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https://doi.org/10.37815/rte.v33n3.799

Artículos originales=

 

Aplicación del método Newton-Raphson por polinomio cúbicos usando el método Tartaglia-Cardano<= span style=3D'font-size:10.0pt'>

A= pplication of the Newton-Raphson method by c= ubic polynomial using the Tartaglia-Cardano method

 

Pedro Fabricio Echeverría Briones1 h= ttps://orcid.org/0000-0002-4921-1111

 

1Universidad Tecnológica Empresarial = de Guayaquil, Guayaquil, Ecuador

coordinadoracademicoposgrado03@uteg.edu.e= c, pecheverr@gmail.com

 

Enviado:         2021/01/12

Aceptado:       2021/10/18

Publicado:      2021/12/30                         

Resumen

Algunos de los métodos usados para encontrar raíces de funciones en las matemáticas aplicadas son = los algoritmos interactivos, que constituyen un campo de interés muy importante dentro de la disciplina debido a que permiten encontrar soluciones para usa= r la menor capacidad de recursos dentro de los sistemas computacionales. La propuesta de estos algoritmos iterativos implica la relación de diferentes áreas de las matemáticas: álgebra, cálculo diferencial, cálculo vectorial y análisis numérico. Esta relación conjuga la teoría de cálculo diferencial y= el análisis numérico realizado por matemáticos ingleses, como Newton y Raphson. Estos propusieron la primera forma de buscar una raíz de una función usando iteraciones, la que a su vez se sustentó en teorías algebraicas desarrollad= as por algunos matemáticos italianos, como Tartaglia y Cardano, quienes anteriormente habían encontrado las soluciones a las raíces de un polinomio= de orden cúbico. Dentro del proceso deductivo que se siguió en el proceso, se utilizaron las aproximaciones recomendadas que se generaron al aplicar las Series de Taylor. Por otro lado, durante el proceso de experimentación se h= an identificado determinados patrones repetitivos, especialmente, cuando el pu= nto de inicio se aleja de la raíz. Esto permitió mejorar el algoritmo iterativo= al identificar el lugar de la curva de convergencia.  Finalmente, se generalizó la ecuación iterativa para muchas variables, a partir de las deducciones utilizadas de = 1 y 2 variables.

 

Palabras clave: iteración, raíz, cero de función, convergenc= ia, cúbico.

3D"Cuadro

Abstract

Some of the methods employed to find roots of functions in applied mathematics are interactive algorithms, which is a very important field of interest within the discipline because they allow finding solutions using t= he least number of resources within computational systems. The proposal of the= se iterative algorithms involves the relationship of different areas of mathematics being algebra, differential calculus, vector calculus, and numerical analysis. This relationship combines the theories of differential calculus and numerical analysis carried out by English mathematicians, such= as Newton and Raphson. They proposed the first way to find a root of a function using iterations with the application of algebraic theories of Italian mathematicians, Tartaglia and Cardano, who had previously found the solutions to the roots of a cubic order polynomial. Al= ong the deductive process, the recommended approximations were followed by using the Taylor series. On the other hand, the experimentation process determined repetitive patterns, especially when the starting point is far from the roo= t. This allowed the improvement of the iterative algorithm by identifying the location of the convergence curve. Finally, the iterative equation was generalized for many variables using the deductions of 1 and 2 variables. <= /span>

 

Keywords: iteration, root, function zero, convergen= ce, cubic.

 

Introducción<= o:p>

El méto= do de Newton-Raphson utiliza demostraciones algebraicas, cálculo diferencial, cál= culo vectorial y análisis numérico para la aproximación de raíces. Cabe señalar = que este método contiene errores de orden cuadrático. Por otro lado, para determinar las raíces de un polinomio de orden cúbico, se utiliza el método Tartaglia-Cardano. Esta es la razón por la que el algoritmo iterativo que se deduce tiene ventajas, pues, al encontrar la raíz de una función usando la = raíz del polinomio cúbico, que se aproxima por series de Taylor, se obtiene un e= rror de orden cuartico. Esto hace que la convergencia sea más segura, a pesar de= que se usan puntos de inicio muy alejados con respecto al valor de la raíz. La forma de programación utilizada es sencilla porque el proceso de experimentación simplemente utiliza una hoja de cálculo. Algunas restriccio= nes del modelo son que la función tenga una tercera derivada continua diferente= de cero y que existan ciertos intervalos en donde se pueda tener un valor inic= ial.

 =

Desarrollo

Definición de= una función de una variable  en serie de Taylor

Para formular = una función f(x), que es m veces derivable, en polinomios por serie de Taylor, = se utilizó la ecuación (1) (Leithold, 1998; Burden, Faires, y Burden, 2017):

 

=

<= span style=3D'mso-bookmark:_Hlk90128094'> 

          (1)

Al llevar la s= erie de Taylor hasta un polinomio cúbico se tiene:

 

=

          (2)

 

Obteniendo:

=

          (3)

 

Definición de una función de dos variables   en serie de Taylor

Para formular = una función la cual puede ser m veces derivable, se utiliza= su descomposición en polinomios por serie de Taylor (Coll= ey, 2012; Conte, 1980):

 

=

          (4)

 

Resolución algebraica de Tartaglia-Cardano para polinomios cúbicos

Para el polino= mio de orden cúbico, se debe llevar al modelo de polinomio hacia donde este satisf= ace a, b y c (Ivorra, 2020):

&nb= sp;

=

    (5)

 

Donde una de l= as raíces reales cumplirá con:

 

=

    (= 6)

 

Siendo:

 

=

    (= 7)

 

=

(8)

 

=

      (9)

 

Convergencia del algoritmo iterativo usando el polinomio de orden cúbico para funciones = de una variable

Al combinar la= serie de Taylor de la ecuación (3) con la resolución de Newton-Raphson  =   (Burd= en, Faires, & Burden, 2017), se obtiene:

&nb= sp;

0 =3Df(xn)+= f'<= /i>= xn1!(<= /m:r>xn+1-xn)+= f''= = xn2!(xn<= /i>+1= -xn<= /i>)2+= f'''= xn3!(xn<= /i>+1= -xn<= /i>)3= = .

        (10)

 

De la ecuación= (10) se tiene que pasar al formato de la ecuación (5), dividiendo para    todo= s los términos del polinomio.

 

=

     = (11)

<= span style=3D'mso-bookmark:_Hlk90128094'> 

<= span style=3D'mso-bookmark:_Hlk90128094'>          (2)

 

Encontrando los valores de a, b y c:

 

=

   (1= 2)

 

=

     = (13)

 

=

    (= 14)

 

Determinando l= os valores de las ecuaciones (7), (8) y (9):

 

=

<= span style=3D'mso-bookmark:_Hlk90128094'> 

     = (15)

 

=

   (1= 6)

 

 

=

     = (17)

 

Al aplicar los valores de las ecuaciones (15), (16) y (17) en la ecuación (5), se determin= a:

 

=

  (18= )

 

Al despejar  se det= ermina el algoritmo iterativo:

 

=

   (1= 9)

 

Convergencia del algoritmo iterativo usando el polinomio de orden cúbico para funciones = de dos variables

Aquí se usa el modelo de Newton-Raphson con el método de Tartaglia-Cardano para una funció= n de dos variables. Para varias variables, se tiene que utilizar la serie de Tay= lor, en donde se realiza una reducción de términos, asumiendo que =  cuando  excepto cuando los índices de las sumato= rias sean iguales. Reduciendo la ecuación (4):

 

=

          (20)

<= span style=3D'mso-bookmark:_Hlk90128094'> 

<= span style=3D'mso-bookmark:_Hlk90128094'>          (2)

 

Donde: j=3D{1,2}, se obtienen los siguientes valores de la ecu= ación (5) :

 

=

   (2= 1)

 

=

    (= 22)

 

Donde: j=3D{1,2}, se obtienen los siguientes valores de las ec= uaciones (6), (7) y (8) :

 

=

   (2= 3)

 

=

   (2= 4)

 

=

  (25= )

 

Al aplicar los valores de las ecuaciones (23), (24) y (25) en la ecuación (5), se determin= a:

 

=

      (26)

 

Al despejar , se determina el algoritmo iterativo para una función de dos variables:

 

=

    (= 27)

&n= bsp;

Análisis de resultados

Validación de la convergencia del algo= ritmo iterativo para funciones de una variable

A continuación, en los siguientes gráficos, se presen= tan los resultados que se obtuvieron de las pruebas que se realizaron con diferentes funciones, cuyas condiciones fueron que estas tengan la tercera derivada diferente de cero y en diferentes puntos de inicio:

 =

Figura 1 en escala logarítmica= por número de iteración para  con diferentes puntos= de inicios alejados de la raíz

 

3D"Gráfico,

 

Figura 2 en escala logarítmica por número de iter= ación para con diferentes puntos de inicios alejado= s de la raíz

 

3D"Gráfico,

 

Figura 3 en escala logarítmica= por número de iteración para con diferentes puntos= de inicios alejados de la raíz

3D"Gráfico,

 

Figura 4 en escala logarítmica= por número de iteración para<= ![if !msEquation]> con diferentes puntos= de inicios alejados de la raíz

3D"Gráfico

Descripción

 

Figura <= /span>5<= /span>

Curv= as de Convergencia de  en escala logarítmica= por número de iteración paracon diferentes puntos de inicios alejados de la raíz

 

3D"Gráfico,

 

A partir de las representaciones gráficas de esta experimentación, se puede deducir que el algoritmo iterativo tiene dos form= as de convergencia. Este patrón se puede observarse visualmente cuando se alej= a el valor de inicio de la raíz. En resumen, se puede concluir: 

 

1.=      Al o= bservar las curvas de convergencia, se verifica la formación de una línea recta decreciente entre . Es= te proceso es lento. Lo que llama la atención es que para diferentes valores d= el punto inicial se obtienen rectas que son casi paralelas.<= /p>

2.     Cuando la curva se acerca a la raíz, su convergencia se acelera cambiando la pendiente de la curva rápidamente por = cada iteración. 

 

Esta es la razón por la que se puede determinar una aceleración al diferenciar cuando esta se encuentra entre los dos patrones.  A continuación, se prese= ntan el análisis de los siguientes casos:

 

a)     Si los , el= patrón se encontraría en los siguientes gráficos:

 

Figura 6 en escala logarítmica= por número de iteración para

3D"Gráfico,

 

Figura 7 por número de iteraci= ón

3D"Imagen

 

 

Para acelerar la convergencia, se predice= el valor de la distancia del valor de inicio con respecto a la raíz. Si se uti= liza la expresión de la recta por número de iteración n:

 

= =

    (28)

 

= =

    (29)

 

= =

    (30)

 

Obtenemos la pendiente de la recta m:

 

= =

    (31)

 

= =

    (32)

 

Obtenemos la intersección de la recta b:

= =

    (33)

 

= =

    (34)

 

Si se establece que y=3D0, se obtiene el número de iteraciones n:

 

= =

    (35)

 

= =

    (36)

 

= =

    (37)

 

Si se convierte n en entero:

 

= =

  (38)

 

De manera que se determina la distancia entre el valor inicial y la raíz al obtener

 

a)=      Si l= os , se encontraría el patrón en los siguientes gráficos.

 

Figura 8 en escala logarítmica= por número de iteración para

3D"Gráfico,

 

Figura 9 por número de iteraci= ón

3D"Interfaz

 

 

Si este patrón se repite, el primer proceso de convergencia puede ser acelerado al considerar el patrón de las iteraciones= , lo que, a su vez, puede determinar un procedimiento para calcular la distancia entre el valor inicial y la raíz.  = Si para esto se utiliza la ecuación de la recta por iteración n:

 

= =

    (39)

 

= =

    (40)

 

= =

    (41)

 

Obtenemos la pendiente de la recta m:

 

= =

   (42)

 

= =

    (43)

 

Obtenemos la intersección de la recta b:

 

= =

    (44)

 

= =

   (45)

 

=

   (46)

 

Por lo que el modelo sería:

 

=

  (47)

 

Si se establece que y=3D0, se tendría a n:=

 

=

   (48)

 

=

    (49)

 

=

    (50)

 

El valor de n debe ser entero y estar antes de la zon= a de convergencia, por lo que se le restará 3 iteraciones.  Obteniendo:

 

=

    (51)

 

De manera que se pueda determinar la distancia entre = el valor inicial y la raíz al obtener . Es= to mejora el algoritmo iterativo de la ecuación (27) y permite obtener la expresión:

 

= =

    (52)

 

      =           Si , entonces   <= /span>

      =           Si , entonces   <= /span>

 

Validación de la convergencia del algo= ritmo iterativo utilizando un predictor de distancia entre el punto inicial y la = raíz para funciones de una variable

Considerando los casos anteriores: =

 

a)=      Si <= /span> en la función , al= aplicar el coeficiente de aceleración, se obtienen menos iteraciones.

 <= /span>

Figura 10 en escala logarítmica= por número de iteración para  con diferentes punto= s de inicios alejados de la raíz

 

3D"Imagen

 

a)     Si  en la función  , al aplicar el coeficiente de aceleraci= ón, se obtienen menos iteraciones.

 <= /span>

Figura 11 en escala logarítmica= por número de iteración para   con diferentes punt= os de inicios alejados de la raíz

 

3D"Gráfico,

Obsérvese cómo, sin importar la distancia del punto de inicio, se disminuyen las iteraciones con respecto a la Figura 5.

 

Validación de la convergencia del algo= ritmo iterativo para funciones de dos variables

Para se repiten las curvas de convergenc= ia (ver Figura 12).

 

Figura <= /span>12=

Curv= as de Convergencia de  en escala logarítmica= por número de iteración para   con diferentes punt= os de inicios alejados de la raíz

 

3D"Gráfico,

 

Conclusiones

Después de las aplicaciones, deducciones algebraicas y aplicaciones de funciones que ayudan a reducir el número de pasos, se puede obtener una generalización del algoritmo iterativo para algunas variables: =

 

=     (53)

Donde j determina las variables y n el número de iteraciones.

 

El método de uso de resolución de una raíz cúbica es un proceso que converge mientras se cumplan las condiciones de los valores , y esto se obtiene si los valores de j  n≥0= . Esto ayuda a definir intervalos de valores en donde no se puede inici= ar.

 

Utilizar la memoria de los valores de las iteraciones, ayuda a definir modelos matemáticos que permitan acelerar ecuaciones iterativas.

 

Las funciones encontradas en las curvas de convergencias podrían convertirse en un patrón que determine ecuaciones iterativas para la búsqueda de una raíz en diferentes funciones.=

 

Referencias

Leitho= ld, L. (1998). El Cálculo. México: Oxf= ord University Press.

Artega= , J., & Malakhaltsev, M. (2013). Cálc= ulo Vectorial. Mexico: Cengage Learning Editores S.A.

Burden= , R., Faires, D., & Burden, A. (2017). Análisis numérico. México: Cengage Learning.

Colley, S. (2012). Vector Calculus. New York: Pearson.

Conte, S. D. (1980). Elementary numerical analysis: algorithmic approach. New York: McGraw-Hill.

Marsden, J., & Tromba, A. (1991). Cálculo vectorial. México DF: Addison-Wesley Iberoamericana.

Taylor, A. E., & Mann, W. R. (1983). Advanced Calculus. New York: Wiley= .

Weintraub, S. (1996). Differential forms, a complement to vector calculus. New Yo= rk: Academic Press.

Ivorra= , C. (2020, 10 01). Las fórmulas de Cardano-Ferrari. Retri= eved from Universidad de Valencia: https://www.uv.es/ivorra/Libros/Ecuaciones.pd= f

 

=  

 

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