Finding a Cluster of Zeros of Univariate Polynomials

A method to compute an accurate approximation for a zero cluster of a complex univariate polynomial is presented. The theoretical background on which this method is based deals with homotopy, Newton's method, and Rouché's theorem. First the homotopy method provides a point close to the zero cluster. Next the analysis of the behaviour of the Newton method in the neighbourhood of a zero cluster gives the number of zeros in this cluster. In this case, it is sufficient to know three points of the Newton sequence in order to generate an open disk susceptible to contain all the zeros of the cluster. Finally, an inclusion test based on a punctual version of the Rouché theorem validates the previous step. A specific implementation of this algorithm is given. Numerical experiments illustrate how this method works and some figures are displayed.     

An adaptive version of a fourth-order iterative method for quadratic equations

A fourth-order iterative method for quadratic equations is presented. A semilocal convergence theorem is performed. A multiresolution transform corresponding to interpolatory technique is used for fast application of the method. In designing this algorithm we apply data compression to the linear and the bilinear forms that appear on the method. Finally, some numerical results are studied.     

An application of multi-objective modeling: The case of the Indian sugar industry

A multi-objective linear programming model is developed for the Indian sugar industry to plan for additional output by production technique, geographical region and forecasted year. Various policy scenarios generated by assigning different values to the policy variables in the model are studied. Thus a useful planning tool which demonstrates the exact impact of the policy parameters on various objectives is provided to the central decision maker. A satisficing multi-objective decision making method is developed based on an existing method of solution and used in policy analysis. The solution method is ideally suited to any general planning problem.     

Algorithm for verifying the equivalence of linear unary recursive programs on ordered semigroup scales

A new method for solving the problem of equivalence of linear unary recursive programs is proposed. The main idea behind this method is to reduce the equivalence problem to known problems on graphs and to problems in group theory. A class of program semantics is singled out for which the problem of the equivalence of the programs in question is solvable in polynomial time by the proposed method.     

A Monte Carlo Method for the Simulation of First Passage Times of Diffusion Processes

A reliable Monte Carlo method for the evaluation of first passage times of diffusion processes through boundaries is proposed. A nested algorithm that simulates the first passage time of a suitable tied-down process is introduced to account for undetected crossings that may occur inside each discretization interval of the stochastic differential equation associated to the diffusion. A detailed analysis of the performances of the algorithm is then carried on both via analytical proofs and by means of some numerical examples. The advantages of the new method with respect to a previously proposed numerical-simulative method for the evaluation of first passage times are discussed. Analytical results on the distribution of tied-down diffusion processes are proved in order to provide a theoretical justification of the Monte Carlo method.     

Reconstruction of boundary regimes in the inverse problem of thermal convection of a high-viscosity fluid

A problem of reconstruction of boundary regimes in a model for free convection of a high-viscosity fluid is considered. A variational method and a quasi-inversion method are suggested for solving the problem in question. The variational method is based on the reduction of the original inverse problem to some equivalent variational minimum problem for an appropriate objective functional and solving this problem by a gradient method. When realizing the gradient method for finding a minimizing element of the objective functional, an iterative process actually reducing the original problem to a series of direct well-posed problems is organized. For the quasi-inversion method, the original differential model is modified by means of introducing special additional differential terms of higher order with small parameters as coefficients. The new perturbed problem is well-posed; this allows one to solve this problem by standard methods. An appropriate choice of small parameters gives an opportunity to obtain acceptable qualitative and quantitative results in solving the inverse problem. A comparison of the methods suggested for solving the inverse problem is made with the use of model examples.     

On the general nonlinear self-adjoint eigenvalue problem for systems of ordinary differential equations with singularities

A nonlinear self-adjoint eigenvalue problem for the general linear system of ordinary differential equations is examined on an unbounded interval. A method is proposed for the approximate reduction of this problem to the corresponding problem on a finite interval. Under the assumption that the initial data are monotone functions of the spectral parameter, a method is given for determining the number of eigenvalues lying on a prescribed interval of this parameter. No direct calculation of eigenvalues is required in this method.     

Numerical solution of some partial differential equations by means of a deterministic method of approximate functional integration

A numerical method of solution of some partial differential equations is presented. The method is based on representation of Green functions of the equations in the form of functional integrals and subsequent approximate calculation of the integrals with the help of a deterministic approach. In this case the solution of the equations is reduced to evaluation of usual (Riemann) integrals of relatively low multiplicity. A procedure allowing one to increase accuracy of the solutions is suggested. The features of the method are investigated on examples of numerical solution of the Schrödinger equation and related diffusion equation.     

On the approximation of derivatives using divided difference operators preserving the local convergence order of iterative methods

A development of an inverse first-order divided difference operator for functions of several variables is presented. Two generalized derivative-free algorithms built up from Ostrowski’s method for solving systems of nonlinear equations are written and analyzed. A direct computation of the local order of convergence for these variants of Ostrowski’s method is given. In order to preserve the local order of convergence, any divided difference operator is not valid. Two counterexamples of computation of a classical divided difference operator without preserving the order are presented. A rigorous study to know a priori if the new method will preserve the order of the original modified method is presented. The conclusion is that this fact does not depend on the method but on the systems of equations and if the associated divided difference verifies a particular condition. A new divided difference operator solving this problem is proposed. Furthermore, a computation that approximates the order of convergence is generated for the examples and it confirms in a numerical way that the order of the methods is well deduced. This study can be applied directly to other Newton’s type methods where derivatives are approximated by divided differences.     

Extended multiscale FEM for design of beams and frames with complex topology

A numerical method for design of beams and frames with complex topology is proposed. The method is based on extended multi-scale finite element method where beam finite elements are used on coarse scale and continuum elements on fine scale. A procedure for calculation of multi-scale base functions, up-scaling and downscaling techniques is proposed by using a modified version of window method that is used in computational homogenization. Coarse scale finite element is embedded into a frame of a material that is representing surrounding structure in a sense of mechanical properties. Results show that this method can capture displacements, shear deformations and local stress-strain gradients with significantly reduced computational time and memory comparing to full scale continuum model. Moreover, this method includes a special hybrid finite elements for precise modelling of structural joints. Hence, the proposed method has a potential application in large scale 2D and 3D structural analysis of non-standard beams and frames where spatial interaction between structural elements is important.     

Parallel FE approximation of the even/odd-parity form of the linear Boltzmann equation

A novel solution method has been developed to solve the linear Boltzmann equation on an unstructured triangular mesh. Instead of tackling the first-order form of the equation, this approach is based on the even/odd-parity form in conjunction with the conventional multigroup discrete-ordinates approximation. The finite element method is used to treat the spatial dependence. The solution method is unique in that the space-direction dependence is solved simultaneously, eliminating the need for the conventional inner iterations, and the method is well suited for massively parallel computers.     

Approximate Solutions of Non-linear Boundary-value Problems

A method for obtaining approximate solutions of non-linear boundaryvalue problems, based on a judicious application of the maximumprinciple, is described. After preliminaries, the method isapplied to generate pointwise bounding approximate solutionsof the Liouville equation. A comparison is also made with themethod based on the existence of complementary variational principles.     

A series method for solving nonlinear two-point boundary value problems

Summary A new method for solving nonlinear boundary value problems based on Taylor-type expansions generated by the use of Lie series is derived and applied to a set of test examples. A detailed discussion is given of the comparative performance of this method under various conditions. The method is of theoretical interest but is not applicable, in its present form, to real life problems; in particular, because of the algebraic complexity of the expressions involved, only scalar second order equations have been discussed, though in principle systems of equations could be similarly treated. A continuation procedure based on this method is suggested for future investigation.     

The numerical solution of non-linear partial differential equations by the method of lines

A new method for solving non-linear parabolic partial differential equations, based on the method of lines, is developed. Second order and fourth order finite difference approximations to the spatial derivatives are used, and in each case the band structure of the associated Jacobian is exploited. This, together with the use of Gear's fourth order stiffly stable method in Nordsieck form, leads to a method which compares favourably with the respected Sincovec and Madsen method on Burgers' equation. The method has been tested on a number of difficult problems in the literature and has proved to be most successful.     

Comparison of the flow of viscoelastic media in shear and in tension

A geometrical method of comparing steady-state flow in uniaxial tension and in simple shear is proposed. This method is based on a comparison of the deformation regimes relative to the principal axes.     

Application of the dual active set algorithm to quadratic network optimization

A new algorithm, the dual active set algorithm, is presented for solving a minimization problem with equality constraints and bounds on the variables. The algorithm identifies the active bound constraints by maximizing an unconstrained dual function in a finite number of iterations. Convergence of the method is established, and it is applied to convex quadratic programming. In its implementable form, the algorithm is combined with the proximal point method. A computational study of large-scale quadratic network problems compares the algorithm to a coordinate ascent method and to conjugate gradient methods for the dual problem. This study shows that combining the new algorithm with the nonlinear conjugate gradient method is particularly effective on difficult network problems from the literature.     

Application of multi-attribute utility theory to measure social preferences for health states

A four-attribute health state classification system designed to uniquely categorize the health status of all individuals two years of age and over is presented. A social preference function defined over the health state classification system is required. Standard multi-attribute utility theory is investigated for the task, problems are identified and modifications to the standard method are proposed. The modified methods is field tested in a survey research project involving 112 home interviews. Results are presented and discussed in detail for both the social preference function and the performance of the modified method. A recommended social preference function is presented, complete with a range of uncertainty. The modified method is found to be applicable to the task--no insurmountable difficulties are encountered. Recommendations are presented, based on our experience, for other investigators who may be interested in reapplying the method in other studies.     

Decomposition Method with a Variable Parameter for a Class of Monotone Variational Inequality Problems

In this paper, we focus on a useful modification of the decomposition method by He et al. (Ref. 1). Experience on applications has shown that the number of iterations of the original method depends significantly on the penalty parameter. The main contribution of our method is that we allow the penalty parameter to vary automatically according to some self-adaptive rules. As our numerical simulations indicate, the modified method is more flexible and efficient in practice. A detailed convergence analysis of our method is also included.