spline cubico.xlsx

i 0 1 2 3

f(Xi) 2 4.4366 6.7134 13.913

X 0 1 1.5 2.25

2.5 0.05

0.05 2.5

hi 1 0.5 0.75

f[Xi,Xi+1] 2.4366 4.5536 9.59946667

si 0 2.292041 11.65

ai 0.38200683 3.11931967 -2.58888889 g0(x)= g1(x)=

12.702 3 0 .2 7 5 2

s1=229 s1=229204 2041 1 s0=s3= s0=s3=0 0 s2=11.651751

i=1 i=1 i=2 i=2

h0 s0+2( s0+2(h0 h0+h +h1) 1)s1 s1+h +h1s 1s2= 2=6[ 6[f[ f[x1 x1,x ,x1+ 1+1] 1]-f -f[x [xii-1, 1,xi xi]) ]) h0 s0+2 s0+2(h (h0+ 0+h1 h1)s )s1+ 1+h1 h1s2 s2=6 =6[f [f[x [x1, 1,x2 x2]]-f[ f[x0 x0,x ,x1] 1]))

a=(Si+1-Si)/6hi bi= si/2 ci=f()

0.059137356 0.059137356 -0.00480491 -0.00480491

bi 0 1.1460 1.1460205 205 5.825 5.825

ci 2.05 2.0545 4593 9317 17 3.2007 3.2007598 5983 3 6.6869 6.6869666 6667 7

G2x^2+I2x+C2 #REF!

0.40016006 -0.0080032

18.526123

-0.0080032 0.40016006

5.08283313 -0.24229852

f(x)=e^x cos x x

f(x)

0

1

0.1 0.23 0.5 0.7 1

1.099649667 1.225456534 1.446889037 1.540203025 1.46869394 (X-0.1)(X-0.23)(X-0.5)

L0=

0.243478261  

(X-0)(X-0.23)(X-0.5) (0.1-0)(0.1-0.23)(0.1-0.5)

L1=

L2=

*1.099649667

-0.888178577  

(X-0)(X-0.1)(X-0.5) (0.23-0)(0.23-0.1)(0.23-0.5)

L2=

L3=

0.3

*1

(0-0.1)(0-0.23)(0-0.5)

L0=

L1=

punto a interpolar

*1.225456534

1.821563039  

(X-0)(X-0.1)(X-0.23) (0.5-0)(0.5-0.1)(0.5-0.23)

L3=

*1.446889037

0.112535814 El valor del punto a interpolar ("0.3") es la suma de las Li es:

1.28939854

valor de ("0.3") valuado en la funcion:

1.28956937 error

f ᶦᵛ(x)= -4e^x cos x

fᶦᵛ(0)= -4e^(0)cos (0)= fᶦᵛ(0.5)= -4e^(0.5)cos (0.5)= cota inferior= cota inferior= cota superior= cota superior=

 

-4 -5.787556146 (fᶦᵛ(0)/4!)(X-X0)(X-X1)(X-X2)(X-X3) 0.00014 (fᶦᵛ(.5)/4!)(X-X0)(X-X1)(X-X2)(X-X3) 0.000202564

0.00017084

i 0 1 2 3 4 5 SISTEMA ECS. 1.22 0.49 0 0

X 0.15 0.27 0.76 0.89 1.07 2.11 0.49 1.24 0.13 0

f(Xi) 0.168 0.2974 0.7175 0.7918 0.8698 0.9972 0 0.13 0.62 0.18

hi 0.12 0.49 0.13 0.18 1.04

f[Xi,Xi+1] 1.07833333 0.85734694 0.57153846 0.43333333 0.1225

si 0 -0.677754 -1.018487 -0.92175 -0.696346

ai -0.941325 -0.11589558 0.12402179 0.20870741 0.11159391 INVERSA 0.97854879 -0.39557047 0.08475747 -0.0062526

0 -1.32591837 0 -1.71485086 0.18 -0.82923077 2.44 -1.865

0.2 0.05 0

0.05 0.2 0.05

bi 0 -0.338877 -0.5092435 -0.460875 -0.348173

ci 1.09188841 1.0512232 0.63564415 0.50952871 0.36389995

-0.39557047 0.08475747 -0.0062526 0.98488974 -0.21102879 0.0155677 -0.21102879 1.69341957 -0.12492439 0.0155677 -0.12492439 0.4190518

0 0.05 0.2

n-2

SI -0.67775386 -1.01848705 -0.92175019 -0.6963463

i 0 1 2 3 4 5 6 7 8 9 SISTEMA ECS. 0

X 0

f(Xi) 0

hi 0

f[Xi,Xi+1] #DIV/0!

0

0

0

#DIV/0!

si 0

ai #DIV/0!

bi 0

ci #DIV/0!

L0=

L0=

L1=

L1=

L2=

L2=

L3=

L3=

L4=

L4=

X

f(X)

1920 1940 1970 1980 1985 1990 1995 2000

14335 19654 48225 66845 75643 81250 91121 97362

Punto interpolar:

1975

(((X-1940)(X-1970)(X-1980)(X-1985)(X-1990)(X-1995)(X-2000)) ((1920-1940)(1920-1970)(1920-1980)(1920-1985)(1920-1990)(1920-1995)(1920-2000))))

* 14335

0.57431891 (((X-1920)(X-1970)(X-1980)(X-1985)(X-1990)(X-1995)(X-2000)) ((1940-1920)(1940-1970)(1940-1980)(1940-1985)(1940-1990)(1940-1995)(1940-2000))))

*19654

-11.37384259 (((X-1920)(X-1940)(X-1980)(X-1985)(X-1990)(X-1995)(X-2000)) ((1970-1920)(1970-1940)(1970-1980)(1970-1985)(1970-1990)(1970-1995)(1970-2000))))

*48225

10314.79167 (((X-1920)(X-1940)(X-1970)(X-1985)(X-1990)(X-1995)(X-2000)) ((1980-1920)(1980-1940)(1980-1970)(1980-1985)(1980-1990)(1980-1995)(1980-2000))))

*66845

134038.151 (((X-1920)(X-1940)(X-1970)(X-1980)(X-1990)(X-1995)(X-2000)) ((1985-1920)(1985-1940)(1985-1970)(1985-1980)(1985-1990)(1985-1995)(1985-2000)))) -165940.4843

*75643

L5=

L5=

L6=

L6=

(((X-1920)(X-1940)(X-1970)(X-1980)(X-1985)(X-1995)(X-2000)) ((1990-1920)(1990-1940)(1990-1970)(1990-1980)(1990-1985)(1990-1995)(1990-2000)))) 111718.75  

(((X-1920)(X-1940)(X-1970)(X-1980)(X-1985)(X-1990)(X-2000))

((1995-1920)(1995-1940)(1995-1970)(1995-1980)(1995-1985)(1995-1990)(1995-2000))))

*91121

-42523.13333

L7=

(((X-1920)(X-1940)(X-1970)(X-1980)(X-1985)(X-1990)(X-1995)) ((2000-1920)(2000-1940)(2000-1970)(2000-1980)(2000-1985)(2000-1990)(2000-1995))))

L7=

*81250

6507.703125 El valor del punto a interpolar ("1975") es la suma de las Li es:

54104.9786

*97362

Σ=

xi

f(xi)

xi*f(xi)

xi^2

Xi^3

xi^4

xi^2*f(xi)

1967 1968 1969 1972 1974 1975 1976

10.5 18.5 22.6 33 44.9 35 27.6

20653.5 36408 44499.4 65076 88632.6 69125 54537.6

3869089 3873024 3876961 3888784 3896676 3900625 3904576

7610498063 7622111232 7633736209 7668682048 7692038424 7703734375 7715442176

1.49698E+13 1.50003E+13 1.50308E+13 1.51226E+13 1.51841E+13 1.52149E+13 1.52457E+13

40625434.5 71650944 87619318.6 128329872 174960752.4 136521875 107766297.6

13801

192.1

378932.1

27209735

53646242527

1.05768E+14

747474494.1

Σ(xi^2) Σ(xi^3) Σ(xi^4) Σ(xi^5)

Σ(xi^3) Σ(xi^4) Σ(xi^5) Σ(xi^6)

Σ f(xi) Σ(xi*f(xi)) Σ(xi^2*f(xi)) Σ(xi^3*f(xi))

m= MATRIZ m Σxi Σ(xi^2) Σ(xi^3)

7

Σxi Σ(xi^2) Σ(xi^3) Σ(xi^4)

MATRIZ 7 13801 27209735 5.3646E+10

13801 27209735 5.3646E+10 192.1 27209735 53646242527 1.0577E+14 50 378932.1 5.3646E+10 1.05768E+14 4.1114E+20 45747474494.1 1.0577E+14 4.11142E+20 4.1114E+20 1.03054E+13

50017.89551 -25.36948555 3.72928E-11 9.45814E-15

40 35

   e 30     l    t    i    T 25    s    i    x    A20

polinomio

-6.40395E-19 X^3+2.14901E-08X^2+2.486312325X-4874.583019 15

1967 1968 1969 1972 1974 1975 1976

16.0764708 18.5628677 21.0492646 28.5084556 33.4812499 35.9676471 38.4540443

10 5 0 1966

1968

1970

1972

1974

1976

1

xi^5

xi^6

5.79197E+19 5.80966E+19 5.82739E+19 5.88087E+19 5.91675E+19 5.93475E+19 5.9528E+19

5.79197E+19 5.80966E+19 5.82739E+19 5.88087E+19 5.91675E+19 5.93475E+19 5.9528E+19

1.57183E+14 2.77506E+14 3.39697E+14 4.99047E+14 6.81765E+14 5.32521E+14 4.20782E+14

4.11142E+20

4.11142E+20

2.03283E+16

 INVERSA

VECTOR SOL.

-25.36948555 0.012867647 -2.83745E-14 -9.59506E-18

3.72928E-11 -2.83745E-14 -6.768E-32 2.43348E-21

Datos de la tabla recta que ajusta los datos

978

xi^3*f(xi)

9.45814E-15 -4874.583019 a3 -9.59506E-18 2.486312325 a2 2.43348E-21 2.14901E-08 a1 -1.9559E-39 -6.40395E-19 a0

X

f(X)

F[1]

F[2]

F[3]

F[4]

F[5]

F[6]

F[7]

5 10 15 20 25 30 35 40

49 105 172 253 352 473 619 793

56 67 81 99 121 146 174

11 14 18 22 25 28

3 4 4 3 3

1 0 -1 0

-1 -1 1

0 2

2

=

-0.2

a) Interpolar para

x=39

Newton regresivo h=

5

x=

39

Xn=

40

P(39)=

Pn(39)= Pn(39)=

793 + (-0.2)(174) +

X-Xn

S=

h

  (0.2)(0.8)(28) 2!

39-40

=

+

5

 

(0.2)(0.8)(1.8)(3) 3!

793 -34.8 -2.24 -0.144 755.816

b) obtener una aproximacion para x=12 Newton Regresivo h=

5

x=

12

x0=

10

P(12)=

Pn(12)= Pn(16)=

105 + (0.4)(67) +

h

  (0.4)(-0.6)(14)

105 +26.8 -1.68 +0.256 130.376

X-X0

S=

2!

  =

+

12-10 5

 

=

0.4

(0.4)(-0.6)(-1.6)(4) 3!

xi

f(xi)

Xi^2

Xi*f(xi)

p(xi)

[f(Xi)-P(xi)]^2

1967 1968 1969 1972 1974 1975 1976 13801

10.5 18.5 22.6 33 44.9 35 27.6 192.1

3869089 3873024 3876961 3888784 3896676 3900625 3904576 27209735

20653.5 36408 44499.4 65076 88632.6 69125 54537.6 378932.1

16.0764706 18.5628676 21.0492647 28.5084559 33.48125 35.9676471 38.4540441

31.09702422 0.003952341 2.404779953 20.17396856 130.3878516 0.93634083 117.8102737

m=

7

a0 a1

-4874.66654 2.48639706

P(x)=

2.48639706X-4874.66654

Xi 1 2 3 4 5 6 7 8 9 10 55

M

f(Xi) 1.3 3.5 4.2 5 7 8.8 10.1 12.5 13 15.6 81

10 A0 A1

-0.36 1.53818182

Xi^2 1 4 9 16 25 36 49 64 81 100 385

f(Xi)*Xi 1.3 7 12.6 20 35 52.8 70.7 100 117 156 572.4

P(Xi) 1.17818182 2.71636364 4.25454545 5.79272727 7.33090909 8.86909091 10.4072727 11.9454545 13.4836364 15.0218182

(f(Xi)*P(Xi))^2 0.014839669 0.61408595 0.002975207 0.628416529 0.109500826 0.004773554 0.094416529 0.307520661 0.233904132 0.334294215

5. ajustar un Spline Cubico a la funcion definida por el siguiente conjunto de datos. TABLA DE DATOS x

f(x)

hi

F[xi, xi+1]

si

ai

bi

0.1 0.15 0.2 0.25 0.3

1.1052 1.1618 1.2214 1.284 1.3499

0.05 0.05 0.05 0.05

1.132 1.192 1.252 1.318

0 1.55571 0.977143 1.73571 0

5.1857 -1.92855667 2.52855667 -5.7857

0 0.777855 0.4885715 0.867855

Sistema de ecuaciones para encontrar los Si i=1 i=2 i=3

Matriz para

0.2s1 + 0.05s2 = 0.36 0.05s1 + 0.2s2 + 0.05s3 = 0.36 0.05s2 + 0.2s3 = 0.396

0.2 0.05 0

0.05 0.2 0.05

0.977143 - 1.55571

a2=

Matriz Inversa

Si

5.35714286 -1.42857143 0.35714286 -1.42857143 5.71428571 -1.42857143 0.35714286 -1.42857143 5.35714286

1.55571 0.977143 1.73571

Encontrar los "ai"

a0=

a0=

a3=

a3=

1.55571 - 0

a1=

6(0.05)

6(0.05)

5.1857

a1=

-1.92855667

0 - 1.73571 6(0.05) -5.7857

Encontrar los "bi"

b0=

b0=

b3=

b3=

0

b1=

2 0

b1=

1.55571

b2=

2 0.777855

b2=

1.73571 2 0.867855

Encontrar los "ci"

c0=

1.132 -

((1.55571 + 2(0)) * 0.05 6

c1=

1.192 -

 

( (0.977

c0=

1.11903575

c2=

1.252 -

c2=

1.22125003

((1.73571 + 2(0.977143)) * 0.05 6

Encontrar los "di" d0=

1.1052

d3=

1.284

c1=

1.15792864

c3=

1.318 -

c3=

1.2890715

((0

di = yi d1=

1.1618

Polinomios del Spline Cubico g0(x)=

 

5.1857(x-0.1)^3 + 1.11903575(x-0.1) + 1.1052

g1(x)=

(-1.928566(x-0.15)^3) + 0.777855(x-0.15)^2 + 1.157928642(x-0.15) + 1.1618

g2(x)=

2.528556(x-0.2)^3 + 0.4885715(x-0.2)^2 + 1.221250033(x-0.2) + 1.2214

g3(x)=

(-5.7857(x-0.25)^3) + 0.867855(x-0.25)^2 + 1.2890715(x-0.25) + 1.284

d2=

ci

di

1.11903575 1.15792864 1.22125003 1.2890715

1.1052 1.1618 1.2214 1.284

allar los Si 0 0.05 0.2

0.36 0.36 0.396

1.73571 - 0.977143 6(0.05) a2=

2.52855667

0.977143 2 0.4885715

43 + 2(1.55571)) * 0.05 6

2(1.73571)) * 0.05 6

1.2214

0.1 ≤ x ≤ 0.15 0.15 ≤ x ≤ 0.2 0.2 ≤ x ≤ 0.25 0.25 ≤ x ≤ 0.3

4. Obtener una estimacion para f(0.25) a partir de los siguientes datos, mediante el polinomio de He

x

f(x) 0.1 -0.62049958 0.2 -0.28398668 0.3 0.00660095 0.4 0.2484244

f´(x)

Interpolar

0.25

3.58502082 3.14003327 2.6666843 2.16529366 TABLA DE DIFERENCIAS DIVIDIDAS

x

f(X) 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4

-0.62049958 -0.62049958 -0.28398668 -0.28398668 0.00660095 0.00660095 0.2484244 0.2484244

1ra Dif

2da Dif

3ra Dif

3.58502082 -2.1989182 3.365129 -2.25095729 3.14003327 -2.34156971 2.9058763 -2.39192 2.6666843 -2.484498 2.4182345 -2.5294084 2.16529366

-0.5203909 -0.4530621 -0.5035029 -0.46289 -0.449104

4ta Dif 0.336644 -0.252204 0.2030645 0.06893

5ta Dif   -2.94424 1.51756167 -0.6706725

Polinomio de Hermite Ph(x)=

(-0.62049958) + 3.58502082(x - 0.1) - 2.1989182(x - 0.1)^2 - 0.520 0.336644(x - 0.1)^2 (x - 0.2)^2 + (-2.94424(x - 0.1)^2 (x - 0. (-73.889287(x -0.1)^2 (x - 0.2)^2 (x - 0.3)^2 (x

Evaluando "0.25" en el polinomio de Hermite P(0.25)=

-0.13277669

 

rmite

6ta Dif

7ma Dif  

14.8726722 -7.29411389

-73.889287

909(x - 0.1)^2 (x-0.2) + .2)^2 (x - 0.3)) + - 0.4))