Métodos Numéricos

f(x) = 1/n (x+4)(x+1)(x-3)+ 1/3
Donde n = 19

1° Raiz: Newton-Raphson

Grafica:

Funcion utilizada: (1/19).*(x+4).*(x+1).*(x-3)+(1/3)
Funcion derivada: (1/19).*((3.*x.^2)+(4.*x)-11)
Valor inicial de X: -5
Vector X: -5:0.01:5

Corrida:

Ingresa la funcion f(x): (1/19).*(x+4).*(x+1).*(x-3)+(1/3)
Ingresa la funcion f(x) derivada: (1/19).*((3.*x.^2)+(4.*x)-11)
Ingresa el vector x:-5:0.01:5
Ingrese el limite inferior x: -5

1 -5.000000000 -1.350877193 2.315789474 -4.416666667
2 -4.416666667 -0.222374513 1.571271930 -4.275141506
3 -4.275141506 -0.011710303 1.406838877 -4.266817665
4 -4.266817665 -0.000039446 1.397364647 -4.266789436
5 -4.266789436 -0.000000000 1.397332554 -4.266789435
>> x

x =

-4.266789435243204

>> y =(1/19).*(x+4).*(x+1).*(x-3)+(1/3)

y =

-4.996003610813204e-016

>> hold on, plot(x, y, '*r')

N | Xa | F(x) | F(d) | X |
1 | -5.000000000 | -1.350877193 | 2.315789474 | -4.416666667 |
2 | -4.416666667 | -0.222374513 | 1.571271930 | -4.275141506 |
3 | -4.275141506 | -0.011710303 | 1.406838877 | -4.266817665 |
4 | -4.266817665 | -0.000039446 | 1.397364647 | -4.266789436 |
5 |-4.266789436 | -0.000000000 | 1.397332554 | -4.266789435 |

La raiz se encuentra en:

X | Y |
-4.266789435243204 | -4.996003610813204e-016 |

Se comprueba sustituyendo el valor de X en la ecuación original, obteniendo:
-4.996003610813204e-016, el cual es un valor muy pequeño aproximado a cero, después se grafica y se obtiene:

El código utilizado fue:

clc, clear
funx =input('Ingresa la funcion f(x): ','s');
funder = input('Ingresa la funcion f(x) derivada: ','s');
x = input('Ingresa el vector x: ');
y = eval(funx);
tabla = [x, y];
plot (x, y, 'g'), grid on, hold on
x = input('Ingrese el limite inferior x: ');
for n = 1:5,
xa = x;
fx = eval(funx);
fd = eval(funder);
x = x - fx/fd;
fprintf('%3d %12.9f %12.9f %12.9f %12.9f\n', n, xa, fx,fd, x)
end

2° Raiz: Punto Fijo

Grafica:

x = [-5:0.01:5]
y = (1/19).*(x+4).*(x+1).*(x-3)+(1/3)
Valor inicial utilizado es -1
Ecuacion despejada = x = ((x.^3)/11)+((2.*x.^2)/11)-(12/11)+(19/33)

Corrida:

>> % Punto Fijo
>> x = [-5:0.01:5];
>> y = (1/19).*(x+4).*(x+1).*(x-3)+(1/3);
>> plot(x,y,'r')
>> grid on, hold on
>> % Despejamos una Xpara obtener la raiz entre -1 y 0
>> % x = ((x.^3)/11)+((2.*x.^2)/11)-(12/11)+(19/33)
>> x = -1

x =

-1

>> x = ((x.^3)/11)+((2.*x.^2)/11)-(12/11)+(19/33)

x =

-0.424242424242424

>> x = ((x.^3)/11)+((2.*x.^2)/11)-(12/11)+(19/33)

x =

-0.489369022051216

>> x = ((x.^3)/11)+((2.*x.^2)/11)-(12/11)+(19/33)

x =

-0.482263425342544>> x = ((x.^3)/11)+((2.*x.^2)/11)-(12/11)+(19/33)

x =

-0.483061319297154

>> x = ((x.^3)/11)+((2.*x.^2)/11)-(12/11)+(19/33)

x =

-0.482971972551413

>> x = ((x.^3)/11)+((2.*x.^2)/11)-(12/11)+(19/33)

x =

-0.482981980616193

>> x = ((x.^3)/11)+((2.*x.^2)/11)-(12/11)+(19/33)

x =

-0.482980859615016

>> x =((x.^3)/11)+((2.*x.^2)/11)-(12/11)+(19/33)

x =

-0.482980985178615

>> x = ((x.^3)/11)+((2.*x.^2)/11)-(12/11)+(19/33)

x =

-0.482980971114214

>> x = ((x.^3)/11)+((2.*x.^2)/11)-(12/11)+(19/33)

x =

-0.482980972689570

>> x = ((x.^3)/11)+((2.*x.^2)/11)-(12/11)+(19/33)

x =

-0.482980972513114

>> x = ((x.^3)/11)+((2.*x.^2)/11)-(12/11)+(19/33)

x =-0.482980972532879

>> x = ((x.^3)/11)+((2.*x.^2)/11)-(12/11)+(19/33)

x =

-0.482980972530665

>> x = ((x.^3)/11)+((2.*x.^2)/11)-(12/11)+(19/33)

x =

-0.482980972530913

>> x = ((x.^3)/11)+((2.*x.^2)/11)-(12/11)+(19/33)

x =

-0.482980972530885

>> x = ((x.^3)/11)+((2.*x.^2)/11)-(12/11)+(19/33)

x =

-0.482980972530889

>> y =...