EJERCICIOS METODOS NUMÉRICOS UNIDAD I

MÉTODOS CERRADOS
1.     Determine las raíces reales de las siguientes funciones: a)     Gráficamente.

f(x) = −0.5x2 + 2.5x + 4.5 [5,10]

b)     Analíticamente

Empleando la fórmula cuadrática para obtener sus raices

b  b 2  4ac
x
2a

  2.5    2.5   4  0.5   4.5 

2

x
2  0.5 
x1  1.405124838 , x2  6.405124838

c)     Usando el método de bisección y de la regla falsa y un criterio de detención de 1%

METODO DE BISECCIÓN

f(x) = −0.5x2 + 2.5x + 4.5

xr 
 xa  xb 
Raices verdaderas en el intervalo
A -1.40512484
f(a)
f(b)
4.50
-20.50
2
B 6.40512484
C 2.5 Teorema de bolzano f(a).f(b) -92.25 < 0 cumple

a 5 Iteracción Xa Xb xr f(xa)
b 10 0 5 10 7.5 4.5
1 5 7.5 6.25 4.5
2 6.25 7.5 6.875 0.59375
3 6.25 6.875 6.5625 0.59375
4 6.25 6.5625 6.40625 0.59375
5 6.25 6.40625 6.328125 0.59375
6 6.328125 6.40625 6.3671875 0.29772949
7 6.3671875 6.40625 6.38671875 0.14743042
8 6.38671875 6.40625 6.39648438 0.07170868
9 6.39648438 6.40625 6.40136719 0.03370476
10 6.40136719 6.40625 6.40380859 0.01466703
11 6.40380859 6.40625 6.4050293 0.00513923
 12 6.4050293 6.40625 6.40563965 0.0003731
13 6.4050293 6.40563965 6.40533447 0.0003731
14 6.4050293 6.40533447 6.40518188 0.0003731

METODO DE LA REGLA FALSA

f(x) = −0.5x2 + 2.5x + 4.5 [5,10]

Raices verdaderas en el intervalo
Entre los puntos
xr 
 xa  xb 
XL 5 A
Xu 10 2 B
C
f(Xl) 4.50 f  xb   xb  xa 
xr  xb 
f(Xu) -20.50 f  xb   f  xa 
Teorema de bolzano f(a).f(b) -92.25 < 0 cumple

Iteracción Xl Xu f(xl) f(xu) xr f(xr) f(xl)f(xr)

0 5 10 4.5 -20.5 5.9 1.85 8.3025
1 5.9 10 1.845 -20.5 6.23853211 0.63668883 1.17469089
2 6.23853211 10 0.6366888309 -20.5 6.35183673 0.20667688 0.13158886
3 6.35183673 10 0.2066768846 -20.5 6.38824968 0.06575723 0.0135905
4 6.38824968 10 0.065757227 -20.5 6.39979794 0.02078803 0.00136696
5 6.39979794 10 0.0207880325 -20.5 6.40344502 0.00655848 0.00013634
6 6.40344502 10 0.0065584761 -20.5 6.40459529 0.00206783 1.35618E-05
7 6.40459529 10 0.0020678291 -20.5 6.40495792 0.00065184 1.34789E-06
8 6.40495792 10 0.0006518366 -20.5 6.40507222 0.00020546 1.33929E-07
9 6.40507222 10 0.0002054638 -20.5 6.40510825 6.47624E-05 1.33063E-08
10 6.40510825 10 6.476241E-05 -20.5 6.40511961 2.04131E-05 1.322E-09
11 6.40511961 10 2.041306E-05 -20.5 6.40512319 6.43417E-06 1.31341E-10
12 6.40512319 10 6.434165E-06 -20.5 6.40512432 2.02804E-06 1.30487E-11
13 6.40512432 10 2.028038E-06 -20.5 6.40512467 6.39234E-07 1.29639E-12
14 6.40512467 10 6.392341E-07 -20.5 6.40512479 2.01485E-07 1.28796E-13
ODOS NUMÉRICOS UNIDAD I

xa  xb 
2

f(xr) f(xa)f(xr) Ea(%) Et(%)

-4.875 -21.9375 17.0937365
0.59375 2.671875 20 2.42188625
-1.9453125 -1.155029297 9.09090909 7.33592512
-0.62695313 -0.372253418 4.76190476 2.45701943
-0.00439453 -0.002609253 2.43902439 0.01756659
0.29772949 0.176776886 1.2345679 1.20215983
0.14743042 0.0438943841 0.61349693 0.59229662
0.07170868 0.0105720407 0.3058104 0.28736502
0.03370476 0.0024169237 0.15267176 0.13489921
0.01466703 0.0004943488 0.07627765 0.05866631
0.00513923 7.537729E-05 0.03812429 0.02054986
0.0003731 1.917423E-06 0.01905851 0.00149163
 -0.00201053 -7.5012E-07 0.00952835 0.00803748
-0.00081867 -3.05443E-07 0.0047644 0.00327292
-0.00022278 -8.31169E-08 0.00238226 0.00089064

aices verdaderas en el intervalo

-1.40512484
6.40512484
2.5

Ea(%) Et(%)
7.8862606231
5.42647059 2.6009286638
1.78380883 0.8319604162
0.56999872 0.2634634294
0.18044724 0.0831662604
0.05695509 0.0262261073
0.01795995 0.0082676403
0.00566172 0.0026060666
0.00178464 0.0008214396
0.00056252 0.0002589175
0.00017731 8.161046E-05
5.5887E-05 2.572348E-05
1.76155E-05 8.107997E-06
5.55237E-06 2.555626E-06
1.7501E-06 8.055286E-07
f (x) = 5x3 − 5x2 + 6x – 2 [0,1]
a) Gráficamente
b) Analiticamente

5 x3  5 x 2  6 x  2  0
ax 3  bx 2  cx  d  0
x1  0.4181006
x2  0.2909496
x3  0.2909496

c)     Usando el método de bisección y de la regla falsa y un criterio de detención de 1%

METODO DE BISECCIÓN
f (x) = 5x3 − 5x2 + 6x – 2
Raices verdaderas en el intervalo
A 0.41810062 f(a) -2
xr 
 xa  xb 
f(b) 4
2
a 0 Teorema de bolzano f(a).f(b) -8 < 0 cumple
b 1
Ieraciones Xa Xb Xr f(Xa)
0 0 1 0.5 -2
1 0 0.5 0.25 -2
2 0.25 0.5 0.375 -0.734375
3 0.375 0.5 0.4375 -0.18945313
4 0.375 0.4375 0.40625 -0.18945313
5 0.40625 0.4375 0.421875 -0.05245972
6 0.40625 0.421875 0.4140625 -0.05245972
7 0.4140625 0.421875 0.41796875 -0.01791334
8 0.41796875 0.421875 0.41992188 -0.00058562
9 0.41796875 0.41992188 0.41894531 -0.00058562
10 0.41796875 0.41894531 0.41845703 -0.00058562
11 0.41796875 0.41845703 0.41821289 -0.00058562
12 0.41796875 0.41821289 0.41809082 -0.00058562
13 0.41809082 0.41821289 0.41815186 -4.3509E-05
14 0.41809082 0.41815186 0.41812134 -4.3509E-05
15 0.41809082 0.41812134 0.41810608 -4.3509E-05
16 0.41809082 0.41810608 0.41809845 -4.3509E-05

METODO DE LA REGLA FALSA

f (x) = 5x3 − 5x2 + 6x – 2
Entre los puntos
xr 
 xa  xb  Raices verdaderas en el intervalo
XL 0 2 A 0.41810062
Xu 1
f  xb   xb  xa 
f(Xl) -2 xr  xb 
f(Xu) 4 f  xb   f  xa 

Teorema de bolzano f(a).f(b) -8 < 0

Iteracción Xl Xu f(xl) f(xu) Xr f(xr) f(xl)f(xr)

0 0 1 -2 4 0.66666667 1.25925926 -2.51851852
1 0 0.66666667 -2 1.25925926 0.25757576 -0.70082714 1.40165428
2 0.25757576 0.66666667 -0.70082714 1.25925926 0.52039658 0.47296622 -0.33146757
3 0.25757576 0.52039658 -0.70082714 0.47296622 0.36347631 -0.23961385 0.16792789
4 0.36347631 0.52039658 -0.23961385 0.47296622 0.4676302 0.22369338 -0.05360003
5 0.36347631 0.4676302 -0.23961385 0.22369338 0.41376375 -0.019237 0.00460945
6 0.41376375 0.4676302 -0.019237 0.22369338 0.46336466 0.20409167 -0.00392611
7 0.41376375 0.46336466 -0.019237 0.20409167 0.45909216 0.18452909 -0.00354979
8 0.41376375 0.45909216 -0.019237 0.18452909 0.45481283 0.1650043 -0.00317419
9 0.41376375 0.45481283 -0.019237 0.1650043 0.45052681 0.1455159 -0.00279929
10 0.41376375 0.45052681 -0.019237 0.1455159 0.44623426 0.12606249 -0.00242506
11 0.41376375 0.44623426 -0.019237 0.12606249 0.44193531 0.10664265 -0.00205148
12 0.41376375 0.44193531 -0.019237 0.10664265 0.43763011 0.08725496 -0.00167852
13 0.41376375 0.43763011 -0.019237 0.08725496 0.43331883 0.06789794 -0.00130615
14 0.41376375 0.43331883 -0.019237 0.06789794 0.42900161 0.04857013 -0.00093434
15 0.41376375 0.42900161 -0.019237 0.04857013 0.4246786 0.02927003 -0.00056307
16 0.41376375 0.4246786 -0.019237 0.02927003 0.42034997 0.00999614 -0.0001923
17 0.41376375 0.42034997 -0.019237 0.00999614 0.41601588 -0.00925307 0.000178
18 0.41601588 0.42034997 -0.00925307 0.00999614 0.41826658 0.0007371 -6.8204E-06
19 0.41601588 0.41826658 -0.00925307 0.0007371 0.41618194 -0.0085164 7.88029E-05
20 0.41618194 0.41826658 -0.0085164 0.0007371 0.416348 -0.0077797 6.62551E-05
21 0.416348 0.41826658 -0.0077797 0.0007371 0.41651405 -0.00704297 5.47922E-05
22 0.41651405 0.41826658 -0.00704297 0.0007371 0.41668008 -0.0063062 4.44144E-05
23 0.41668008 0.41826658 -0.0063062 0.0007371 0.41684611 -0.0055694 3.51217E-05
24 0.41684611 0.41826658 -0.0055694 0.0007371 0.41701214 -0.00483256 2.69145E-05
25 0.41701214 0.41826658 -0.00483256 0.0007371 0.41717815 -0.00409569 1.97927E-05
26 0.41717815 0.41826658 -0.00409569 0.0007371 0.41734416 -0.00335878 1.37565E-05
27 0.41734416 0.41826658 -0.00335878 0.0007371 0.41751016 -0.00262184 8.80621E-06
28 0.41751016 0.41826658 -0.00262184 0.0007371 0.41767615 -0.00188487 4.94183E-06
29 0.41767615 0.41826658 -0.00188487 0.0007371 0.41784214 -0.00114786 2.16356E-06
30 0.41784214 0.41826658 -0.00114786 0.0007371 0.41800811 -0.00041081 4.71556E-07
31 0.41800811 0.41826658 -0.00041081 0.0007371 0.41817408 0.00032627 -1.3403E-07
32 0.41800811 0.41817408 -0.00041081 0.00032627 0.41808158 -8.4556E-05 3.47368E-08
33 0.41808158 0.41817408 -8.4556E-05 0.00032627 0.41815504 0.00024171 -2.0438E-08
34 0.41808158 0.41815504 -8.4556E-05 0.00024171 0.418136 0.00015715 -1.3288E-08
35 0.41808158 0.418136 -8.4556E-05 0.00015715 0.41811696 7.25943E-05 -6.1383E-09
36 0.41808158 0.41811696 -8.4556E-05 7.25943E-05 0.41809792 -1.1962E-05 1.01147E-09
37 0.41809792 0.41811696 -1.1962E-05 7.25943E-05 0.41811427 6.0632E-05 -7.2529E-10
38 0.41809792 0.41811427 -1.1962E-05 6.0632E-05 0.41811158 4.86698E-05 -5.822E-10
39 0.41809792 0.41811158 -1.1962E-05 4.86698E-05 0.41810888 3.67076E-05 -4.391E-10
40 0.41809792 0.41810888 -1.1962E-05 3.67076E-05 0.41810619 2.47454E-05 -2.9601E-10
41 0.41809792 0.41810619 -1.1962E-05 2.47454E-05 0.4181035 1.27833E-05 -1.5292E-10
42 0.41809792 0.4181035 -1.1962E-05 1.27833E-05 0.4181008 8.21095E-07 -9.8221E-12
43 0.41809792 0.4181008 -1.1962E-05 8.21095E-07 0.41809811 -1.1141E-05 1.33271E-10
44 0.41809811 0.4181008 -1.1141E-05 8.21095E-07 0.41809829 -1.032E-05 1.14975E-10
45 0.41809829 0.4181008 -1.032E-05 8.21095E-07 0.41809848 -9.4989E-06 9.80282E-11
46 0.41809848 0.4181008 -9.4989E-06 8.21095E-07 0.41809866 -8.6778E-06 8.24292E-11
47 0.41809866 0.4181008 -8.6778E-06 8.21095E-07 0.41809885 -7.8567E-06 6.81787E-11
48 0.41809885 0.4181008 -7.8567E-06 8.21095E-07 0.41809903 -7.0356E-06 5.52765E-11
49 0.41809903 0.4181008 -7.0356E-06 8.21095E-07 0.41809922 -6.2145E-06 4.37227E-11
xa  xb 
2

f(Xr) f(Xa)*f(Xb) Ea% Et%

0.375 -0.75 19.5884386
-0.734375 1.46875 100 40.2057807
-0.18945313 0.13912964 33.3333333 10.308671
0.08666992 -0.01641989 14.2857143 4.6398838
-0.05245972 0.00993866 7.69230769 2.83439362
0.01678085 -0.00088032 3.7037037 0.90274509
-0.01791334 0.00093973 1.88679245 0.96582426
-0.00058562 1.04903E-05 0.93457944 0.03153959
0.00809266 -4.7392E-06 0.46511628 0.43560275
0.0037523 -2.1974E-06 0.23310023 0.20203158
0.00158304 -9.2705E-07 0.11668611 0.085246
0.00049864 -2.9201E-07 0.05837712 0.02685321
-4.3509E-05 2.54797E-08 0.02919708 0.00234319
0.00022756 -9.9009E-09 0.01459641 0.01225501
9.20233E-05 -4.0039E-09 0.00729874 0.00495591
2.42567E-05 -1.0554E-09 0.0036495 0.00130636
-9.6263E-06 4.18834E-10 0.00182478 0.00051841
as en el intervalo

Ea(%) Et(%)
59.4512504
158.823529 38.3938351
50.5039486 24.4668275
43.1720742 13.0648709
22.2727033 11.8463304
13.0186478 1.03727786
10.7045081 10.8261123
0.93064198 9.80422802
0.94089933 8.78071105
0.95133414 7.75559528
0.96195114 6.72891526
0.97275523 5.70070611
0.98375149 4.67100355
0.99494524 3.63984386
1.00634199 2.60726388
1.01794749 1.57330101
1.02976774 0.53799319
1.04180898 0.49862111
0.53810156 0.03969404
0.50089555 0.45890288
0.03988355 0.41918651
0.03986578 0.37947201
0.03984803 0.33975937
0.03983028 0.3000486
0.03981255 0.26033969
0.03979483 0.22063266
0.03977712 0.18092751
0.03975943 0.14122423
0.03974175 0.10152283
0.03972408 0.0618233
0.03970642 0.02212567
0.03968878 0.01757009
 0.02212555 0.00455446
0.01756855 0.01301638
0.00455334 0.00846265
0.00455358 0.0039089
0.00455381 0.00064488
0.00390943 0.00326468
0.0006442 0.00262046
0.00064421 0.00197623
0.00064421 0.00133201
0.00064422 0.00068779
0.00064422 4.35633E-05
0.00064423 0.00060066
4.42204E-05 0.00055644
4.42204E-05 0.00051222
4.42204E-05 0.000468
4.42204E-05 0.00042378
4.42204E-05 0.00037956
4.42203E-05 0.00033534
f(x) = −25+82x−90x2+44x3−8x4 + 0.7x5 [0.5,1.0] a) Gráficamente

b) Analiticamente
f  0.5 
25  82  0.5   90  0.5   44  0.5   8  0.5   0.7  0.5 
2 3 4 5

1.478125
f  1.0 
25  82  1.0   90  1.0   44  1.0   8  1.0   0.7  1.0 
2 3 4 5

3.7
c)     Usando el método de bisección y de la regla falsa y un criterio de detención de 1%
METODO DE BISECCIÓN
f(x) = −25+82x−90x2+44x3−8x4 + 0.7x5
A 0.57940934
xr 
 xa  xb 
f(a) -1.478125
a 0.5 f(b) 3.7
2
b 1
Teorema de bolzano f(a).f(b) -5.4690625 < 0 cumple

Iteracción Xa Xb xr f(xa) f(xr) f(xa)f(xr) Ea(%)

0 0.5 1 0.75 -1.478125 2.07236328 -3.063211975
1 0.5 0.75 0.625 -1.478125 0.68199158 -1.0080688 20
2 0.5 0.625 0.5625 -1.478125 -0.28199167 0.4168189409 11.1111111
3 0.5625 0.625 0.59375 -0.28199167 0.22645251 -0.063857722 5.26315789
4 0.5625 0.59375 0.578125 -0.28199167 -0.02084103 0.005876997 2.7027027
5 0.578125 0.59375 0.5859375 -0.02084103 0.10449812 -0.002177848 1.33333333
6 0.578125 0.5859375 0.58203125 -0.02084103 0.04225659 -0.000880671 0.67114094
7 0.578125 0.58203125 0.58007813 -0.02084103 0.01081541 -0.000225404 0.33670034
8 0.578125 0.58007813 0.57910156 -0.02084103 -0.00498582 0.0001039097 0.16863406
9 0.57910156 0.58007813 0.57958984 -0.00498582 0.00292153 -1.45662E-05 0.084246
10 0.57910156 0.57958984 0.5793457 -0.00498582 -0.00103046 5.137704E-06 0.04214075
11 0.5793457 0.57958984 0.57946777 -0.00103046 0.00094595 -9.74771E-07 0.02106594
12 0.5793457 0.57946777 0.57940674 -0.00103046 -4.2148E-05 4.343241E-08 0.01053408
13 0.57940674 0.57946777 0.57943726 -4.2148E-05 0.00045193 -1.90481E-08 0.00526676
14 0.57940674 0.57943726 0.579422 -4.2148E-05 0.0002049 -8.6361E-09 0.00263345

METODO DE LA REGLA FALSA

f(x) = −25+82x−90x2+44x3−8x4 + 0.7x5

Entre los puntos Raiz verdadera

XL 0.5 xr
 x  xb 
 a A 0.579409341
2
 xr 
 xa  xb 
Xu 1 2
f  xb   xb  xa 
f(Xl) -1.478125 xr  xb 
f(Xu) 3.7 f  xb   f  xa 

Teorema de bolzano f(a).f(b) -5.4690625 < 0 cumple

Iteracción Xl Xu f(xl) f(xu) xr f(xr) f(xl)f(xr)

0 0.5 1 -1.478125 3.7 0.64272782 0.9187886202 -1.35808443
1 0.64272782 1 0.91878862 3.7 0.52470097 -0.974986275 -0.89580629
2 0.64272782 0.52470097 0.91878862 -0.97498627 0.58546562 0.0970243502 0.08914487
3 0.58546562 0.52470097 0.09702435 -0.97498627 0.579966 0.0090038838 0.0008736
4 0.579966 0.52470097 0.00900388 -0.97498627 0.5794603 0.0008250137 7.42833E-06
5 0.5794603 0.52470097 0.00082501 -0.97498627 0.57941401 7.550646E-05 6.22939E-08
6 0.57941401 0.52470097 7.55065E-05 -0.97498627 0.57940977 6.909721E-06 5.21729E-10
7 0.57940977 0.52470097 6.90972E-06 -0.97498627 0.57940938 6.323138E-07 4.36911E-12
8 0.57940938 0.52470097 6.32314E-07 -0.97498627 0.57940935 5.786346E-08 3.65879E-14
9 0.57940935 0.52470097 5.78635E-08 -0.97498627 0.57940934 5.295129E-09 3.06394E-16
10 0.57940934 0.52470097 5.29513E-09 -0.97498627 0.57940934 4.845541E-10 2.56578E-18
11 0.57940934 0.52470097 4.84554E-10 -0.97498627 0.57940934 4.434453E-11 2.14873E-20
12 0.57940934 0.52470097 4.43445E-11 -0.97498627 0.57940934 4.061119E-12 1.7987E-22
13 0.57940934 0.52470097 4.06112E-12 -0.97498627 0.57940934 3.699818E-13 1.50254E-24
14 0.57940934 0.52470097 3.68511E-13 -0.97498627 0.57940934 4.150152E-14 1.52938E-26
) Gráficamente

Et(%)
29.4421658
7.86847152
2.91837563
2.47504795
0.22166384
1.12669205
0.45251411
0.11542513
0.05311935
0.03115289
0.01098323
0.01008483
0.0004492
0.00481781
0.00218431
 Ea(%) Et(%)
10.928108314
22.4941164 9.4420927499
10.3788578 1.0452500177
0.94826577 0.0960732151
0.08727017 0.0087953727
0.00799033 0.000804983
0.00073123 7.374748E-05
6.69161E-05 6.831385E-06
6.12354E-06 7.078466E-07
5.60369E-07 1.474775E-07
5.12798E-08 9.619776E-08
4.69258E-09 9.150514E-08
4.29443E-10 9.10757E-08
3.93381E-11 9.103642E-08
3.564E-12 9.103285E-08
ln (x2) = 0.7 [0.5,2] a) Gráficamente

Usando el método de la bisección

ln  x   0.7
2

f  x   ln  x   0.7
2

xa  0.5 xb  2
f  xa   2.086 f  xb   0.686
 x  xa 
xr   b   1.25
 2 
f  xr   0.254
c)     Usando el método de bisección y de la regla falsa y un criterio de detención de 1%
METODO DE BISECCIÓN

ln (x2) = 0.7
a 0.5 f(a) -0.21954699
b 2 f(b) -0.21954699 B 1.41906755
Teorema de bolzano f(a).f(b) 0.04820088

Iteracción Xa Xb xr f(xa) f(xr) f(xa)f(xr) Ea(%)

0 0.5 2 1.25 -0.21954699 -0.65020696 0.14275098
1 0.5 2 1.25 -0.21954699 -0.65020696 0.14275098
2
Et(%)
f (x) = (0.8 − 0.3x)/x [1.0,3.0] a) Gráficamente

b) Analiticamente

f  x 
 0.8  0.3x 
x
para cualquier raiz real xr , f  xr   0


 0.8  0.3 x  0
x
1
ya que xr se supone que es una raiz real, entonces  
 xr 
0.8  0.3x  0
0.8
x  2.667
0.3
xr 
 xa  xb 
c)     Usando el método de bisección y de la regla falsa y un criterio de detención de 1%
METODO DE BISECCIÓN 2
f (x) = (0.8 − 0.3x)/x
a 1 f(a) 0.5 Raiz verdadera en el intervalo
b 3 f(b) -0.03333333 A 2.67
Teorema de bolzano f(a).f(b) -0.01666667 < 0 cumple
Iteracción Xa Xb xr f(xa) f(xr) f(xa)f(xr) Ea(%)
0 1 3 2 0.5 -0.03333333 -0.01666667
1 1 2 1.5 0.5 0.1 0.05 33.3333333
2 1.5 2 1.75 0.23333333 0.1 0.02333333 14.2857143
3 1.75 2 1.875 0.15714286 0.1 0.01571429 6.66666667
en el intervalo

Et(%)
25.093633
43.8202247
#DIV/0!
#VALUE!
(x está en radianes)