ruta de convergencia solucion numerica de ecuaciones no lineales

FUNDACIÓN UNIVERSITARIA KONRAD LORENZFACULTAD DE MATEMÁTICAS E INGENIERÍAS

PROGRAMA DE INGENIERÍA DE SISTEMAS

MÉTODOS NUMÉRICOS

TRAYECTORIA DE CONVERGENCIA DE LOS PRINCIPALES MÉTODOS PARA RESOLVER ECUACIONES NO LINEALES

PROCEDIMIENTO ITERATIVO DEL METODO DE PUNTO FIJO

i xi g(xi) error1 -3 4.48168907 166.9390482 4.48168907 0.10636863 4113.355843 0.10636863 0.94820523 88.78210844 0.94820523 0.62244338 52.33598085 0.62244338 0.73255146 15.03076396 0.73255146 0.69331161 5.659771197 0.69331161 0.70704865 1.942870728 0.70704865 0.70220891 0.68921639 0.70220891 0.70391022 0.24169436

10 0.70391022 0.70331169 0.0851017611 0.70331169 0.7035222 0.0299220512 0.7035222 0.70344816 0.0105259713 0.70344816 0.7034742 0.0037021714 0.7034742 0.70346504 0.001302215 0.70346504 0.70346826 0.0004580216 0.70346826 0.70346713 0.000161117 0.70346713 0.70346753 5.6665E-0518 0.70346753 0.70346739 1.9931E-0519 0.70346739 0.70346744 7.0104E-0620 0.70346744 0.70346742 2.4658E-0621 0.70346742 0.70346742 8.6731E-0722 0.70346742 0.70346742 3.0506E-0723 0.70346742 0.70346742 1.073E-0724 0.70346742 0.70346742 3.7741E-0825 0.70346742 0.70346742 1.3275E-0826 0.70346742 0.70346742 4.6692E-0927 0.70346742 0.70346742 1.6423E-0928 0.70346742 0.70346742 5.7766E-1029 0.70346742 0.70346742 2.0318E-1030 0.70346742 0.70346742 7.1462E-1131 0.70346742 0.70346742 2.5125E-1132 0.70346742 0.70346742 8.838E-1233 0.70346742 0.70346742 3.1091E-1234 0.70346742 0.70346742 1.089E-1235 0.70346742 0.70346742 3.7877E-13

f ( x )=x2−e−x f ( x )=x2−e−x=0g( x )=√e−x

36 0.70346742 0.70346742 037 0.70346742 0.70346742 038 0.70346742 0.70346742 039 0.70346742 0.70346742 040 0.70346742 0.70346742 041 0.70346742 0.70346742 042 0.70346742 0.70346742 043 0.70346742 0.70346742 044 0.70346742 0.70346742 045 0.70346742 0.70346742 046 0.70346742 0.70346742 047 0.70346742 0.70346742 048 0.70346742 0.70346742 049 0.70346742 0.70346742 050 0.70346742 0.70346742 051 0.70346742 0.70346742 052 0.70346742 0.70346742 053 0.70346742 0.70346742 054 0.70346742 0.70346742 055 0.70346742 0.70346742 056 0.70346742 0.70346742 057 0.70346742 0.70346742 058 0.70346742 0.70346742 059 0.70346742 0.70346742 060 0.70346742 0.70346742 061 0.70346742 0.70346742 062 0.70346742 0.70346742 0 -4 -3 -2 -1 0 1 2 3 4 5 6

-5

-4

-3

-2

-1

0

1

2

3

4

5

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

x

y

raíz

f(x)g(x)

punto fijo



MÉTODOS NUMÉRICOS


tabulación para grafica a g(x) y f(x)

x G(X)-3 4.48168907 -11.0855369

-2.9 4.26311452 -9.76414537-2.8 4.05519997 -8.60464677-2.7 3.85742553 -7.58973172-2.6 3.66929667 -6.70373804-2.5 3.49034296 -5.93249396-2.4 3.32011692 -5.26317638-2.3 3.15819291 -4.68418245-2.2 3.00416602 -4.1850135

-2.1 2.85765112 -3.75616991-2 2.71828183 -3.3890561

-1.9 2.58570966 -3.07589444-1.8 2.45960311 -2.80964746-1.7 2.33964685 -2.58394739-1.6 2.22554093 -2.39303242-1.5 2.11700002 -2.23168907-1.4 2.01375271 -2.09519997-1.3 1.91554083 -1.97929667-1.2 1.8221188 -1.88011692-1.1 1.73325302 -1.79416602

-1 1.64872127 -1.71828183-0.9 1.56831219 -1.64960311-0.8 1.4918247 -1.58554093-0.7 1.41906755 -1.52375271-0.6 1.34985881 -1.4621188-0.5 1.28402542 -1.39872127-0.4 1.22140276 -1.3318247-0.3 1.16183424 -1.25985881-0.2 1.10517092 -1.18140276-0.1 1.0512711 -1.09517092

1.5266E-15 1 -10.1 0.95122942 -0.894837420.2 0.90483742 -0.778730750.3 0.86070798 -0.650818220.4 0.81873075 -0.51032005

f ( x )=x2−e−x=0 x2=e−x x=√e−x

0.5 0.77880078 -0.356530660.6 0.74081822 -0.188811640.7 0.70468809 -0.00658530.8 0.67032005 0.190671040.9 0.63762815 0.40343034

1 0.60653066 0.632120561.1 0.57694981 0.877128921.2 0.54881164 1.138805791.3 0.52204578 1.417468211.4 0.4965853 1.713403041.5 0.47236655 2.026869841.6 0.44932896 2.358103481.7 0.42741493 2.707316481.8 0.40656966 3.074701111.9 0.38674102 3.46043138

2 0.36787944 3.864664722.1 0.34993775 4.287543572.2 0.33287108 4.729196842.3 0.31663677 5.189741162.4 0.30119421 5.669282052.5 0.2865048 6.1679152.6 0.27253179 6.685726422.7 0.25924026 7.222794492.8 0.24659696 7.779189942.9 0.23457029 8.35497678

3 0.22313016 8.950212933.1 0.21224797 9.56495083.2 0.20189652 10.19923783.3 0.19204991 10.85311683.4 0.18268352 11.52662673.5 0.17377394 12.21980263.6 0.16529889 12.93267633.7 0.15723717 13.66527653.8 0.14956862 14.41762923.9 0.14227407 15.1897581

4 0.13533528 15.98168444.1 0.1287349 16.79342734.2 0.12245643 17.62500444.3 0.11648416 18.47643144.4 0.11080316 19.34772274.5 0.10539922 20.2388914.6 0.10025884 21.14994824.7 0.09536916 22.08090474.8 0.09071795 23.03177034.9 0.08629359 24.0025534

5 0.082085 24.9932621

-4 -3 -2 -1 0 1 2 3 4 5 6

-5

-4

-3

-2

-1

0

1

2

3

4

5


x

y

raíz

f(x)g(x)

punto fijo

ruta de convergencia ruta de convergencia: claves raiz y punto fijo

X Y-2 0 Xo 0 0.70346742-2 2.71828183 Xo X1 0.70346742

2.71828183 2.71828183 x1 x12.71828183 0.25688137 x1 x20.25688137 0.25688137 x2 x20.25688137 0.87946573 x2 x30.87946573 0.879465730.87946573 0.644208490.64420849 0.64420849 Recuérdese que X1=g(Xo), X2=g(X1)

0.64420849 0.724622650.72462265 0.724622650.72462265 0.696065630.69606563 0.696065630.69606563 0.706075710.70607571 0.706075710.70607571 0.7025506

0.7025506 0.70255060.7025506 0.70378997

0.70378997 0.703789970.70378997 0.70335398

y en general Xi+1=g(xi)

-4 -3 -2 -1 0 1 2 3 4 5 6

-5

-4

-3

-2

-1

0

1

2

3

4

5


x

y

raíz

f(x)g(x)

punto fijo

raiz y punto fijo

0.703467420



MÉTODOS NUMÉRICOS


MÉTODO DE NEWTON RAPHSON

K Xi Xi+1 error1 -2.5 -5.93249396 7.18249396 -1.67403426 49.33983522 -1.67403426 -2.53125104 1.98557322 -0.39921295 319.3336523 -0.39921295 -1.33128004 0.69222512 1.5239765 126.195484 1.5239765 2.10466045 3.2657969 0.87952109 73.27344475 0.87952109 0.35857575 2.17402379 0.71458465 23.08144246 0.71458465 0.02123588 1.91856464 0.70351602 1.573330167 0.70351602 9.2425E-05 1.90187441 0.70346742 0.006908158 0.70346742 1.7773E-09 1.90180126 0.70346742 1.3285E-079 0.70346742 0 1.90180126 0.70346742 0

10 0.70346742 0 1.90180126 0.70346742 011 0.70346742 0 1.90180126 0.70346742 012 0.70346742 0 1.90180126 0.70346742 013 0.70346742 0 1.90180126 0.70346742 0

-4 -3 -2 -1 0 1 2 3 4 5 6

-15

-10

-5

0

5

10

15

20

25

30

TRAYECTORIA DE CONVERGENCIA METODO NEWTON RAPHSON. f(x)=x2-e-x

x

y

f ( x )=x2−e−x

-4 -3 -2 -1 0 1 2 3 4 5 6

-15

-10

-5

0

5

10

15

20

25

30


x

y



MÉTODOS NUMÉRICOS


tabulación gráfica función trayectoria de convergencia trayectoria de convergencia: claves

x f(x) X Y -3 -11.0855369 5 0 xo

-2.9 -9.76414537 5 24.9932621 xo-2.8 -8.60464677 2.50235669 0 x1-2.7 -7.58973172 2.50235669 6.17989724 x1-2.6 -6.70373804 1.28742118 0 x2-2.5 -5.93249396 1.28742118 1.38147173 x2-2.4 -5.26317638 0.80283435 0-2.3 -4.68418245 0.80283435 0.19648579-2.2 -4.1850135 0.70716151 0 Recuerde que-2.1 -3.75616991 0.70716151 0.0070357

-2 -3.3890561 0.70347281 0-1.9 -3.07589444 0.70347281 1.0248E-05-1.8 -2.80964746 0.70346742 0 En general: -1.7 -2.58394739 0.70346742 2.1852E-11-1.6 -2.39303242 0.70346742 0-1.5 -2.23168907 0.70346742 0-1.4 -2.09519997 0.70346742 0-1.3 -1.97929667 0.70346742 0-1.2 -1.88011692-1.1 -1.79416602

-1 -1.71828183-0.9 -1.64960311-0.8 -1.58554093-0.7 -1.52375271-0.6 -1.4621188-0.5 -1.39872127-0.4 -1.3318247-0.3 -1.25985881-0.2 -1.18140276-0.1 -1.09517092

1.5266E-15 -10.1 -0.894837420.2 -0.778730750.3 -0.650818220.4 -0.510320050.5 -0.356530660.6 -0.18881164

-4 -3 -2 -1 0 1 2 3 4 5 6

-15

-10

-5

0

5

10

15

20

25

30


x

y

0.7 -0.00658530.8 0.190671040.9 0.40343034

1 0.632120561.1 0.877128921.2 1.138805791.3 1.417468211.4 1.713403041.5 2.026869841.6 2.358103481.7 2.707316481.8 3.074701111.9 3.46043138

2 3.864664722.1 4.287543572.2 4.729196842.3 5.189741162.4 5.669282052.5 6.1679152.6 6.685726422.7 7.222794492.8 7.779189942.9 8.35497678

3 8.950212933.1 9.56495083.2 10.19923783.3 10.85311683.4 11.52662673.5 12.21980263.6 12.93267633.7 13.66527653.8 14.41762923.9 15.1897581

4 15.98168444.1 16.79342734.2 17.62500444.3 18.47643144.4 19.34772274.5 20.2388914.6 21.14994824.7 22.08090474.8 23.03177034.9 24.0025534

5 24.9932621

-4 -3 -2 -1 0 1 2 3 4 5 6

-15

-10

-5

0

5

10

15

20

25

30


x

y

trayectoria de convergencia: claves

0f(xo)

0f(x1)

0f(x2)

Recuerde que x1= x0−f (x0)f ' (x0 )

x2=x1−f ( x1 )f ' (x1)

x i+1=x i−f ( xi )f ' (x i )



MÉTODOS NUMÉRICOSTRAYECTORIA DE CONVERGENCIA DE LOS PRINCIPALES MÉTODOS PARA RESOLVER ECUACIONES NO LINEALES

MÉTODO DE LA SECANTE

K Xi-1 Xi Xi+1 f(Xi-1) f(Xi) error1 -2 2.5 -0.40422742 -3.3890561 6.1679152 2.5 -0.40422742 0.11244301 6.167915 -1.334744813 -0.40422742 0.11244301 1.11563688 -1.33474481 -0.881004844 0.11244301 1.11563688 0.60401486 -0.88100484 0.916939145 1.11563688 0.60401486 0.68866078 0.91693914 -0.181778696 0.60401486 0.68866078 0.70406956 -0.18177869 -0.027994567 0.68866078 0.70406956 0.70346388 -0.02799456 0.001145428 0.70406956 0.70346388 0.70346742 0.00114542 -6.7369E-069 0.70346388 0.70346742 0.70346742 -6.7369E-06 -1.605E-09

10 0.70346742 0.70346742 0.70346742 -1.605E-09 2.1094E-1511 0.70346742 0.70346742 0.70346742 2.1094E-15 012 0.70346742 0.70346742 #DIV/0! 0 013 0.70346742 #DIV/0! #DIV/0! 0 #DIV/0!

-4 -3 -2 -1 0 1 2 3 4 5 6

-15

-10

-5

0

5

10

15

20

25

30

TRAYECTORIA DE CONVERGENCIA METODO DE LA SECANTE. f(x)=x2-e-x

x

y

-4 -3 -2 -1 0 1 2 3 4 5 6

-15

-10

-5

0

5

10

15

20

25

30


x

y



MÉTODOS NUMÉRICOSTRAYECTORIA DE CONVERGENCIA DE LOS PRINCIPALES MÉTODOS PARA RESOLVER ECUACIONES NO LINEALES

tabulación para graficar la función ruta de convergencia

x f(x)-3 -11.0855369 -3 0 xo 0

-2.9 -9.76414537 -3 -11.0855369 x0 f(xo)-2.8 -8.60464677 5 24.9932621 x1 f(x1)-2.7 -7.58973172 5 0 x1 0-2.6 -6.70373804 -0.54192773 0 x2 0-2.5 -5.93249396 -0.54192773 -1.42563238 x2 f(x2)-2.4 -5.26317638 5 24.9932621 x1 f(x1)-2.3 -4.68418245 -0.24287087 0 x3 0-2.2 -4.1850135 -0.24287087 -1.21591772 x3 f(x3)-2.1 -3.75616991 -0.54192773 -1.42563238 x2 f(x2)

-2 -3.3890561 1.49104957 0 x4 0-1.9 -3.07589444 1.49104957 1.99809258 x4 f(x4)-1.8 -2.80964746 -0.24287087 -1.21591772-1.7 -2.58394739 0.41310232 0-1.6 -2.39303242 0.41310232 -0.49094106-1.5 -2.23168907 1.49104957 1.99809258-1.4 -2.09519997 0.6257184 0-1.3 -1.97929667 0.6257184 -0.14335353-1.2 -1.88011692 0.41310232 -0.49094106-1.1 -1.79416602 0.71340642 0

-1 -1.71828183 0.71340642 0.01897642-0.9 -1.64960311-0.8 -1.58554093-0.7 -1.52375271-0.6 -1.4621188-0.5 -1.39872127-0.4 -1.3318247-0.3 -1.25985881-0.2 -1.18140276-0.1 -1.09517092

1.5266E-15 -10.1 -0.894837420.2 -0.778730750.3 -0.650818220.4 -0.510320050.5 -0.356530660.6 -0.188811640.7 -0.00658530.8 0.190671040.9 0.40343034

1 0.632120561.1 0.87712892

-4 -3 -2 -1 0 1 2 3 4 5 6

-15

-10

-5

0

5

10

15

20

25

30


x

y

1.2 1.138805791.3 1.417468211.4 1.713403041.5 2.026869841.6 2.358103481.7 2.707316481.8 3.074701111.9 3.46043138

2 3.864664722.1 4.287543572.2 4.729196842.3 5.189741162.4 5.669282052.5 6.1679152.6 6.685726422.7 7.222794492.8 7.779189942.9 8.35497678

3 8.950212933.1 9.56495083.2 10.19923783.3 10.85311683.4 11.52662673.5 12.21980263.6 12.93267633.7 13.66527653.8 14.41762923.9 15.1897581

4 15.98168444.1 16.79342734.2 17.62500444.3 18.47643144.4 19.34772274.5 20.2388914.6 21.14994824.7 22.08090474.8 23.03177034.9 24.0025534

5 24.9932621

-4 -3 -2 -1 0 1 2 3 4 5 6

-15

-10

-5

0

5

10

15

20

25

30


x

y



MÉTODOS NUMÉRICOS


MÉTODO DE REGLA FALSA

K A B XR f(A) f(B) f(XR)-2 2.5 -0.40422742 -3.3890561 6.167915 -1.33474481

-0.40422742 2.5 0.11244301 -1.33474481 6.167915 -0.881004840.11244301 2.5 0.41085033 -0.88100484 6.167915 -0.494288170.41085033 2.5 0.5658504 -0.49428817 6.167915 -0.24769034

0.5658504 2.5 0.64052304 -0.24769034 6.167915 -0.116746940.64052304 2.5 0.67506559 -0.11674694 6.167915 -0.053409480.67506559 2.5 0.69073248 -0.05340948 6.167915 -0.024097460.69073248 2.5 0.6977736 -0.02409746 6.167915 -0.01080413

0.6977736 2.5 0.70092498 -0.01080413 6.167915 -0.004830350.70092498 2.5 0.70233281 -0.00483035 6.167915 -0.002156840.70233281 2.5 0.70296121 -0.00215684 6.167915 -0.000962520.70296121 2.5 0.7032416 -0.00096252 6.167915 -0.00042943

0.7032416 2.5 0.70336669 -0.00042943 6.167915 -0.000191570.70336669 2.5 0.70342249 -0.00019157 6.167915 -8.5455E-050.70342249 2.5 0.70344738 -8.5455E-05 6.167915 -3.8119E-050.70344738 2.5 0.70345848 -3.8119E-05 6.167915 -1.7004E-050.70345848 2.5 0.70346343 -1.7004E-05 6.167915 -7.5846E-060.70346343 2.5 0.70346564 -7.5846E-06 6.167915 -3.3832E-060.70346564 2.5 0.70346663 -3.3832E-06 6.167915 -1.5091E-060.70346663 2.5 0.70346707 -1.5091E-06 6.167915 -6.7316E-07

-3 -2 -1 0 1 2 3

-4

-2

0

2

4

6

8

TRAYECTORIA DE CONVERGENCIA METODO REGLA FALSA. f(x)=x2-e-x

x

y

-3 -2 -1 0 1 2 3

-4

-2

0

2

4

6

8


x

y



MÉTODOS NUMÉRICOS


TABULACIÓN PARA GRAFICAR A f(x) TRAYECTORIA DE CONVERGENCIA

x f(x) Xf(A)*f(XR) error -2 -3.3890561 2.54.52352503 -1.9 -3.07589444 2.51.17591664 459.495369 -1.8 -2.80964746 -20.43547028 72.6316366 -1.7 -2.58394739 -20.12243041 27.3924119 -1.6 -2.39303242 -0.404227420.02891709 11.658072 -1.5 -2.23168907 -0.404227420.00623539 5.11691737 -1.4 -2.09519997 2.50.00128703 2.26815574 -1.3 -1.97929667 0.112443010.00026035 1.00908475 -1.2 -1.88011692 0.112443015.2188E-05 0.44960293 -1.1 -1.79416602 2.51.0418E-05 0.20045031 -1 -1.71828183 0.41085033

2.076E-06 0.08939331 -0.9 -1.64960311 0.410850334.1334E-07 0.03987095 -0.8 -1.58554093 2.58.2266E-08 0.0177841 -0.7 -1.52375271 0.56585041.6371E-08 0.00793264 -0.6 -1.4621188 0.56585043.2575E-09 0.00353841 -0.5 -1.39872127 2.56.4816E-10 0.00157834 -0.4 -1.3318247 0.640523041.2897E-10 0.00070404 -0.3 -1.25985881 0.64052304

2.566E-11 0.00031404 -0.2 -1.18140276 2.55.1057E-12 0.00014008 -0.1 -1.09517092 0.675065591.0159E-12 6.2485E-05 6.3838E-16 -1 0.67506559

0.1 -0.89483742 2.50.2 -0.77873075 0.690732480.3 -0.65081822 0.690732480.4 -0.51032005 2.50.5 -0.35653066 0.69777360.6 -0.18881164 0.69777360.7 -0.0065853 2.50.8 0.19067104 0.700924980.9 0.40343034 0.70092498

1 0.63212056 2.51.1 0.87712892 0.702332811.2 1.13880579 0.702332811.3 1.41746821 2.51.4 1.71340304 0.702961211.5 2.02686984 0.702961211.6 2.35810348 2.51.7 2.70731648 0.70324161.8 3.07470111 0.70324161.9 3.46043138 2.5

2 3.86466472 0.703366692.1 4.28754357 0.70336669

-3 -2 -1 0 1 2 3

-4

-2

0

2

4

6

8


x

y

2.2 4.72919684 2.52.3 5.18974116 0.703422492.4 5.66928205 0.703422492.5 6.167915 2.5

0.703447380.70344738

2.50.703458480.70345848

-3 -2 -1 0 1 2 3

-4

-2

0

2

4

6

8


x

y

TRAYECTORIA DE CONVERGENCIA

Y0 a 0

6.167915 a f(a)-3.3890561 b f(b)

0 b 00 xr 0

-1.33474481 xr f(xr)6.167915 si(f(a)*f(xr)<0,b,a) f(P13)

0 xr 0-0.88100484 xr f(xr)

6.167915 SI(Q12*Q15>0,P13,P12) f(P16)0 xr 0

-0.49428817 xr f(xr)6.167915

0-0.24769034

6.1679150

-0.116746946.167915

0-0.05340948

6.1679150

-0.024097466.167915

0-0.01080413

6.1679150

-0.004830356.167915

0-0.00215684

6.1679150

-0.000962526.167915

0-0.00042943

6.1679150

-0.00019157

6.1679150

-8.5455E-056.167915

0-3.8119E-05

6.1679150

-1.7004E-05

ruta de convergencia solucion numerica de ecuaciones no lineales

Documents

trayectoria de convergencia

ruta de convergencia

xi 1

7 div 0

ecuaciones

claves

tray

lineales