Fazlollah Soleymani

A real-time mathematical computer method for potato inspection using machine vision

Detection of external defects on potatoes is the most important technology in the realization of automatic potato sorting stations. This paper presents a hierarchical grading method applied to the potatoes. In this work a potato defect detection combining with size sorting system using the machine vision will be proposed. This work also will focus on the mathematics methods used in automation with a particular emphasis on the issues associated with designing, implementing and using classification algorithms to solve equations. In the first step, a simple size sorting based on mathematical binarization is described, and the second step is to segment the defects; to do this, color based classifiers are used. All the detection standards for this work are referenced from the United States Agriculture Department, and Canadian Food Industries. Results show that we have a high accuracy in both size sorting and classification. Experimental results show that support vector machines have very high accuracy and speed between classifiers for defect detection.

A new Bernoulli matrix method for solving high-order linear and nonlinear Fredholm integro-differential equations with piecewise intervals

Abstract This article develops an efficient direct solver for solving numerically the high-order linear Fredholm integro-differential equations (FIDEs) with piecewise intervals under initial-boundary conditions. A Bernoulli matrix approach is implemented for solving linear and nonlinear FIDEs with piecewise intervals under initial boundary conditions. The main characteristic behind this approach is that it reduces such problem to those of solving a system of algebraic equations. A small number of Bernoulli polynomials is needed to obtain a satisfactory result. Numerical results with comparisons are given to confirm the reliability of the proposed method for solving FIDEs with piecewise intervals.

Two new classes of optimal Jarratt-type fourth-order methods

Abstract In this paper, we investigate the construction of some two-step without memory iterative classes of methods for finding simple roots of nonlinear scalar equations. The classes are built through the approach of weight functions and these obtained classes reach the optimal order four using one function and two first derivative evaluations per full cycle. This shows that our classes can be considered as Jarratt-type schemes. The accuracy of the classes is tested on a number of numerical examples. And eventually, it is observed that our contributions take less number of iterations than the compared existing methods of the same type to find more accurate approximate solutions of the nonlinear equations.

An Improvement of Ostrowski’s and King’s Techniques with Optimal Convergence Order Eight

In this paper, we first establish a new class of three-point methods based on the two-point optimal method of Ostrowski. Analysis of convergence shows that any method of our class arrives at eighth order of convergence by using three evaluations of the function and one evaluation of the first derivative per iteration. Thus, this order agrees with the conjecture of Kung and Traub (J. ACM 643–651, 1974) for constructing multipoint optimal iterations without memory. We second present another optimal eighth-order class based on the King’s fourth-order family and the first attained class. To support the underlying theory developed in this work, we examine some methods of the proposed classes by comparison with some of the existing optimal eighth-order methods in literature. Numerical experience suggests that the new classes would be valuable alternatives for solving nonlinear equations.

A multi-step class of iterative methods for nonlinear systems

In this article, the numerical solution of nonlinear systems using iterative methods are dealt with. Toward this goal, a general class of multi-point iteration methods with various orders is constructed. The error analysis is presented to prove the convergence order. Also, a thorough discussion on the computational complexity of the new iterative methods will be given. The analytical discussion of the paper will finally be upheld through solving some application-oriented problems.

Accurate fourteenth-order methods for solving nonlinear equations

We establish new iterative methods of local order fourteen to approximate the simple roots of nonlinear equations. The considered three-step eighth-order construction can be viewed as a variant of Newton’s method in which the concept of Hermite interpolation is used at the third step to reduce the number of evaluations. This scheme includes three evaluations of the function and one evaluation of the first derivative per iteration, hence its efficiency index is 1.6817. Next, the obtained approximation for the derivative of the Newton’s iteration quotient is again taken into consideration to furnish novel fourteenth-order techniques consuming four function and one first derivative evaluations per iteration. In providing such new fourteenth-order methods, we also take a special heed to the computational burden. The contributed four-step methods have 1.6952 as their efficiency index. Finally, various numerical examples are given to illustrate the accuracy of the developed techniques.

Optimal Steffensen-type methods with eighth order of convergence

This paper proposes two classes of three-step without memory iterations based on the well known second-order method of Steffensen. Per computing step, the methods from the developed classes reach the order of convergence eight using only four evaluations, while they are totally free from derivative evaluation. Hence, they agree with the optimality conjecture of Kung-Traub for providing multi-point iterations without memory. As things develop, numerical examples are employed to support the underlying theory developed for the contributed classes of optimal Steffensen-type eighth-order methods.

On a New Method for Computing the Numerical Solution of Systems of Nonlinear Equations

We consider a system of nonlinear equations . A new iterative method for solving this problem numerically is suggested. The analytical discussions of the method are provided to reveal its sixth order of convergence. A discussion on the efficiency index of the contribution with comparison to the other iterative methods is also given. Finally, numerical tests illustrate the theoretical aspects using the programming package Mathematica.

Finding the solution of nonlinear equations by a class of optimal methods

This paper is devoted to the study of an iterative class for numerically approximating the solution of nonlinear equations. In fact, a general class of iterations using two evaluations of the first order derivative and one evaluation of the function per computing step is presented. It is also proven that the class reaches the fourth-order convergence. Therefore, the novel methods from the class are Jarratt-type iterations, which agree with the optimality hypothesis of Kung-Traub. The derived class is further extended for multiple roots. That is to say, a general optimal quartic class of iterations for multiple roots is contributed, when the multiplicity of the roots is available. Numerical experiments are employed to support the theory developed in this work.

A higher order iterative method for computing the Drazin inverse.

A method with high convergence rate for finding approximate inverses of nonsingular matrices is suggested and established analytically. An extension of the introduced computational scheme to general square matrices is defined. The extended method could be used for finding the Drazin inverse. The application of the scheme on large sparse test matrices alongside the use in preconditioning of linear system of equations will be presented to clarify the contribution of the paper.