FANDOM


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 ] $