique is based on an additional functional evaluation that allows us to increase the order of convergence from three to five. With the second technique, we make some changes aimed at minimizing the calculus of inverses. Finally, looking for a better efficiency, we eliminate terms that contribute to the error equation from sixth order onwards. The paper contains a comparative study of the asymptotic error constants of the methods and some theoretical and numerical examples that illustrate the given results. We also analyse the efficiency of the aforementioned methods, by showing some numerical examples with a set of test functions and by using adaptive multi-precision arithmetic in the computation. [ABSTRACT FROM AUTHOR]

This paper presents a new hybrid algorithm for unconstrained optimization problems, which combines the idea of the IMPBOT algorithm with the nonmonotone line search technique. A feature of the proposed method is that at each iteration, a system of linear equations is solved only once to obtain a trial step, via a modified limited-memory BFGS two loop recursion that requires only matrix–vector products, thus reducing the computations and storage. Furthermore, when the trial step is not accepted, the proposed method performs a line search along it using a modified nonmonotone scheme, thus a larger stepsize can be yielded in each line search procedure. Under some reasonable assumptions, the convergence properties of the proposed algorithm are analysed. Numerical results are also reported to show the efficiency of this proposed method. [ABSTRACT FROM PUBLISHER]

In this paper, an iterative method is presented for finding the bisymmetric solutions of a pair of consistent matrix equations A1XB1=C1, A2XB2=C2, by which a bisymmetric solution can be obtained in finite iteration steps in the absence of round-off errors. Moreover, the solution with least Frobenius norm can be obtained by choosing a special kind of initial matrix. In the solution set of the matrix equations, the optimal approximation bisymmetric solution to a given matrix can also be derived by this iterative method. The efficiency of the proposed algorithm is shown by some numerical examples. [ABSTRACT FROM AUTHOR]

A novel approach to simulate cellular neural networks (CNN) is presented in this paper. The approach, time-multiplexing simulation, is prompted by the need to simulate hardware models and test hardware implementations of CNN. For practical applications, due to hardware limitations, it is impossible to have a one-to-one mapping between the CNN hardware processors and all the pixels of the image. This simulator provides a solution by processing the input image block by block, with the number of pixels in a block being the same as the number of CNN processors in the hardware. The algorithm for implementing this simulator is presented along with popular numerical integration algorithms. Some simulation results and comparisons are also presented. [ABSTRACT FROM AUTHOR]

In this paper, the variational iteration method is used to solve a system of third-order boundary value problem associated with obstacle, unilateral and contact problems. Numerical solution obtained by the method is of high accuracy. The numerical example compared with those considered by other authors shows that the method is more efficient. [ABSTRACT FROM AUTHOR]

This paper describes a cubically convergent iteration method for finding the multiple roots of nonlinear equations, f(x)=0, where f:→ is a continuous function. This work is the extension of our earlier work [P.K. Parida, and D.K. Gupta, An improved regula-falsi method for enclosing simple zeros of nonlinear equations, Appl. Math. Comput. 177 (2006), pp. 769-776] where we have developed a cubically convergent improved regula-falsi method for finding simple roots of f(x)=0. First, by using some suitable transformation, the given function f(x) with multiple roots is transformed to F(x) with simple roots. Then, starting with an initial point x0 near the simple root x* of F(x)=0, the sequence of iterates {xn}, n=0, 1, ... and the sequence of intervals {[an, bn]}, with x*∈{[an, bn]} for all n are generated such that the sequences {(xn-x*)} and {(bn-an)} converges cubically to 0 simultaneously. The convergence theorems are established for the described method. The method is tested on a number of numerical examples and the results obtained are compared with those obtained by King [R.F. King, A secant method for multiple roots, BIT 17 (1977), pp. 321-328.]. [ABSTRACT FROM AUTHOR]

In this paper, we present and analyse a new predictor-corrector iterative method for solving non-linear single variable equations. The convergence analysis establishes that the new method is cubically convergent. Numerical tests show that the method is comparable with the well-known existing methods and in many cases gives better results. [ABSTRACT FROM AUTHOR]

The extended version with the analysis of dynamic system for Wilkinson's iteration improvement of solution is presented in this paper. It turns out that the iteration improvement can be viewed as applying explicit Euler method with step size h=1 to a dynamic system which has a unique globally asymptotically stable equilibrium point, that is, the solution x*=A-1b of linear system Ax=b with non-singular matrix A. As a result, an extended iterative improvement process for solving ill-conditioned linear system of algebraic equations with non-singular coefficients matrix is proposed by following the solution curve of a linear system of ordinary differential equations. We prove the unconditional convergence and derive the roundoff results for the extended iterative refinement process. Several numerical experiments are given to show the effectiveness and competition of the extended iteration refinement in comparison with Wilkinson's. [ABSTRACT FROM AUTHOR]