### METODE NUMERICE MATEMATICE APLICABILE ÎN PROBLEME ALE ECONOMIEI AGRARE ŞI MANAGEMENTULUI AGRICOL

#### Abstract

The mathematical numerical methods can consistently serve the economical and managerial phenomena, including those dedicated to zootechnical and agricultural areas.

The paper fully turns to good account a series of numerical methods implemented on the computer toward the author, using it in the implementation of improvements of classical mathematical methods. The author has built his own algorithms, improved in terms of convergence of the implemented numerical methods. The mathematical numerical methods whose implementation is to be used in this paper refers to solving non-linear equations, non-algebraic (transcendent) equations, and also to solving non-linear systems of equations.

Enough to mention the connection between the size of economical activity (production capacity, capital, etc.) on the one hand, and economical performance (unit cost, unit of income or profit, etc.) on the other hand. There are corresponding phenomena, mathematically, through a nonlinear dependence – parabolic, exponential, hyperbolic one. Mathematically treatment of these phenomena is done through specific functions, such as the Gompertz function: (exponential), then the logistic function: (of exponential and hyperbolic kind).

In problems of technical performance optimization of a production process, with or without financial restrictions, with one or more inputs, the rate of technical substitution, and the rate of economic substitution, both of these rates, are obtained by solving a system of differential equations with partial derivatives. Numerical mathematical methods meant to solve these differential systems categories, involve solving the equations and systems of equations, both linear and non linear.

We also point out the Cobb-Douglass nonlinear equation focused on throughout almost all the paper, through which the author wants to shape up economical issues in the field of animal husbandry and agriculture.

Unlike algebraic equations and linear systems, where the solving methods are direct, the non-linear equations and systems do not allow resolution through direct methods. Here, the solving methods are iterative, but even if it offers approximate solutions, they are sufficiently precise. We also should say that, if in solving non-linear equations, the insurance of the convergence is a simple issue, the solving of the non-linear systems the convergence conditions are more severe. Here, the author, after a detailed study of the issue, starts the implementation with an initial approximation closely related to the exactly solution (mathematical solution). Otherwise, the majority of the methods can follow the unwanted way, the divergence one, due to the rather weak global convergence.

Regarding the used methods in the implementation of solving non-linear equations, besides the classical methods (the chord method, the tangent method, etc.), the author also followed more convergent methods, such as Bailey's Method, the Lagrange or Rational Interpolation Method, Jarrat’s Method, Wegstein’s Method, the method of Steffensen and Aiken’s one, as well as his own generical method.

Regarding the methods used in the implementation, concerning the solving of the non-linear systems of equations, we mention the classical unperformant Gauss-Seidel Method, but mostly the Gauss-Newton-Raphson Method, adapted by the author.

We must mention that the author is the owner of the computer implementation in an acknowledged programming environment, of all the methods listed above, and more. The implementation has been organized in an Integrated environment for numerical methods in mathematics and statistics. The mentioned integrated environment has been the object of a scientific research contract funded by the former Minister of Science and Technology (MCT), the contract drawn up between the parties: ITC (Institute of Technical and Scientifical Computer) Cluj-Napoca, and the mentioned ministry.

From all the implemented numerical methods in the integrated environment, in order to serve the economical and managerial phenomena of Animal Husbandry and Agriculture, the paper appeals to the non-linear equations and the non-linear systems of equations.

The paper fully turns to good account a series of numerical methods implemented on the computer toward the author, using it in the implementation of improvements of classical mathematical methods. The author has built his own algorithms, improved in terms of convergence of the implemented numerical methods. The mathematical numerical methods whose implementation is to be used in this paper refers to solving non-linear equations, non-algebraic (transcendent) equations, and also to solving non-linear systems of equations.

Enough to mention the connection between the size of economical activity (production capacity, capital, etc.) on the one hand, and economical performance (unit cost, unit of income or profit, etc.) on the other hand. There are corresponding phenomena, mathematically, through a nonlinear dependence – parabolic, exponential, hyperbolic one. Mathematically treatment of these phenomena is done through specific functions, such as the Gompertz function: (exponential), then the logistic function: (of exponential and hyperbolic kind).

In problems of technical performance optimization of a production process, with or without financial restrictions, with one or more inputs, the rate of technical substitution, and the rate of economic substitution, both of these rates, are obtained by solving a system of differential equations with partial derivatives. Numerical mathematical methods meant to solve these differential systems categories, involve solving the equations and systems of equations, both linear and non linear.

We also point out the Cobb-Douglass nonlinear equation focused on throughout almost all the paper, through which the author wants to shape up economical issues in the field of animal husbandry and agriculture.

Unlike algebraic equations and linear systems, where the solving methods are direct, the non-linear equations and systems do not allow resolution through direct methods. Here, the solving methods are iterative, but even if it offers approximate solutions, they are sufficiently precise. We also should say that, if in solving non-linear equations, the insurance of the convergence is a simple issue, the solving of the non-linear systems the convergence conditions are more severe. Here, the author, after a detailed study of the issue, starts the implementation with an initial approximation closely related to the exactly solution (mathematical solution). Otherwise, the majority of the methods can follow the unwanted way, the divergence one, due to the rather weak global convergence.

Regarding the used methods in the implementation of solving non-linear equations, besides the classical methods (the chord method, the tangent method, etc.), the author also followed more convergent methods, such as Bailey's Method, the Lagrange or Rational Interpolation Method, Jarrat’s Method, Wegstein’s Method, the method of Steffensen and Aiken’s one, as well as his own generical method.

Regarding the methods used in the implementation, concerning the solving of the non-linear systems of equations, we mention the classical unperformant Gauss-Seidel Method, but mostly the Gauss-Newton-Raphson Method, adapted by the author.

We must mention that the author is the owner of the computer implementation in an acknowledged programming environment, of all the methods listed above, and more. The implementation has been organized in an Integrated environment for numerical methods in mathematics and statistics. The mentioned integrated environment has been the object of a scientific research contract funded by the former Minister of Science and Technology (MCT), the contract drawn up between the parties: ITC (Institute of Technical and Scientifical Computer) Cluj-Napoca, and the mentioned ministry.

From all the implemented numerical methods in the integrated environment, in order to serve the economical and managerial phenomena of Animal Husbandry and Agriculture, the paper appeals to the non-linear equations and the non-linear systems of equations.

#### Full Text:

PDFDOI: http://dx.doi.org/10.15835/arspa.v67i3-4.2989

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