Problema 1. Encuentre el mínimo la función

y=2x2+16x

por cada uno de los siguientes métodos:

a) Método analítico

b) Método de la sección dorada

c) Método de Newton

d) Método de Quasi-Newton

e) Método de la secante

f) Método de interpolación cuadrática

Método Analítico

y=2x2+16x

dydx

=4 x−16x2

=0

4 x=16x2

x3=4

k x1 f1 s x3 f3 t s' x2 f2 x4 f4 error

1 1.4 -15.3485714 0.1 1.5 -15.1666667 0.618034 0.1618034 1.6618034 -15.1512755

2 1.5 -15.1666667 1.6618034 -15.1512755 0.26180339 1.92360679 -15.7182344 1.7618034 -15.289506

3 1.7618034 -15.289506 1.6618034 -15.1512755 1.5 -15.1666667 1.6 -15.12 1.12107127

4 1.6618034 -15.1512755 1.6 -15.12 1.5 -15.1666667 1.5618034 -15.123027 0.02001581

5 1.5618034 -15.123027 1.6 -15.12 1.6618034 -15.1512755 1.6236068 -15.1268008 0.02494785

6 1.6236068 -15.1268008 1.6 -15.12 1.5618034 -15.123027 1.58541019 -15.1190764 0.05109048

7 1.6 -15.12 1.58541019 -15.1190764 1.5618034 -15.123027 1.5763932 -15.119783 0.00467348

8 1.5763932 -15.119783 1.58541019 -15.1190764 1.6 -15.12 1.590983 -15.1191295 0.0043227

9 1.590983 -15.1191295 1.58541019 -15.1190764 1.5763932 -15.119783 1.58196601 -15.1192302 0.00066655

x=3√4x=1.587401

Método directo Sección DoradaEncuentre el mínimo de la función f(x)=2x^2+16/x

Métodos indirectos: Newton

Encuentre el mínimo de la función f(x)=2x^2+16/x

f(x)=2x^2+16/x f'(x)=4x-16/x^2 f''(x)=4+32/x^3

k xk f'xk f''xk error

1 1.4 -2.5632653115.661807

6

2 1.56366344 -0.2891974912.369897

426.612267

9

3 1.58704257 -0.0043027112.005422

33.0359203

7

4 1.58740097 -9.7159E-0712.000001

2 0.0451756

5 1.58740105 -4.885E-14 12 1.0201E-05

Métodos indirectos: Quasi Newtonh= 0.001k xk f(x) x+h x-h f(x-h) f(x+h) error

1 1.4 15.3485714 1.401 1.399 15.3511425 15.346016 2 1.56316364 15.1226137 1.56416364 1.56216364 15.1229153 15.1223245 1.479213333 1.58652745 15.1190572 1.58752745 1.58552745 15.1190737 15.1190527 0.021640614 1.58690078 15.1190541 1.58790078 1.58590078 15.1190661 15.1190541 9.2744E-065 1.5869011 15.1190541 1.5879011 1.5859011 15.1190661 15.1190541 1.2821E-08

Métodos indirectos: La secante

f(x)=2x^2+16/x f'(x)=4x-16/x^2

k xp f'xp xq f'xq error1 1 -12 1.4 -2.56326531

2 1.5086505

2 -0.9951934 7.20183486

3 1.5546491

1 -0.40136126 2.95877662

4 1.5738422

9 -0.16411094 1.21951051

5 1.5817989

2 -0.067464 0.5030115

6 1.5850882

8 -0.02779375 0.20751944

7 1.5864465

8 -0.01146058 0.085618758 1.5870072 -0.00472742 0.03532562

9 1.5872385

4 -0.00195032 0.0145752210 1.587334 -0.00080467 0.0060137

11 1.5873733

9 -0.000332 0.00248124

12 1.5873896

4 -0.00013698 0.00102376

13 1.5873963

4 -5.6518E-05 0.0004224

Métodos directosInterpolación Cuadráticaf(x)=2x^2+16/x

k x1 x2 x3 f1 f2 f3 x* f* error1 1.4 1.5 1.6 15.3485714 15.1666667 15.12 1.58450704 15.1191029 2 1.4 1.5 1.58450704 15.3485714 15.1666667 15.1191029 1.58358719 15.11914 0.058086433 1.4 1.5 1.58358719 15.3485714 15.1666667 15.11914 1.58353241 15.1191425 0.003459334 1.4 1.5 1.58353241 15.3485714 15.1666667 15.1191425 1.58352915 15.1191427 0.000206055 1.4 1.5 1.58352915 15.3485714 15.1666667 15.1191427 1.58352896 15.1191427 1.2273E-05

Problema 2. Determine los puntos críticos de la función

f (x1 , x2 )=27 x1−19x13−2x2

2+x24

y clasifíquelos para determinar si son máximos o mínimos relativos.

∇ f=⌊

dfd x1dfd x2

⌋= 0

dfd x1

=27−13x12

dfd x2

=−4 x2+4 x23

x1=[9 ,−9 ]

x2=[1 ,−1,0 ]

Di=⌊

∂2 f∂ x1

2=−23x1

∂2 f∂ x2∂ x1

=0

∂2 f∂ x1∂ x2

=0 ∂2 f∂ x2

2=−4+12 x22

⌋

D1=⌊−23x1 ⌋ D2=⌊

−23x1 0

0 −4+12 x22

⌋

Para x1=9 y x2=1

D1=−6D 2=−48

Para x1=9 y x2=−1

D1=−6D 2=−48

Para x1=9 y x2=0

D1=−6D 2=24 Máximo

Para x1=−9 y x2=1

D1=6D2=48 Mínimo relativo

Para x1=−9 y x2=−1

D1=6D2=48 Mínimo relativo

Para x1=−9 y x2=0

D1=6D2=−24 Máximo

Problema 3 Minimizar la función

f (x1 x2 )=x13+x23−3x1 x2

por cada uno de los siguientes métodos:

a) Método de Newton

b) Método de Gradiente.

Método del gradientef(x1,x2)=x1^3+x2^3-3x1x2 alpha= 0.05

df/dx1= 3x1^2-3x2 df/dx2= 3x2^2-3x1

k x1 x2 df/dx1 df/dx2 x1(i+1) x2(i+1) ||dF||1 1.4 1.8 0.48 5.52 1.376 1.524 5.540830262 1.376 1.524 1.108128 2.839728 1.3205936 1.3820136 3.048278663 1.3205936 1.3820136 1.08586157 1.76810397 1.26630052 1.2936084 2.074918554 1.26630052 1.2936084 0.92972583 1.22136652 1.21981423 1.23254008 1.53496785

5 1.21981423 1.23254008 0.76622004 0.89802242 1.18150323 1.18763895 1.180481866 1.18150323 1.18763895 0.62493277 0.68694917 1.15025659 1.1532915 0.928676557 1.15025659 1.1532915 0.50939618 0.53947405 1.12478678 1.12631779 0.741968148 1.12478678 1.12631779 0.41648253 0.43141497 1.10396265 1.10474704 0.599647049 1.10396265 1.10474704 0.34195949 0.34951013 1.08686468 1.08727154 0.48897201

10 1.08686468 1.08727154 0.28200988 0.28588415 1.07276419 1.07297733 0.4015710711 1.07276419 1.07297733 0.233537 0.2355485 1.06108734 1.06119991 0.3316965912 1.06108734 1.06119991 0.19411928 0.19517371 1.05138137 1.05144122 0.2752727313 1.05138137 1.05144122 0.1618847 0.1624418 1.04328714 1.04331913 0.2293338114 1.04328714 1.04331913 0.13538676 0.13568301 1.0365178 1.03653498 0.1916753815 1.0365178 1.03653498 0.1135025 0.11366089 1.03084267 1.03085193 0.1606288216 1.03084267 1.03085193 0.09535405 0.09543911 1.02607497 1.02607998 0.1349111517 1.02607497 1.02607998 0.0802496 0.08029546 1.02206249 1.02206521 0.113522518 1.02206249 1.02206521 0.06763959 0.06766438 1.01868051 1.01868199 0.0956743619 1.01868051 1.01868199 0.05708399 0.05709744 1.01582631 1.01582711 0.0807384620 1.01582631 1.01582711 0.04822794 0.04823525 1.01341491 1.01341535 0.0682097821 1.01341491 1.01341535 0.04078331 0.04078729 1.01137575 1.01137599 0.0576791222 1.01137575 1.01137599 0.03451475 0.03451692 1.00965001 1.00965014 0.0488127723 1.00965001 1.00965014 0.02922901 0.02923019 1.00818856 1.00818863 0.041336924 1.00818856 1.00818863 0.02476663 0.02476727 1.00695023 1.00695027 0.0350257625 1.00695023 1.00695027 0.02099549 0.02099584 1.00590046 1.00590048 0.0296923626 1.00590046 1.00590048 0.01780575 0.01780594 1.00501017 1.00501018 0.0251812727 1.00501017 1.00501018 0.01510577 0.01510588 1.00425488 1.00425489 0.0213628728 1.00425488 1.00425489 0.01281893 0.01281899 1.00361393 1.00361394 0.0181287529 1.00361393 1.00361394 0.01088097 0.010881 1.00306988 1.00306989 0.0153880430 1.00306988 1.00306989 0.00923792 0.00923794 1.00260799 1.00260799 0.013064431 1.00260799 1.00260799 0.00784437 0.00784438 1.00221577 1.00221577 0.0110936232 1.00221577 1.00221577 0.00666204 0.00666204 1.00188267 1.00188267 0.0094215533 1.00188267 1.00188267 0.00565864 0.00565864 1.00159974 1.00159974 0.0080025234 1.00159974 1.00159974 0.00480689 0.00480689 1.00135939 1.00135939 0.0067979635 1.00135939 1.00135939 0.00408372 0.00408372 1.00115521 1.00115521 0.0057752536 1.00115521 1.00115521 0.00346962 0.00346962 1.00098172 1.00098172 0.0049067937 1.00098172 1.00098172 0.00294807 0.00294807 1.00083432 1.00083432 0.004169238 1.00083432 1.00083432 0.00250505 0.00250505 1.00070907 1.00070907 0.0035426839 1.00070907 1.00070907 0.00212872 0.00212872 1.00060263 1.00060263 0.0030104640 1.00060263 1.00060263 0.00180899 0.00180899 1.00051218 1.00051218 0.002558341 1.00051218 1.00051218 0.00153734 0.00153734 1.00043532 1.00043532 0.0021741242 1.00043532 1.00043532 0.00130652 0.00130652 1.00036999 1.00036999 0.001847743 1.00036999 1.00036999 0.00111038 0.00111038 1.00031447 1.00031447 0.0015703244 1.00031447 1.00031447 0.00094371 0.00094371 1.00026729 1.00026729 0.0013346145 1.00026729 1.00026729 0.00080207 0.00080207 1.00022718 1.00022718 0.001134346 1.00022718 1.00022718 0.0006817 0.0006817 1.0001931 1.0001931 0.0009640747 1.0001931 1.0001931 0.0005794 0.0005794 1.00016413 1.00016413 0.000819448 1.00016413 1.00016413 0.00049246 0.00049246 1.0001395 1.0001395 0.0006964549 1.0001395 1.0001395 0.00041857 0.00041857 1.00011858 1.00011858 0.0005919550 1.00011858 1.00011858 0.00035577 0.00035577 1.00010079 1.00010079 0.0005031351 1.00010079 1.00010079 0.00030239 0.00030239 1.00008567 1.00008567 0.00042765

52 1.00008567 1.00008567 0.00025702 0.00025702 1.00007282 1.00007282 0.0003634953 1.00007282 1.00007282 0.00021846 0.00021846 1.00006189 1.00006189 0.0003089654 1.00006189 1.00006189 0.00018569 0.00018569 1.00005261 1.00005261 0.0002626155 1.00005261 1.00005261 0.00015783 0.00015783 1.00004472 1.00004472 0.0002232156 1.00004472 1.00004472 0.00013416 0.00013416 1.00003801 1.00003801 0.0001897357 1.00003801 1.00003801 0.00011403 0.00011403 1.00003231 1.00003231 0.0001612658 1.00003231 1.00003231 9.6925E-05 9.6925E-05 1.00002746 1.00002746 0.0001370759 1.00002746 1.00002746 8.2386E-05 8.2386E-05 1.00002334 1.00002334 0.0001165160 1.00002334 1.00002334 7.0027E-05 7.0027E-05 1.00001984 1.00001984 9.9033E-05

Método de Newton

f(x1,x2)=x1^3+x2^3-3x1x2

df/dx1= 3x1^2-3x2 df/dx2= 3x2^2-3x1i x df/dx1 df/dx2 H inv H-1 ||df||

1 1.4 1.8 0.48 5.52 8.4 -30.1321585

90.0367107

25.5408302

6

-3 10.80.0367107

20.1027900

1

21.133920

7 1.2149780.2123945

71.0267523

16.8035242

3 -30.1795675

80.0738974

61.0484902

3

-37.2898678

40.0738974

60.1675877

3

31.019907

11.027211

50.0389972

80.1057687

8 6.1194428 -3 0.21463010.1044722

10.1127289

8

-36.1632687

90.1044722

1 0.2131039

41.000487

21.000597

6 0.00113140.0021248

96.0029234

6 -30.2220337

70.1109505

80.0024073

3

-36.0035855

50.1109505

80.2220092

9

51.000000

31.000000

3 7.1141E-07 1.0702E-066.0000016

6 -30.2222221

20.1111110

2 1.2851E-06

-3 6.00000190.1111110

20.2222221

1

Problema 4 Obtenga el máximo valor de la función por el método simplex

Z=f (x1 , x2)=6 x1+5 x2

s . a :2x1+5 x2≤20

−5 x1−x2≤−5

−3 x1−11 x2≤−33

Convertir las desigualdades en igualdades

2 x1+5 x2+s1=20

−5 x1−x2+s2=−5

−3 x1−11 x2+s3=−33

Igualar la función objetivo a cero y después agregar las variables de holgura del sistema anterior:

Z−6 x1−5 x2=0

Tablero Inicial

Base

Variable de decisión

Variable de holgura Solución

X1 X2 S1 S2 S3

S1 2 5 1 0 0 20

S2 -5 -1 0 1 0 -5

S3 -3 -11 0 0 1 -33

Z -6 -5 0 0 0 0

Tablero Inicial

Base

Variable de decisión

Variable de holgura Solución

X1 X2 S1 S2 S3

S1 2 5 1 0 0 20

S2 -5 -1 0 1 0 -5

S3 1 -11/3 0 0 1/6 -11

Z -1 -5 0 0 0 0