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Xi f(Xi) f'(Xi) X(i+1) Xi f(xi) f''(Xi)

1 -0.01846518 -0.1849452 0.90015866 1 -0.01846518 -0.00026294

0.90015866 -0.01662868 -0.18496866 0.81025867 -69.226367 -25.9878513 0.00074302

0.81025867 -0.01497314 -0.18498773 0.7293174 34906.7514 -7.1125E+13 -9.7399E+33

0.7293174 -0.01348118 -0.18500323 0.65644742 34906.7514 -7.1125E+13 -9.7399E+33

0.65644742 -0.01213698 -0.18501582 0.59084775 34906.7514 -7.1125E+13 -9.7399E+33

0.59084775 -0.01092614 -0.18502605 0.53179587 34906.7514 -7.1125E+13 -9.7399E+33

0.53179587 -0.00983561 -0.18503435 0.47864029 34906.7514 -7.1125E+13 -9.7399E+33

0.47864029 -0.00885357 -0.18504109 0.43079377 34906.7514 -7.1125E+13 -9.7399E+33

0.43079377 -0.00796933 -0.18504656 0.38772717 34906.7514 -7.1125E+13 -9.7399E+33

0.38772717 -0.00717321 -0.18505099 0.34896378 34906.7514 -7.1125E+13 -9.7399E+33

0.34896378 -0.00645648 -0.18505459 0.3140742 34906.7514 -7.1125E+13 -9.7399E+33

0.3140742 -0.00581126 -0.18505751 0.28267174 34906.7514 -7.1125E+13 -9.7399E+33

0.28267174 -0.00523045 -0.18505987 0.25440818 34906.7514 -7.1125E+13 -9.7399E+33

0.25440818 -0.00470764 -0.18506179 0.22897 34906.7514 -7.1125E+13 -9.7399E+33

0.22897 -0.00423704 -0.18506335 0.20607492 34906.7514 -7.1125E+13 -9.7399E+33

0.20607492 -0.00381346 -0.18506461 0.18546883 34906.7514 -7.1125E+13 -9.7399E+33

0.18546883 -0.0034322 -0.18506563 0.16692297 34906.7514 -7.1125E+13 -9.7399E+33

0.16692297 -0.00308905 -0.18506646 0.15023142 34906.7514 -7.1125E+13 -9.7399E+33

0.15023142 -0.00278019 -0.18506713 0.13520882 34906.7514 -7.1125E+13 -9.7399E+33

0.13520882 -0.0025022 -0.18506768 0.12168834 34906.7514 -7.1125E+13 -9.7399E+33

0.12168834 -0.00225201 -0.18506812 0.10951979 34906.7514 -7.1125E+13 -9.7399E+33

Met

od

o N

ewto

n

Rap

hso

n En esta primera iteracion aplicamos el metodo de forma tradicional sin alterar la formulacion del metodo

Met

od

o N

ewto

n

Rap

hso

n En la segunda realizamos el metodo utilizando de la segunda derivada en vez de la primera, donde el valor si converge en la tercera iteracion pero en funcion de el valor de f(x)

X(i+1) Xi f(xi) f''(Xi) X(i+1) Xi f'(xi)

-69.226367 1 -0.01846518 -0.00026294 -69.226367 29 -0.14774472

34906.7514 -69.226367 -25.9878513 0.00074302 34906.7514 18.0295028 -0.15998881

34906.7514 34906.7514 -7.1125E+13 -9.7399E+33 34906.7514 -4.69329033 -0.18198938

34906.7514 34906.7514 -7.1125E+13 -9.7399E+33 34906.7514 174.179507 -16.6590845

34906.7514 34906.7514 -7.1125E+13 -9.7399E+33 34906.7514 58.717223 -0.36641024

34906.7514 34906.7514 -7.1125E+13 -9.7399E+33 34906.7514 48.8399257 -0.23047613

34906.7514 34906.7514 -7.1125E+13 -9.7399E+33 34906.7514 40.7239649 -0.17254591

34906.7514 34906.7514 -7.1125E+13 -9.7399E+33 34906.7514 32.8212599 -0.14997112

34906.7514 34906.7514 -7.1125E+13 -9.7399E+33 34906.7514 23.4693241 -0.15141027

34906.7514 34906.7514 -7.1125E+13 -9.7399E+33 34906.7514 8.39475203 -0.17769881

34906.7514 34906.7514 -7.1125E+13 -9.7399E+33 34906.7514 -57.6995945 0.67690603

34906.7514 34906.7514 -7.1125E+13 -9.7399E+33 34906.7514 -18.2213735 -0.12734502

34906.7514 34906.7514 -7.1125E+13 -9.7399E+33 34906.7514 51.6162525 -0.26056247

34906.7514 34906.7514 -7.1125E+13 -9.7399E+33 34906.7514 43.1481608 -0.18562151

34906.7514 34906.7514 -7.1125E+13 -9.7399E+33 34906.7514 35.3270954 -0.15419792

34906.7514 34906.7514 -7.1125E+13 -9.7399E+33 34906.7514 26.6673166 -0.14845466

34906.7514 34906.7514 -7.1125E+13 -9.7399E+33 34906.7514 14.2752606 -0.16713717

34906.7514 34906.7514 -7.1125E+13 -9.7399E+33 34906.7514 -17.9127124 -0.12952419

34906.7514 34906.7514 -7.1125E+13 -9.7399E+33 34906.7514 52.3954856 -0.27007642

34906.7514 34906.7514 -7.1125E+13 -9.7399E+33 34906.7514 43.8070154 -0.18975813

34906.7514 34906.7514 -7.1125E+13 -9.7399E+33 34906.7514 35.9848724 -0.15572251

En la segunda realizamos el metodo utilizando de la segunda derivada en vez de la primera, donde el valor si converge en la tercera iteracion pero en funcion de el valor de M

eto

do

New

ton

R

aph

son En este recuadro

encontramos que si la funcion la hacemos utilizando la segunda derivada en vez de la primera pero en funcion de Xi, el valor es divergente monotonicamente

Met

od

o N

ewto

n

Rap

hso

n En este metododo remplazamos a f(x) por f'(xi) y a f'(x) por f''(xi), de igual manera los resultados no son los esperados para ninguna combinacion, exepto la segunda q por obvias razones no es la correcta

f''(x) X(i+1)

-0.01346746 18.0295028

-0.0070409 -4.69329033

0.00101742 174.179507

-0.14428161 58.717223

-0.0370962 48.8399257

-0.02839789 40.7239649

-0.02183378 32.8212599

-0.01603637 23.4693241

-0.01004408 8.39475203

-0.00268856 -57.6995945

-0.01714632 -18.2213735

0.00182344 51.6162525

-0.03076992 43.1481608

-0.02373353 35.3270954

-0.01780622 26.6673166

-0.01197983 14.2752606

-0.00519253 -17.9127124

0.00184223 52.3954856

-0.03144639 43.8070154

-0.0242591 35.9848724

-0.0182816 27.46688

En este metododo remplazamos a f(x) por

f''(xi), de igual manera los resultados no son los esperados para ninguna combinacion, exepto la segunda q por obvias razones no es la correcta

X g(x) Error X

1 6 0 2

6 -344 101.744186 1.29099445

-344 40233906 100.000855 1.37477412

40233906 -6.5129E+22 100 1.36401734

-6.5129E+22 2.7627E+68 100 1.36538433

2.7627E+68 -2.109E+205 100 1.36521038

-2.109E+205 #NUM! #NUM! 1.36523251

#NUM! #NUM! #NUM! 1.3652297

#NUM! #NUM! #NUM! 1.36523005

#NUM! #NUM! #NUM! 1.36523001

1.36523001

1.36523001

1.36523001

1.36523001

1.36523001

1.36523001

1.36523001

1.36523001

1.36523001

1.36523001

1.36523001

1.36523001

1.36523001

Se quiere aproximar una raíz de la ecuación x3 - 30x2 + 2400 = 0, que

sabemos se encuentra en el intervalo (10,15), mediante el método del

punto fijo. ¿Cuál de las siguientes funciones utilizarías para poder

esperar convergencia en el proceso de iteración? Justifique su respuesta.

h(x) Error

1.29099445 0

1.37477412 6.09406791

1.36401734 0.78860996

1.36538433 0.10011736

1.36521038 0.01274138

1.36523251 0.00162102

1.3652297 0.00020624

1.36523005 2.624E-05

1.36523001 3.3385E-06

1.36523001 4.2476E-07

1.36523001 5.4042E-08

1.36523001 6.8757E-09

1.36523001 8.7479E-10

1.36523001 1.113E-10

1.36523001 1.4166E-11

1.36523001 1.8053E-12

1.36523001 2.277E-13

1.36523001 3.2529E-14

1.36523001 0

1.36523001 0

1.36523001 0

1.36523001 0

1.36523001 0

f(Xi)*f(Xs)= -100

Biseccion

Iteraciones Xi Xr Xs f(Xi) f(Xr)

1 0 50 100 1 -50

2 0 25 50 1 -25

3 0 12.5 25 1 -12.5

4 0 6.25 12.5 1 -6.24999627

5 0 3.125 6.25 1 -3.12306955

6 0 1.5625 3.125 1 -1.51856307

7 0 0.78125 1.5625 1 -0.57163861

8 0 0.390625 0.78125 1 0.06720836

9 0.390625 0.5859375 0.78125 0.06720836 -0.27615195

10 0.390625 0.48828125 0.5859375 0.06720836 -0.1116778

11 0.390625 0.439453125 0.48828125 0.06720836 -0.0242163

12 0.390625 0.415039063 0.43945313 0.06720836 0.02097616

13 0.415039063 0.427246094 0.43945313 0.02097616 -0.00174688

14 0.415039063 0.421142578 0.42724609 0.02097616 0.00958255

15 0.421142578 0.424194336 0.42724609 0.00958255 0.00390986

Grafica Biseccion

Secante

Iteracion Xi f(Xi) f(Xi-1) Xi+1 Error

0 110 -110

1 100 -100 -110 0 0

2 0 1 -100 0.99009901 #DIV/0!

3 0.99009901 -0.85205711 1 0.53459421 100

4 0.534594211 -0.1913072 -0.85205711 0.40271171 85.205711

5 0.402711712 0.044186946 -0.1913072 0.42745749 32.7486126

0

20

40

60

80

100

120

0 5 10 15 20

Biseccion

Error vs Iteraciones

6 0.427457486 -0.00213813 0.04418695 0.42631535 5.78906073

7 0.426315346 -2.3333E-05 -0.00213813 0.42630274 0.26790971

8 0.426302744 1.23969E-08 -2.3333E-05 0.42630275 0.00295596

9 0.426302751 -7.1942E-14 1.2397E-08 0.42630275 1.5697E-06

Grafico Secante

Punto Fijo

Iteracion Xi f(x) Error

1 100 1.3839E-87 0

2 1.3839E-87 1 7.226E+90

3 1 0.135335283 100

4 0.135335283 0.762867769 638.90561

5 0.762867769 0.217461047 82.2596669

6 0.217461047 0.647315095 250.806629

7 0.647315095 0.273999173 66.4056888

8 0.273999173 0.57810582 136.247098

9 0.57810582 0.314676031 52.603976

10 0.314676031 0.532936999 83.7146025

11 0.532936999 0.344426695 40.9543657

12 0.344426695 0.502151511 54.7316182

13 0.502151511 0.366299849 31.4098061

14 0.366299849 0.480657799 37.0875562

15 0.480657799 0.382389484 23.7919682

16 0.382389484 0.465436796 25.6984881

17 0.465436796 0.394209182 17.8428765

18 0.394209182 0.454563181 18.0684819

19 0.454563181 0.402876038 13.277362

20 0.402876038 0.446751809 12.82954

-20

0

20

40

60

80

100

120

0 2 4 6 8 10

Secante


-1E+90

0

1E+90

2E+90

3E+90

4E+90

5E+90

6E+90

7E+90

8E+90

0

21 0.446751809 0.40921949 9.82106159

22 0.40921949 0.441119714 9.1716841

23 0.441119714 0.413855074 7.2316479

24 0.413855074 0.437048918 6.58796803

25 0.437048918 0.417238267 5.30692176

26 0.417238267 0.43410166 4.74804282

27 0.43410166 0.419704948 3.8846644

28 0.419704948 0.431965353 3.43019816

29 0.431965353 0.421502022 2.83828423

30 0.421502022 0.430415592 2.48239164

31 0.430415592 0.422810503 2.07092188

32 0.422810503 0.429290684 1.79869933

33 0.429290684 0.42376282 1.50950888

34 0.42376282 0.42847382 1.30447115

35 0.42847382 0.424455699 1.09948375

36 0.424455699 0.42788047 0.94665268

Falsa Posicion f(Xi)*f(Xs)=

f(Xi)*f(Xr) Error Iteraciones Xi Xr

-50 0 1 0 0.99009901

-25 100 2 0 0.534594211

-12.5 100 3 0 0.44874589

-6.24999627 100 4 0 0.431007689

-3.12306955 100 5 0 0.427291289

-1.51856307 100 6 0 0.426510545

-0.57163861 100 7 0 0.426346434

0.06720836 100 8 0 0.426311934

-0.01855972 33.3333333 9 0 0.426304682

-0.00750568 20 10 0 0.426303157

-0.00162754 11.1111111 11 0 0.426302836

0.00140977 5.88235294 12 0 0.426302769

-3.6643E-05 2.85714286 13 0 0.426302755

0.000201 1.44927536 14 0 0.426302752

3.7466E-05 0.71942446 15 0 0.426302751

Grafica Falsa Posicion

Newton Raphson

Iteracion Xi f(Xi)

1 100 -100

2 0 1

3 0.33333333 0.180083786

4 0.42218312 0.007646564

5 0.42629493 1.44945E-05

6 0.42630275 5.21899E-11


0

10

20

30

40

50

60

70

80

90

0 5

7 0.42630275 0

8 0.42630275 0

9 0.42630275 0

10 0.42630275 0

Grafica Newton Raphson


-20

0

20

40

60

80

100

120

0 2 4 6 8

Newton Raphson

10 20 30 40

Punto Fijo


Xs f(Xi) f(Xr) f(Xs) f(Xi)*f(Xr) Error

100 1 -0.85205711 -100 -0.85205711 0

0.99009901 1 -0.1913072 -0.85205711 -0.1913072 85.205711

0.53459421 1 -0.04115518 -0.1913072 -0.04115518 19.1307203

0.44874589 1 -0.00869758 -0.04115518 -0.00869758 4.11551845

0.43100769 1 -0.00183054 -0.00869758 -0.00183054 0.86975798

0.42729129 1 -0.00038492 -0.00183054 -0.00038492 0.18305385

0.42651055 1 -8.0926E-05 -0.00038492 -8.0926E-05 0.03849237

0.42634643 1 -1.7013E-05 -8.0926E-05 -1.7013E-05 0.00809262

0.42631193 1 -3.5767E-06 -1.7013E-05 -3.5767E-06 0.00170132

0.42630468 1 -7.5192E-07 -3.5767E-06 -7.5192E-07 0.00035767

0.42630316 1 -1.5808E-07 -7.5192E-07 -1.5808E-07 7.5192E-05

0.42630284 1 -3.3232E-08 -1.5808E-07 -3.3232E-08 1.5808E-05

0.42630277 1 -6.9864E-09 -3.3232E-08 -6.9864E-09 3.3232E-06

0.42630275 1 -1.4687E-09 -6.9864E-09 -1.4687E-09 6.9864E-07

0.42630275 1 -3.0877E-10 -1.4687E-09 -3.0877E-10 1.4687E-07

f'(Xi) Xi+1 Error

-1 0 0

-3 0.33333333 #DIV/0!

-2.02683424 0.42218312 100

-1.85965936 0.42629493 21.0453192

-1.85261884 0.42630275 0.96454564

-1.8526055 0.42630275 0.00183526

10 15 20

Falsa Posicion


-1.8526055 0.42630275 6.6082E-09

-1.8526055 0.42630275 0

-1.8526055 0.42630275 0

-1.8526055 0.42630275 0

8 10 12

Newton Raphson


-20

0

20

40

60

80

100

120

0 5 10 15 20

Biseccion

Newton Raphson

Secante

Falsa Posicion

Biseccion

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