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Autor: Alfredo Andrés Valenzuela Riquelme

MODELO DE FUNCIÓN LOGÍSTICA


Ejercicio: especificación de la matriz


-Especificar la matriz g(P_i) , en termino de solamente a, b, c, d o sea:

-Especificar g(P_i) en terminos de f ,\frac{\partial f}{\partial a},\frac{\partial f}{\partial b},\frac{\partial f}{\partial c},\frac{\partial f}{\partial d}

-Calcular derivadas parciales de f .

g(P_i) , en terminos de f,a ,b ,c ,d:

Tenemos; g(p_i)=\left ( 
      \begin{matrix} 
         g_{11}(p_i) &  g_{12}(p_i) &  g_{13}(p_i) &  g_{14}(p_i)\\
         g_{21}(p_i) &  g_{22}(p_i) &  g_{23}(p_i) &  g_{24}(p_i)\\
         g_{31}(p_i) &  g_{32}(p_i) &  g_{33}(p_i) &  g_{34}(p_i)\\
         g_{41}(p_i) &  g_{42}(p_i) &  g_{43}(p_i) &  g_{44}(p_i)
      \end{matrix}
   \right )


      \begin{matrix} 
         g_{11}(p_i)\frac{\partial F_1}{\partial a} &  g_{12}(p_i)\frac{\partial F_1}{\partial b} &  g_{13}(p_i)\frac{\partial F_1}{\partial c} &  g_{14}(p_i)\frac{\partial F_1}{\partial c}\\
         g_{21}(p_i)\frac{\partial F_1}{\partial a} &  g_{22}(p_i)\frac{\partial F_1}{\partial b} &  g_{23}(p_i)\frac{\partial F_1}{\partial c} &  g_{24}(p_i)\frac{\partial F_1}{\partial c}\\
         g_{31}(p_i)\frac{\partial F_1}{\partial a} &  g_{32}(p_i)\frac{\partial F_1}{\partial b} &  g_{33}(p_i)\frac{\partial F_1}{\partial c} &  g_{34}(p_i)\frac{\partial F_1}{\partial c}\\
         g_{41}(p_i)\frac{\partial F_1}{\partial a} &  g_{42}(p_i)\frac{\partial F_1}{\partial b} &  g_{43}(p_i)\frac{\partial F_1}{\partial c} &  g_{44}(p_i)\frac{\partial F_1}{\partial c}
      \end{matrix}

\to calculamos:

(1) Para g_{11}(P_i) ;

g_{11}(P_i) = \frac{\partial F_1}{\partial a}; \to se aplica la definicion de:

F_1=f_1(a,b,c,d):= \frac{\partial S}{\partial a}(a,b,c,d)

\therefore g_{11}(P_i)= \frac{\partial F_1}{\partial a}

=\frac{\partial }{\partial a}\left [  \sum_{i=1}^n  2(f(x_i)-f_i)  \frac{\partial }{\partial a} f(x_i) \right ]

= \sum_{i=1}^n \frac{\partial }{\partial a}\left [2(f(x_i)-f_i)  \frac{\partial }{\partial a} f(x_i) \right ]

=\sum_{i=1}^n \left [2(f(x_i)-f_i)\frac{\partial }{\partial a}\left (\frac{\partial }{\partial a}f(x_i)\right )+\frac{\partial }{\partial a}f(x_i)\frac{\partial }{\partial a} 2(f(x_i)-f_i)\right ]

=\left [2(f(x_i)-f_i)\frac{\partial ^2}{\partial a^2}f(x_i) + 2\left (\frac{\partial }{\partial a}f(x_i)\right )^2\right ]

g_{11}(P_i) =\sum_{i=1}^n 2\left [f(x_i)\frac{\partial ^2}{\partial a^2}f(x_i) + \left (\frac{\partial }{\partial a}f(x_i)\right )^2\right ]


(2) Para g_{12}(P_i) ;

\frac{\partial F_1}{\partial b}=2f(x_i)\frac{\partial }{\partial b}\left (\frac{\partial }{\partial a}f(x_i)\right )+\frac{\partial }{\partial a} f(x_i)\frac{\partial }{\partial b}2f(x_i)

(3) Para g_{13}(P_i) ;

\frac{\partial F_1}{\partial c}=2f(x_i)\frac{\partial }{\partial c}\left (\frac{\partial }{\partial a}f(x_i)\right )+\frac{\partial }{\partial a} f(x_i)\frac{\partial }{\partial c}2f(x_i)

(4) Para g_{14}(P_i) ;

\frac{\partial F_1}{\partial d}=2f(x_i)\frac{\partial }{\partial d}\left (\frac{\partial }{\partial a}f(x_i)\right )+\frac{\partial }{\partial a} f(x_i)\frac{\partial }{\partial d}2f(x_i)

(5) Para g_{21}(P_i) ;

\frac{\partial F_2}{\partial a}=2f(x_i)\frac{\partial }{\partial a}\left (\frac{\partial }{\partial b}f(x_i)\right )+\frac{\partial }{\partial b} f(x_i)\frac{\partial }{\partial a}2f(x_i)

(6) Para g_{22}(P_i) ;

\frac{\partial F_2}{\partial b}=2\left [f(x_i)\frac{\partial ^2}{\partial b^2}f(x_i) + \left (\frac{\partial }{\partial b}f(x_i)\right )^2\right ]

(7) Para g_{23}(P_i) ;

\frac{\partial F_2}{\partial c}=2f(x_i)\frac{\partial }{\partial c}\left (\frac{\partial }{\partial b}f(x_i)\right )+\frac{\partial }{\partial b} f(x_i)\frac{\partial }{\partial c}2f(x_i)

(8) Para g_{24}(P_i) ;

\frac{\partial F_2}{\partial d}=2f(x_i)\frac{\partial }{\partial d}\left (\frac{\partial }{\partial b}f(x_i)\right )+\frac{\partial }{\partial b} f(x_i)\frac{\partial }{\partial d}2f(x_i)

(9) Para g_{31}(P_i) ;

\frac{\partial F_3}{\partial a}=2f(x_i)\frac{\partial }{\partial a}\left (\frac{\partial }{\partial c}f(x_i)\right )+\frac{\partial }{\partial c} f(x_i)\frac{\partial }{\partial a}2f(x_i)

(10) Para g_{32}(P_i) ;

\frac{\partial F_3}{\partial b}=2f(x_i)\frac{\partial }{\partial b}\left (\frac{\partial }{\partial c}f(x_i)\right )+\frac{\partial }{\partial c} f(x_i)\frac{\partial }{\partial b}2f(x_i)

(11) Para g_{33}(P_i) ;

\frac{\partial F_3}{\partial c}=2\left [f(x_i)\frac{\partial ^2}{\partial c^2}f(x_i) + \left (\frac{\partial }{\partial c}f(x_i)\right )^2\right ]

(12) Para g_{34}(P_i) ;

\frac{\partial F_3}{\partial d}=2f(x_i)\frac{\partial }{\partial d}\left (\frac{\partial }{\partial c}f(x_i)\right )+\frac{\partial }{\partial c} f(x_i)\frac{\partial }{\partial d}2f(x_i)

(13) Para g_{41}(P_i) ;

\frac{\partial F_4}{\partial a}=2f(x_i)\frac{\partial }{\partial a}\left (\frac{\partial }{\partial d}f(x_i)\right )+\frac{\partial }{\partial d} f(x_i)\frac{\partial }{\partial a}2f(x_i)

(14) Para g_{42}(P_i) ;

\frac{\partial F_4}{\partial b}=2f(x_i)\frac{\partial }{\partial b}\left (\frac{\partial }{\partial d}f(x_i)\right )+\frac{\partial }{\partial d} f(x_i)\frac{\partial }{\partial b}2f(x_i)

(15) Para g_{43}(P_i) ;

\frac{\partial F_4}{\partial c}=2f(x_i)\frac{\partial }{\partial c}\left (\frac{\partial }{\partial d}f(x_i)\right )+\frac{\partial }{\partial d} f(x_i)\frac{\partial }{\partial c}2f(x_i)

(16) Para g_{44}(P_i) ;

\frac{\partial F_4}{\partial c}=2\left [f(x_i)\frac{\partial ^2}{\partial d^2}f(x_i) + \left (\frac{\partial }{\partial d}f(x_i)\right )^2\right ]

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