Viewing a response to: @xeliram/ecuaciones-diofanticas-lineal. spanish · @ · 58 days ago. @xeliram, go and place your daily vote for Steem on . Optimización por Enjambre de Partículas Discreto en la Solución Numérica de un Sistema de Ecuaciones Diofánticas Lineales. Iván Amaya a, Luis Gómez b. Ecuaciones diofánticas lineales, soluciones cor- tas, algoritmo de reducción de la base. 1. Introduction. One can solve the linear Diophantine equation. aT x = b.

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February 25th, Received in revised form: April 3th, Abstract This article proposes the use of a discrete version of the well known Particle Swarm Optimization, DPSO, a metaheuristic optimization algorithm for numerically solving a system of linear Diophantine equations. Likewise, the transformation of this type of problem i. The current algorithm is able to find all the integer roots in a given search domain, at least for the examples shown. Simple problems are used to show its efficacy.

Moreover, aspects related to the processing time, as well as to the effect of increasing the population and the search space, are discussed.

It was found that the strategy shown ecuaciknes represents a good approach when dealing with systems that have more unknowns than equations, or ceuaciones it becomes of considerable size, since a big search domain is required. Linear Diophantine equations; objective function; optimization; particle swarm.

Se utilizan algunos problemas sencillos para verificar su eficacia. Introduction With each passing day is easier to linealds the boom that the modeling and description of systems have generated in science and engineering, especially through Diophantine equations. Areas such as cryptography, integer factorization, number theory, algebraic geometry, control theory, data dependence on supercomputers, communications, and so on, are some examples [1].

Moreover, there is a strong mathematical foundation for this type of equations and their solutions both, at a fundamental and at an applied level.

Ecuadiones vary from the fanciest and most systematic approaches, up to the most recursive ones, but it is evident that there is no unified solution process, nor a single alternative for doing so. Furthermore, some equations may have a single solution, while others may have an infinite number, or, possibly, may not even have a solution in the integer or rational domains. This also applies for linear systems with this kind of equations i.

Matiyasevich, during the early 90s, proved that it was not possible to have an analytic algorithm that allows to foresee if a given Diophantine equation has, an integer solution, or not [3]. This problem may have been one of the engines that have boosted the search for numerical alternatives.

In order to solve a system of linear Diophantine equations, a diofantivas elimination method which is quite similar to Gauss’s is a good approach for small systems, but it becomes demanding for bigger ones. The specialized literature report some methods like those based on the The authors; licensee Universidad Nacional de Colombia. Some authors have previously proposed the solution of a Diophantine equation through artificial intelligence algorithms [6], [7].


This article proposes to solve, in case the solution exists in the given search domain, a linear system of Diophantine equations. Initially, some basic and necessary related concepts are laid out, and then the viability of using the numeric strategy is shown through some examples.

Fundamentals A linear Diophantine diofantocas, with n unknowns, is defined by eq. It is said that the integers t 1, t n are a solution for eq. Thus, the problem of determining whether a linear Diophantine equation has a solution or not, is reduced to showing if the greatest common divisor of the a i coefficients divide b or not.

Consider the case of two unknowns, for example, with an equation as the one shown by eq. Even so, the problem now transforms in finding a particular solution, which can be done using the 3 following method. Let X be a non-empty subset of R n and consider eq. X R is a function. X R be defined by: Therefore, a is a solution for eq.


An immediate consequence of the previous theorem is that if eq. If the function g defined in 5 has a global minimum in X and this value is zero, then eq.

Moreover, all global minimizers of g are solutions of ecuacionws. Thus, the choice siofanticas the region is quite important to enclose, lnieales least, a solution with integer coordinates System of linear equations Consider the following system of m linear Diophantine equations, with unknowns x 1, x n. Then, and in the same way that with systems of equations in real variables, the fact that one of the equations of a system has a solution, does not imply that the whole system also has.

Even so, a method that generalizes finding all eciaciones roots 7 Suppose that the system of equations 8 has a solution in X, and let a X. Then, a is a solution of linealse system 8 if, and only if, a minimizes the function g diofanricas in 9. The general condition of the theorem 4 about the feasibility of solving the system 8 is important, since it is possible that the function g defined in 9 can be globally minimized but that the system 8 does not have a solution.

An immediate consequence of theorem 4 is that if the system 8 has a solution in X, then the global minimum of g defined in 9 exists and it is zero; moreover, the following result exists: If the function g defined in 9 has a global minimum in X and this value is zero, then the system 8 has a solution in X.

Moreover, all global minimizers of g are solutions of the system 8. Therefore, for the function g defined in 9if there does not exist a global minimum in X or if it exists but is different from zero, then the system of equations 8 does not have a solution in X.

A basic result of the mathematical analysis of the algorithm establishes that if X is a compact set i. Now, for g to be continuous ecuxciones X it is enough that each f i is continuous in X.

For the case of systems of Diophantine equations, unlike the particular case of an equation with two unknowns, the fact that linwales solution exists does not imply that others do, and even less that an infinite number exists. For the search of possible solutions of a system of Diophantine equations, it must hold that the set X have points with integer coordinates, i. A first stage linealez given by the random assignation of a swarm of user defined integers.

Any size can be used here. Likewise, the definition of these values is subject to previous knowledge of the objective function fitnessas well as to the presence of restrictions. Moreover, an initial speed of zero can be defined for the diofatnicas. After that, the algorithm evaluates, in the given search space, the objective function.

With it, local and global best values are established, and both, speed and position, of each particle, are reevaluated as shown below. This procedure is iterative and is repeated until the convergence criteria are met, or until all solutions in the search domain are found. An algorithm, considered as a variant of the traditional PSO, was used, [9]. In the same fashion as said PSO, its version for discrete solutions includes two vectors X i and V i, related to the position and speed of each particle, for every iteration.

The first one is a vector of random numbers, initially, in a valid solution interval. The second one can also be a random libeales, but it can be assumed as zero for the first iteration, in order to keep it simple. When the problems become multidimensional, the vectors transform into a position and a speed matrices, since there is a value for dkofanticas unknown, [9], [11]. Discrete PSO differs from its traditional version in which the new speed and position depend on both, an equation and a ecuacionfs rule, which chooses among the local and global best values for the ecuacoines iteration.

Thus, position update is done according to eq. Results and Analysis This section shows the results achieved after solving some systems of linear Diophantine equations, as an example of the method.

Ecuaciones diofánticas lineales. Tabla con algoritmo de Euclides.

During all the examples, the following parameters were used: These values were chosen based on some preliminary tests and on the information available in the literature [1], [9] System of equations A It is required to solve the system given by eq. The full statement of the problem is as follows: In order to solve this problem with the discrete PSO algorithm, the following objective function is created: Their duration, however, varied from s, ecuaciohes iterations, and up to s, with iterations.


It can then be concluded that, for this system, the algorithm delivers an answer with excellent precision and accuracy, even though the number of iterations and the duration were variable. Convergence time as a function of iterations for system B.

Ecuaciones Diofánticas by Otto Mauricio Sajchè on Prezi

In order to solve it through the algorithm, the following objective function was defined: As a result, the same answer is achieved, so it is important to remark the excellent quality of the results in terms of accuracy and precisionas well as, the variability in time and iterations, ecucaiones looking for all the solutions in the integer domain.

When compared to the previous system, it can be seen that the convergence time increased, and an almost linear relation between iterations and time can be seen in Fig System of equations C For this case a system of 12 linear Diophantine equations was selected: A search space evuaciones and 10 was defined, and particles were used.

On the same computer, an excellent quality answer in terms of accuracy and precision was found, but it required an average time of s around 36 hours and iterations. It is worth mentioning that it was not possible to find these roots by using diiofanticas software nor through traditional means. However, in this case they do not have a solution in the set of integers, e. Convergence time as a function of the search domain. It was also observed that if a system, e. Conclusions 17 This research proved that it is possible to numerically solve a system of linear Diophantine equations through an optimization algorithm.

Also, it was observed that it is possible to solve this optimization problem without using conventional approaches. It was shown, through some simple examples, that, at least for these systems, solutions with high precision and accuracy are achieved.

Moreover, it was found that the convergence time and the number of iterations are random variables that mainly depend on factors such as the algorithm parameters, the initial swarm and the size of the system. Obviously, when solving a squared, small system, traditional approaches, including the ones found in most of the commercial mathematical software, are far quicker, even those that find all the roots of the system.

However, in case that it is required to solve a system with more unknowns than equations, a typical situation, they are out of the question. Likewise, if the system is of a considerable size, the convergence time drastically increases, since a big search domain is required a case found during the current researchso the numerical strategy proposed here gains importance as a possible solution alternative. References [1] Abraham, S. Time-varying feedback systems design via Diophantine equation order reduction, thesis Ph.

Tools and Diophantine Equations and Vol. Analytic and Modern Tools. Structures and Strategies for Complex Problem Solving. Addison-Wesley, Boston, [7] Abraham, S. His research interests include global optimization and microwaves. His research interests include global optimization and Diophantine equations.

His research interests include microwave heating, global optimisation, heat transfer and polymers. In this note we show. The set of integers is Z. It contains all integral numbers from negative infinity to positive infinity. Modular Arithmetic, Congruence, and Matrices Objectives This chapter is intended to prepare the reader for the next few chapters in cryptography.