Ruta De Convergencia Solucion Numerica De Ecuaciones No Lineales
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FUNDACIÓN UNIVERSITARIA KONRAD LORENZ FACULTAD DE MATEMÁTICAS E INGENIERÍAS PROGRAMA DE INGENIERÍA DE SISTEMAS MÉTODOS NUMÉRICOS
TRAYECTORIA DE CONVERGENCIA DE LOS PRINCIPALES MÉTODOS PARA RESOLVER ECUACIONES NO LINEA
f ( x) = x 2 − e −x
g ( x) =
f ( x ) = x 2 − e− x = 0
x2 = e− x
e− x
PROCEDIMIENTO ITERATIVO DEL METODO DE PUNTO FIJO i
xi
g(xi)
error
1 2 3 4 5 6 7 8 9
-3 4.48 0.11 0.95 0.62 0.73 0.69 0.71 0.7
4.48 0.11 0.95 0.62 0.73 0.69 0.71 0.7 0.7
166.94 4113.36 88.78 52.34 15.03 5.66 1.94 0.69 0.24
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37
0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7
0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7
0.09 0.03 0.01 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
TRAY
5
0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7
0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
TRAY
5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 y
38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62
0 -0.5 -1 -1.5 -2 -2.5 -3 -3.5 -4 -4.5 -5
------------3 22 2 2 2 2 22 2 2 1 1 .. . . . . .. . . .
ECUACIONES NO LINEALES
5
x2 = e− x
x=
e− x
tabulación para grafica a g(x) y f(x)
ruta de convergencia
x
X
G(X)
Y
-3 -2.9 -2.8 -2.7 -2.6 -2.5 -2.4 -2.3 -2.2
4.48 4.26 4.06 3.86 3.67 3.49 3.32 3.16 3
-11.09 -9.76 -8.6 -7.59 -6.7 -5.93 -5.26 -4.68 -4.19
-2 -2 2.72 2.72 0.26 0.26 0.88 0.88 0.64
0 2.72 2.72 0.26 0.26 0.88 0.88 0.64 0.64
-2.1 -2 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6
2.86 2.72 2.59 2.46 2.34 2.23 2.12 2.01 1.92 1.82 1.73 1.65 1.57 1.49 1.42 1.35 1.28 1.22 1.16 1.11 1.05 1 0.95 0.9 0.86 0.82 0.78 0.74
-3.76 -3.39 -3.08 -2.81 -2.58 -2.39 -2.23 -2.1 -1.98 -1.88 -1.79 -1.72 -1.65 -1.59 -1.52 -1.46 -1.4 -1.33 -1.26 -1.18 -1.1 -1 -0.89 -0.78 -0.65 -0.51 -0.36 -0.19
0.64 0.72 0.72 0.7 0.7 0.71 0.71 0.7 0.7 0.7 0.7
0.72 0.72 0.7 0.7 0.71 0.71 0.7 0.7 0.7 0.7 0.7
TRAYECTORIA DE CONVERGENCIA METODO PUNTO FIJO. f(x)=x2-e-x Y g(x)=raiz(e-x) Se observa que el punto fijo coincide con la raíz
5
5
4
5
3
5
2
5
1
5
0
5
1
5
2
5
3
5
4
5
5
0.7
0.7
-0.01
TRAYECTORIA METODO PUNTO FIJO. f(x)=x2-e-x Y g(x)=raiz(e-x) 0.8 0.67DE CONVERGENCIA 0.19 Se observa que el punto fijo coincide con la raíz 0.9 0.64 0.4 1 0.61 0.63 1.1 0.58 0.88 1.2 0.55 1.14 1.3 0.52 1.42 1.4 0.5 1.71 1.5 0.47 2.03 1.6 0.45 2.36 1.7 0.43 2.71 1.8 0.41 3.07 1.9 0.39 3.46 2 0.37 3.86 2.1 0.35 4.29 2.2 0.33 4.73 2.3 0.32 5.19 2.4 0.3 5.67 2.5 0.29 6.17 2.6 0.27 6.69 2.7 0.26 7.22 2.8 0.25 7.78 2.9 0.23 8.35 3 0.22 8.95 3.1 0.21 9.56 3.2 0.2 10.2 3.3 0.19 10.85 3.4 0.18 11.53 3.5 0.17 12.22 3.6 0.17 12.93 3.7 0.16 13.67 3.8 0.15 14.42 3.9 0.14 15.19 4 0.14 15.98 4.1 0.13 16.79 4.2 0.12 17.63 4.3 0.12 18.48 4.4 0.11 19.35 4.5 0.11 20.24 4.6 0.1 - - - - - - - - - - - - - - - - - - - - - - - - 21.15 - - - - - - 0 0 0 00 0 0 0 0 01 1 1 1 1 1 11 1 1 2 2 22 2 2 2 2 22 3 3 3 3 3 33 3 3 3 2 2 2 2 2 2 2 2 2 24.7 1 1 1 1 1 1 1 0.1 1 1 1 0 0 022.08 0 0 00 0 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 0.09 23.03 . . . . . . . . . 4.9 . . . . . . .0.09 . . . . . . .24. . . . 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 5 0.08 24.99 x
ruta de convergencia: claves
Xo Xo x1 x1 x2 x2
0 X1 x1 x2 x2 x3
Recuérdese que X1=g(Xo), X2=g(X1) y en general Xi+1=g(xi)
Y g(x)=raiz(e-x)
raiz y punto fijo
0.7 0.7
0.7 0
Y g(x)=raiz(e-x)
2 2 22 3 3 3 3 3 33 3 3 3 4 44 4 4 4 4 4 44 5 . . .. . . . . .. . . . .. . . . . . .. 6 7 89 1 2 3 4 56 7 8 9 12 3 4 5 6 7 89
FUNDACIÓN UNIVERSITARIA KONRAD LORENZ FACULTAD DE MATEMÁTICAS E INGENIERÍAS PROGRAMA DE INGENIERÍA DE SISTEMAS MÉTODOS NUMÉRICOS
TRAYECTORIA DE CONVERGENCIA DE LOS PRINCIPALES MÉTODOS PARA RESOLVER ECUACIONES NO LINEAL
f ( x) = x 2 − e −x MÉTODO DE NEWTON RAPHSON
K
tabulación gráfica función
Xi 1 2 3 4 5 6 7 8 9 10 11 12 13
Xi+1 -2.5 -1.67 -0.4 1.52 0.88 0.71 0.7 0.7 0.7 0.7 0.7 0.7 0.7
-5.93 -2.53 -1.33 2.1 0.36 0.02 0 0 0 0 0 0 0
7.18 1.99 0.69 3.27 2.17 1.92 1.9 1.9 1.9 1.9 1.9 1.9 1.9
error -1.67 -0.4 1.52 0.88 0.71 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7
x 49.34 319.33 126.2 73.27 23.08 1.57 0.01 0 0 0 0 0 0
TRAYECTORIA DE CONVERGENCIA METODO NEWTON RAPHSON. f(x)=x2-e-x 25 22.5 20 17.5 15 12.5 10
y
7.5 5 2.5 0
-3 -2.9 -2.8 -2.7 -2.6 -2.5 -2.4 -2.3 -2.2 -2.1 -2 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
15 12.5 10
y
7.5 5 2.5 0 -2.5 -5 -7.5 -10 -12.5 -3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1 x
1.5
2
2.5
3
3.5
4
4.5
0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 5 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5
ECUACIONES NO LINEALES
abulación gráfica función
f(x) -11.09 -9.76 -8.6 -7.59 -6.7 -5.93 -5.26 -4.68 -4.19 -3.76 -3.39 -3.08 -2.81 -2.58 -2.39 -2.23 -2.1 -1.98 -1.88 -1.79 -1.72 -1.65 -1.59 -1.52 -1.46 -1.4 -1.33 -1.26 -1.18 -1.1 -1 -0.89 -0.78 -0.65 -0.51 -0.36 -0.19 -0.01 0.19
trayectoria de convergencia
X 5 5 2.5 2.5 1.29 1.29 0.8 0.8 0.71 0.71 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7
Y 0 24.99 0 6.18 0 1.38 0 0.2 0 0.01 0 0 0 0 0 0 0 0
trayectoria de convergencia: claves
xo xo x1 x1 x2 x2
0 f(xo) 0 f(x1) 0 f(x2)
Recuerde que x1 = x 0 −
En general:
f ( x0 ) f ′( x 0 )
xi +1 = xi −
x 2 = x1 −
f ( xi ) f ′( xi )
f f
0.4 0.63 0.88 1.14 1.42 1.71 2.03 2.36 2.71 3.07 3.46 3.86 4.29 4.73 5.19 5.67 6.17 6.69 7.22 7.78 8.35 8.95 9.56 10.2 10.85 11.53 12.22 12.93 13.67 14.42 15.19 15.98 16.79 17.63 18.48 19.35 20.24 21.15 22.08 23.03 24 24.99
x 2 = x1 −
−
f ( xi ) f ′( xi )
f ( x1 ) f ′( x1 )
FUNDACIÓN UNIVERSITARIA KONRAD LORENZ FACULTAD DE MATEMÁTICAS E INGENIERÍAS PROGRAMA DE INGENIERÍA DE SISTEMAS
MÉTODOS NUMÉRICOS TRAYECTORIA DE CONVERGENCIA DE LOS PRINCIPALES MÉTODOS PARA RESOLVER ECUACIONES NO LINEAL MÉTODO DE LA SECANTE tabulación para graficar la K
Xi-1 1 2 3 4 5 6 7 8 9 10 11 12 13
Xi -2 2.5 -0.4 0.11 1.12 0.6 0.69 0.7 0.7 0.7 0.7 0.7 0.7
Xi+1 2.5 -0.4 0.11 1.12 0.6 0.69 0.7 0.7 0.7 0.7 0.7 0.7 #DIV/0!
f(Xi-1) -0.4 0.11 1.12 0.6 0.69 0.7 0.7 0.7 0.7 0.7 0.7 #DIV/0! #DIV/0!
f(Xi) -3.39 6.17 -1.33 -0.88 0.92 -0.18 -0.03 0 0 0 0 0 0
error
x
6.17 -1.33 -0.88 0.92 -0.18 -0.03 0 0 0 0 0 0 #DIV/0!
-3 -2.9 -2.8 -2.7 -2.6 -2.5 -2.4 -2.3 -2.2 -2.1 -2 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3
TRAYECTORIA DE CONVERGENCIA METODO DE LA SECANTE. f(x)=x2-e-x 25 22.5 20 17.5 15 12.5 10
y
7.5 5 2.5 0 -2.5 -5 -7.5 -10 -12.5 -3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1 x
1.5
2
2.5
3
3.5
4
4.5
5
0 -2.5 -5 -7.5 -10 -12.5 -3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1 x
1.5
2
2.5
3
3.5
4
4.5
5
1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5
ECUACIONES NO LINEALES
abulación para graficar la función
ruta de convergencia
f(x) -11.09 -9.76 -8.6 -7.59 -6.7 -5.93 -5.26 -4.68 -4.19 -3.76 -3.39 -3.08 -2.81 -2.58 -2.39 -2.23 -2.1 -1.98 -1.88 -1.79 -1.72 -1.65 -1.59 -1.52 -1.46 -1.4 -1.33 -1.26 -1.18 -1.1 -1 -0.89 -0.78 -0.65 -0.51 -0.36 -0.19 -0.01 0.19 0.4 0.63 0.88 1.14 1.42
-3 -3 5 5 -0.54 -0.54 5 -0.24 -0.24 -0.54 1.49 1.49 -0.24 0.41 0.41 1.49 0.63 0.63 0.41 0.71 0.71
0 -11.09 24.99 0 0 -1.43 24.99 0 -1.22 -1.43 0 2 -1.22 0 -0.49 2 0 -0.14 -0.49 0 0.02
xo x0 x1 x1 x2 x2 x1 x3 x3 x2 x4 x4
0 f(xo) f(x1) 0 0 f(x2) f(x1) 0 f(x3) f(x2) 0 f(x4)
1.71 2.03 2.36 2.71 3.07 3.46 3.86 4.29 4.73 5.19 5.67 6.17 6.69 7.22 7.78 8.35 8.95 9.56 10.2 10.85 11.53 12.22 12.93 13.67 14.42 15.19 15.98 16.79 17.63 18.48 19.35 20.24 21.15 22.08 23.03 24 24.99
FUNDACIÓN UNIVERSITARIA KONRAD LORENZ FACULTAD DE MATEMÁTICAS E INGENIERÍAS PROGRAMA DE INGENIERÍA DE SISTEMAS MÉTODOS NUMÉRICOS
TRAYECTORIA DE CONVERGENCIA DE LOS PRINCIPALES MÉTODOS PARA RESOLVER ECUACIONES NO LINEALES MÉTODO DE REGLA FALSA
A
B -2 -0.4 0.11 0.41 0.57 0.64 0.68 0.69 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7
XR 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5
f(A) -0.4 0.11 0.41 0.57 0.64 0.68 0.69 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7
f(B) -3.39 -1.33 -0.88 -0.49 -0.25 -0.12 -0.05 -0.02 -0.01 0 0 0 0 0 0 0 0 0 0 0
f(XR) 6.17 6.17 6.17 6.17 6.17 6.17 6.17 6.17 6.17 6.17 6.17 6.17 6.17 6.17 6.17 6.17 6.17 6.17 6.17 6.17
-1.33 -0.88 -0.49 -0.25 -0.12 -0.05 -0.02 -0.01 0 0 0 0 0 0 0 0 0 0 0 0
f(A)*f(XR) 4.52 1.18 0.44 0.12 0.03 0.01 0 0 0 0 0 0 0 0 0 0 0 0 0 0
TRAYECTORIA DE CONVERGENCIA METODO REGLA FALSA. f(x)=x2-e-x 6.5 6 5.5 5 4.5 4 3.5 3 2.5 2 y
K
1.5 1 0.5 0 -0.5 -1
3.5 3 2.5
y
2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -2.5 -3 -3.5 -2
-1.7 -1.5 -1.2 5 5
-1
-0.7 -0.5 -0.2 5 5
0
0.25 0.5 0.75 x
1
1.25 1.5 1.75
2
2.25 2.5 2
ECUACIONES NO LINEALES
A. f(x)=x2-e-x
error 459.5 72.63 27.39 11.66 5.12 2.27 1.01 0.45 0.2 0.09 0.04 0.02 0.01 0 0 0 0 0 0
TABULACIÓN PARA GRAFICAR A f(x)
TRAYECTORIA DE CONVERGENCIA
x
X
f(x) -2 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
-3.39 -3.08 -2.81 -2.58 -2.39 -2.23 -2.1 -1.98 -1.88 -1.79 -1.72 -1.65 -1.59 -1.52 -1.46 -1.4 -1.33 -1.26 -1.18 -1.1 -1 -0.89 -0.78 -0.65 -0.51 -0.36 -0.19 -0.01 0.19 0.4 0.63 0.88 1.14 1.42 1.71 2.03 2.36 2.71 3.07 3.46
Y 2.5 2.5 -2 -2 -0.4 -0.4 2.5 0.11 0.11 2.5 0.41 0.41 2.5 0.57 0.57 2.5 0.64 0.64 2.5 0.68 0.68 2.5 0.69 0.69 2.5 0.7 0.7 2.5 0.7 0.7 2.5 0.7 0.7 2.5 0.7 0.7 2.5 0.7 0.7 2.5
0 6.17 -3.39 0 0 -1.33 6.17 0 -0.88 6.17 0 -0.49 6.17 0 -0.25 6.17 0 -0.12 6.17 0 -0.05 6.17 0 -0.02 6.17 0 -0.01 6.17 0 0 6.17 0 0 6.17 0 0 6.17 0 0 6.17
5 1.5 1.75
2 2.1 2.2 2.3 2.4 2.5
2
2.25 2.5 2.75
3.86 4.29 4.73 5.19 5.67 6.17
0.7 0.7 2.5 0.7 0.7 2.5 0.7 0.7 2.5 0.7 0.7
0 0 6.17 0 0 6.17 0 0 6.17 0 0
A DE CONVERGENCIA
a a b b xr xr si(f(a)*f(xr)<0,b,a) xr xr SI(Q12*Q15>0,P13,P12) xr xr
0 f(a) f(b) 0 0 f(xr) f(P13) 0 f(xr) f(P16) 0 f(xr)
TRAYECTORIA DE CONVERGENCIA DE LOS PRINCIPALES MÉTODOS PARA RESOLVER ECUACIONES NO LINEA
f ( x) = x 2 − e −x
g ( x) =
f ( x ) = x 2 − e− x = 0
x2 = e− x
e− x
PROCEDIMIENTO ITERATIVO DEL METODO DE PUNTO FIJO i
xi
g(xi)
error
1 2 3 4 5 6 7 8 9
-3 4.48 0.11 0.95 0.62 0.73 0.69 0.71 0.7
4.48 0.11 0.95 0.62 0.73 0.69 0.71 0.7 0.7
166.94 4113.36 88.78 52.34 15.03 5.66 1.94 0.69 0.24
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37
0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7
0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7
0.09 0.03 0.01 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
TRAY
5
0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7
0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
TRAY
5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 y
38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62
0 -0.5 -1 -1.5 -2 -2.5 -3 -3.5 -4 -4.5 -5
------------3 22 2 2 2 2 22 2 2 1 1 .. . . . . .. . . .
ECUACIONES NO LINEALES
5
x2 = e− x
x=
e− x
tabulación para grafica a g(x) y f(x)
ruta de convergencia
x
X
G(X)
Y
-3 -2.9 -2.8 -2.7 -2.6 -2.5 -2.4 -2.3 -2.2
4.48 4.26 4.06 3.86 3.67 3.49 3.32 3.16 3
-11.09 -9.76 -8.6 -7.59 -6.7 -5.93 -5.26 -4.68 -4.19
-2 -2 2.72 2.72 0.26 0.26 0.88 0.88 0.64
0 2.72 2.72 0.26 0.26 0.88 0.88 0.64 0.64
-2.1 -2 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6
2.86 2.72 2.59 2.46 2.34 2.23 2.12 2.01 1.92 1.82 1.73 1.65 1.57 1.49 1.42 1.35 1.28 1.22 1.16 1.11 1.05 1 0.95 0.9 0.86 0.82 0.78 0.74
-3.76 -3.39 -3.08 -2.81 -2.58 -2.39 -2.23 -2.1 -1.98 -1.88 -1.79 -1.72 -1.65 -1.59 -1.52 -1.46 -1.4 -1.33 -1.26 -1.18 -1.1 -1 -0.89 -0.78 -0.65 -0.51 -0.36 -0.19
0.64 0.72 0.72 0.7 0.7 0.71 0.71 0.7 0.7 0.7 0.7
0.72 0.72 0.7 0.7 0.71 0.71 0.7 0.7 0.7 0.7 0.7
TRAYECTORIA DE CONVERGENCIA METODO PUNTO FIJO. f(x)=x2-e-x Y g(x)=raiz(e-x) Se observa que el punto fijo coincide con la raíz
5
5
4
5
3
5
2
5
1
5
0
5
1
5
2
5
3
5
4
5
5
0.7
0.7
-0.01
TRAYECTORIA METODO PUNTO FIJO. f(x)=x2-e-x Y g(x)=raiz(e-x) 0.8 0.67DE CONVERGENCIA 0.19 Se observa que el punto fijo coincide con la raíz 0.9 0.64 0.4 1 0.61 0.63 1.1 0.58 0.88 1.2 0.55 1.14 1.3 0.52 1.42 1.4 0.5 1.71 1.5 0.47 2.03 1.6 0.45 2.36 1.7 0.43 2.71 1.8 0.41 3.07 1.9 0.39 3.46 2 0.37 3.86 2.1 0.35 4.29 2.2 0.33 4.73 2.3 0.32 5.19 2.4 0.3 5.67 2.5 0.29 6.17 2.6 0.27 6.69 2.7 0.26 7.22 2.8 0.25 7.78 2.9 0.23 8.35 3 0.22 8.95 3.1 0.21 9.56 3.2 0.2 10.2 3.3 0.19 10.85 3.4 0.18 11.53 3.5 0.17 12.22 3.6 0.17 12.93 3.7 0.16 13.67 3.8 0.15 14.42 3.9 0.14 15.19 4 0.14 15.98 4.1 0.13 16.79 4.2 0.12 17.63 4.3 0.12 18.48 4.4 0.11 19.35 4.5 0.11 20.24 4.6 0.1 - - - - - - - - - - - - - - - - - - - - - - - - 21.15 - - - - - - 0 0 0 00 0 0 0 0 01 1 1 1 1 1 11 1 1 2 2 22 2 2 2 2 22 3 3 3 3 3 33 3 3 3 2 2 2 2 2 2 2 2 2 24.7 1 1 1 1 1 1 1 0.1 1 1 1 0 0 022.08 0 0 00 0 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 0.09 23.03 . . . . . . . . . 4.9 . . . . . . .0.09 . . . . . . .24. . . . 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 5 0.08 24.99 x
ruta de convergencia: claves
Xo Xo x1 x1 x2 x2
0 X1 x1 x2 x2 x3
Recuérdese que X1=g(Xo), X2=g(X1) y en general Xi+1=g(xi)
Y g(x)=raiz(e-x)
raiz y punto fijo
0.7 0.7
0.7 0
Y g(x)=raiz(e-x)
2 2 22 3 3 3 3 3 33 3 3 3 4 44 4 4 4 4 4 44 5 . . .. . . . . .. . . . .. . . . . . .. 6 7 89 1 2 3 4 56 7 8 9 12 3 4 5 6 7 89
FUNDACIÓN UNIVERSITARIA KONRAD LORENZ FACULTAD DE MATEMÁTICAS E INGENIERÍAS PROGRAMA DE INGENIERÍA DE SISTEMAS MÉTODOS NUMÉRICOS
TRAYECTORIA DE CONVERGENCIA DE LOS PRINCIPALES MÉTODOS PARA RESOLVER ECUACIONES NO LINEAL
f ( x) = x 2 − e −x MÉTODO DE NEWTON RAPHSON
K
tabulación gráfica función
Xi 1 2 3 4 5 6 7 8 9 10 11 12 13
Xi+1 -2.5 -1.67 -0.4 1.52 0.88 0.71 0.7 0.7 0.7 0.7 0.7 0.7 0.7
-5.93 -2.53 -1.33 2.1 0.36 0.02 0 0 0 0 0 0 0
7.18 1.99 0.69 3.27 2.17 1.92 1.9 1.9 1.9 1.9 1.9 1.9 1.9
error -1.67 -0.4 1.52 0.88 0.71 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7
x 49.34 319.33 126.2 73.27 23.08 1.57 0.01 0 0 0 0 0 0
TRAYECTORIA DE CONVERGENCIA METODO NEWTON RAPHSON. f(x)=x2-e-x 25 22.5 20 17.5 15 12.5 10
y
7.5 5 2.5 0
-3 -2.9 -2.8 -2.7 -2.6 -2.5 -2.4 -2.3 -2.2 -2.1 -2 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
15 12.5 10
y
7.5 5 2.5 0 -2.5 -5 -7.5 -10 -12.5 -3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1 x
1.5
2
2.5
3
3.5
4
4.5
0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 5 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5
ECUACIONES NO LINEALES
abulación gráfica función
f(x) -11.09 -9.76 -8.6 -7.59 -6.7 -5.93 -5.26 -4.68 -4.19 -3.76 -3.39 -3.08 -2.81 -2.58 -2.39 -2.23 -2.1 -1.98 -1.88 -1.79 -1.72 -1.65 -1.59 -1.52 -1.46 -1.4 -1.33 -1.26 -1.18 -1.1 -1 -0.89 -0.78 -0.65 -0.51 -0.36 -0.19 -0.01 0.19
trayectoria de convergencia
X 5 5 2.5 2.5 1.29 1.29 0.8 0.8 0.71 0.71 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7
Y 0 24.99 0 6.18 0 1.38 0 0.2 0 0.01 0 0 0 0 0 0 0 0
trayectoria de convergencia: claves
xo xo x1 x1 x2 x2
0 f(xo) 0 f(x1) 0 f(x2)
Recuerde que x1 = x 0 −
En general:
f ( x0 ) f ′( x 0 )
xi +1 = xi −
x 2 = x1 −
f ( xi ) f ′( xi )
f f
0.4 0.63 0.88 1.14 1.42 1.71 2.03 2.36 2.71 3.07 3.46 3.86 4.29 4.73 5.19 5.67 6.17 6.69 7.22 7.78 8.35 8.95 9.56 10.2 10.85 11.53 12.22 12.93 13.67 14.42 15.19 15.98 16.79 17.63 18.48 19.35 20.24 21.15 22.08 23.03 24 24.99
x 2 = x1 −
−
f ( xi ) f ′( xi )
f ( x1 ) f ′( x1 )
FUNDACIÓN UNIVERSITARIA KONRAD LORENZ FACULTAD DE MATEMÁTICAS E INGENIERÍAS PROGRAMA DE INGENIERÍA DE SISTEMAS
MÉTODOS NUMÉRICOS TRAYECTORIA DE CONVERGENCIA DE LOS PRINCIPALES MÉTODOS PARA RESOLVER ECUACIONES NO LINEAL MÉTODO DE LA SECANTE tabulación para graficar la K
Xi-1 1 2 3 4 5 6 7 8 9 10 11 12 13
Xi -2 2.5 -0.4 0.11 1.12 0.6 0.69 0.7 0.7 0.7 0.7 0.7 0.7
Xi+1 2.5 -0.4 0.11 1.12 0.6 0.69 0.7 0.7 0.7 0.7 0.7 0.7 #DIV/0!
f(Xi-1) -0.4 0.11 1.12 0.6 0.69 0.7 0.7 0.7 0.7 0.7 0.7 #DIV/0! #DIV/0!
f(Xi) -3.39 6.17 -1.33 -0.88 0.92 -0.18 -0.03 0 0 0 0 0 0
error
x
6.17 -1.33 -0.88 0.92 -0.18 -0.03 0 0 0 0 0 0 #DIV/0!
-3 -2.9 -2.8 -2.7 -2.6 -2.5 -2.4 -2.3 -2.2 -2.1 -2 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3
TRAYECTORIA DE CONVERGENCIA METODO DE LA SECANTE. f(x)=x2-e-x 25 22.5 20 17.5 15 12.5 10
y
7.5 5 2.5 0 -2.5 -5 -7.5 -10 -12.5 -3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1 x
1.5
2
2.5
3
3.5
4
4.5
5
0 -2.5 -5 -7.5 -10 -12.5 -3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1 x
1.5
2
2.5
3
3.5
4
4.5
5
1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5
ECUACIONES NO LINEALES
abulación para graficar la función
ruta de convergencia
f(x) -11.09 -9.76 -8.6 -7.59 -6.7 -5.93 -5.26 -4.68 -4.19 -3.76 -3.39 -3.08 -2.81 -2.58 -2.39 -2.23 -2.1 -1.98 -1.88 -1.79 -1.72 -1.65 -1.59 -1.52 -1.46 -1.4 -1.33 -1.26 -1.18 -1.1 -1 -0.89 -0.78 -0.65 -0.51 -0.36 -0.19 -0.01 0.19 0.4 0.63 0.88 1.14 1.42
-3 -3 5 5 -0.54 -0.54 5 -0.24 -0.24 -0.54 1.49 1.49 -0.24 0.41 0.41 1.49 0.63 0.63 0.41 0.71 0.71
0 -11.09 24.99 0 0 -1.43 24.99 0 -1.22 -1.43 0 2 -1.22 0 -0.49 2 0 -0.14 -0.49 0 0.02
xo x0 x1 x1 x2 x2 x1 x3 x3 x2 x4 x4
0 f(xo) f(x1) 0 0 f(x2) f(x1) 0 f(x3) f(x2) 0 f(x4)
1.71 2.03 2.36 2.71 3.07 3.46 3.86 4.29 4.73 5.19 5.67 6.17 6.69 7.22 7.78 8.35 8.95 9.56 10.2 10.85 11.53 12.22 12.93 13.67 14.42 15.19 15.98 16.79 17.63 18.48 19.35 20.24 21.15 22.08 23.03 24 24.99
FUNDACIÓN UNIVERSITARIA KONRAD LORENZ FACULTAD DE MATEMÁTICAS E INGENIERÍAS PROGRAMA DE INGENIERÍA DE SISTEMAS MÉTODOS NUMÉRICOS
TRAYECTORIA DE CONVERGENCIA DE LOS PRINCIPALES MÉTODOS PARA RESOLVER ECUACIONES NO LINEALES MÉTODO DE REGLA FALSA
A
B -2 -0.4 0.11 0.41 0.57 0.64 0.68 0.69 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7
XR 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5
f(A) -0.4 0.11 0.41 0.57 0.64 0.68 0.69 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7
f(B) -3.39 -1.33 -0.88 -0.49 -0.25 -0.12 -0.05 -0.02 -0.01 0 0 0 0 0 0 0 0 0 0 0
f(XR) 6.17 6.17 6.17 6.17 6.17 6.17 6.17 6.17 6.17 6.17 6.17 6.17 6.17 6.17 6.17 6.17 6.17 6.17 6.17 6.17
-1.33 -0.88 -0.49 -0.25 -0.12 -0.05 -0.02 -0.01 0 0 0 0 0 0 0 0 0 0 0 0
f(A)*f(XR) 4.52 1.18 0.44 0.12 0.03 0.01 0 0 0 0 0 0 0 0 0 0 0 0 0 0
TRAYECTORIA DE CONVERGENCIA METODO REGLA FALSA. f(x)=x2-e-x 6.5 6 5.5 5 4.5 4 3.5 3 2.5 2 y
K
1.5 1 0.5 0 -0.5 -1
3.5 3 2.5
y
2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -2.5 -3 -3.5 -2
-1.7 -1.5 -1.2 5 5
-1
-0.7 -0.5 -0.2 5 5
0
0.25 0.5 0.75 x
1
1.25 1.5 1.75
2
2.25 2.5 2
ECUACIONES NO LINEALES
A. f(x)=x2-e-x
error 459.5 72.63 27.39 11.66 5.12 2.27 1.01 0.45 0.2 0.09 0.04 0.02 0.01 0 0 0 0 0 0
TABULACIÓN PARA GRAFICAR A f(x)
TRAYECTORIA DE CONVERGENCIA
x
X
f(x) -2 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
-3.39 -3.08 -2.81 -2.58 -2.39 -2.23 -2.1 -1.98 -1.88 -1.79 -1.72 -1.65 -1.59 -1.52 -1.46 -1.4 -1.33 -1.26 -1.18 -1.1 -1 -0.89 -0.78 -0.65 -0.51 -0.36 -0.19 -0.01 0.19 0.4 0.63 0.88 1.14 1.42 1.71 2.03 2.36 2.71 3.07 3.46
Y 2.5 2.5 -2 -2 -0.4 -0.4 2.5 0.11 0.11 2.5 0.41 0.41 2.5 0.57 0.57 2.5 0.64 0.64 2.5 0.68 0.68 2.5 0.69 0.69 2.5 0.7 0.7 2.5 0.7 0.7 2.5 0.7 0.7 2.5 0.7 0.7 2.5 0.7 0.7 2.5
0 6.17 -3.39 0 0 -1.33 6.17 0 -0.88 6.17 0 -0.49 6.17 0 -0.25 6.17 0 -0.12 6.17 0 -0.05 6.17 0 -0.02 6.17 0 -0.01 6.17 0 0 6.17 0 0 6.17 0 0 6.17 0 0 6.17
5 1.5 1.75
2 2.1 2.2 2.3 2.4 2.5
2
2.25 2.5 2.75
3.86 4.29 4.73 5.19 5.67 6.17
0.7 0.7 2.5 0.7 0.7 2.5 0.7 0.7 2.5 0.7 0.7
0 0 6.17 0 0 6.17 0 0 6.17 0 0
A DE CONVERGENCIA
a a b b xr xr si(f(a)*f(xr)<0,b,a) xr xr SI(Q12*Q15>0,P13,P12) xr xr
0 f(a) f(b) 0 0 f(xr) f(P13) 0 f(xr) f(P16) 0 f(xr)
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