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ME 3 1 -3 5 -1 -2 1 2 1 1 3 12 1 1 3 12 -1 -2 1 2 3 1 -3 5 1 1 3 12 1 F1+F2 0 -1 4 14 -3 F1+F3 0 -2 -12 -31 1 1 3 12 -1 F2 0 1 -4 -14 0 -2 -12 -31 -1 F2+F1 1 0 7 26 0 1 -4 -14 2 F2+F3 0 0 -20 -59 1 0 7 26 0 1 -4 -14 - 1/20 F3 0 0 1 2 19/20 X3= 2 19/20 X2= -2 1/5 X1= 5 7/20 PRUEBA F3↔F1

# Metodos Numericos Gauss y Gauss Jordan

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METODO GAUSS

3 1 -3 5-1 -2 1 2

1 1 3 12

1 1 3 12-1 -2 1 23 1 -3 5

1 1 3 121 F1+F2 0 -1 4 14

-3 F1+F3 0 -2 -12 -31

1 1 3 12-1 F2 0 1 -4 -14

0 -2 -12 -31

-1 F2+F1 1 0 7 260 1 -4 -14

2 F2+F3 0 0 -20 -59

1 0 7 260 1 -4 -14

- 1/20 F3 0 0 1 2 19/20

X3= 2 19/20X2= -2 1/5 X1= 5 7/20

PRUEBA1. 5 = 52. 2 = 23. 12 = 12

F3↔F1

METODO GAUSS

METODO GAUSS JORDAN1 1 1 102 -1 0 155 -3 1 20

1 1 1 10-2 F1+F2 0 -3 -2 -5-5 F1+F3 0 -8 -4 -30

1 1 1 10- 1/3 F2 0 1 2/3 1 2/3

0 -8 -4 -30

-1 F2+F1 1 0 1/3 8 1/30 1 2/3 1 2/3

8 F2+F3 0 0 1 1/3 -16 2/3

1 0 1/3 8 1/30 1 2/3 1 2/3

3/4 F3 0 0 1 -12 1/2

- 1/3 F3+F1 1 0 0 12 1/2- 2/3 F3+F2 0 1 0 10

0 0 1 -12 1/2

X1= 12 1/2X2= 10 X3= -12 1/2

PRUEBA1. 10 = 10

2. 15 = 15

3. 20 = 20

METODO GAUSS JORDAN

METODO GAUSS

4 5 -6 282 0 -7 29-5 -8 0 -64

1/4 F1 1 1.25 -1.5 72 0 -7 29-5 -8 0 -64

1 1.25 -1.5 7-2 F1+F2 0 -2.5 -4 155 F1+F3 0 -1.75 -7.5 -29

1 1.25 -1.5 7- 2/5 F2 0 1 1.6 -6 420000

0 -1.75 -7.5 -29 60 25,200,000.00

1 1.25 -1.5 70 1 1.6 -6

1.75 F2+F3 0 0 -4.7 -39.5

1 1.25 -1.5 70 1 1.6 -6

- 10/47 F3 0 0 1 8.404255319

X1= 43.914X2= -19.447X3= 8.404

PRUEBA1. Ecuacion 28.00 = 282. Ecuacion 29.00 = 293. Ecuacion -64.00 = -64

METODO GAUSS JORDAN

10 -3 6 24.51 8 -2 -9-2 4 -9 -50

1 8 -2 -910 -3 6 24.5-2 4 -9 -50

1 8 -2 -9-10 F1+F2 0 -83 26 114.5

2 F1+F3 0 20 -13 -68

1 8 -2 -9- 1/83 F2 0 1 -0.31 -1.38

0 20 -13 -68

-8 F2+F1 1 0 0.51 2.040 1 -0.31 -1.38

-20 F2+F3 0 0 -6.73 -40.41

1 0 0.51 2.040 1 -0.31 -1.38

F2↔F1

- 11/74 F3 0 0 1 6

-0.51 F3+F1 1 0 0 -10.31 F3+F2 0 1 0 0.48

0 0 1 6

X1= -1X2= 0.48X3= 6

PRUEBA1. Ecuacion 24.3 = 24.52. Ecuacion -9 = -93. Ecuacion -50 = -50

METODO DE LA INVERSA

-12 1 -7 SARRUS -0.08 0.00 -0.051 -6 4 761 -0.02 -0.18 0.06-2 -1 10 -0.02 -0.02 0.10

METODO CRAMER

1 7 -3 -51

4 -4 9 61 = A = 53712 -1 3 8

-51 7 -3 -61 7 -3 -4 -51 -3 -9 -511= 61 -4 9 -1 3 8 3 8

8 -1 3-1098 516 45

1 -51 -32= 4 61 9 -4 -51 -3 61 1 -3 -9 1

12 8 3 8 3 12 3 12

516 2379 -5580

1 7 -513= 4 -4 61 -4 7 -51 -4 1 -51 -61 1

12 -1 8 -1 8 12 8 12

-20 -2480 5185

X1= -1 X2= -5 X3= 5

PRUEBA

1. -51 = -512. 61 = 613. 8 = 8

METODO COFACTORES

COFACTORES-6 0 12

4 -1 -1 -4 0 12 -1 -6 12 1 -66 8 0 8 0 6 0 6

384 72 -48

METODO JACOBI

ES 0.05 %3 Cifras Significativas

4 -2 -1 39 4 > 31 -6 2 -28 6 > 3 CUMPLE REGLA1 -3 12 -86 12 > 4

X1= X2= X3= 0

ITERACION 1 ITERACION 2 ITERACION 3Ecuacion 1 X1 9.750000 X2 4.66667 X3 (7.16667)Ecuacion 2 10.291667 3.90278 (6.81250)Ecuacion 3 9.998264 4.11111 (7.04861)

10.043403 3.98351 (6.97208) 9.998734 4.01654 (7.00774) 10.006336 3.99721 (6.99576) 9.999665 4.00247 (7.00123) 10.000928 3.99954 (6.99935)

9.999929 4.00037 (7.00019) 10.000137 3.99992 (6.99990)

ITERACIONESX1 X2 X3 X1 X2 X30 0 0 - - -

10.291667 3.902778 -6.812500 5.263158 -19.572954 -5.1987779.998264 4.111111 -7.048611 -2.934537 5.067568 3.349754

10.043403 3.983507 -6.972078 0.449438 -3.203312 -1.0977159.998734 4.016541 -7.007740 -0.446743 0.822457 0.508903

10.006336 3.997209 -6.995759 0.075967 -0.483645 -0.1712619.999665 4.002470 -7.001226 -0.066711 0.131433 0.078080

10.000928 3.999536 -6.999355 0.012635 -0.073359 -0.0267329.999929 4.000370 -7.000193 -0.009992 0.020855 0.011983

METODO SEIDEL

ES 0.05 %3 Cifras Significativas

1 -3 12 10 1 > 155 -12 2 -33 12 > 7 NO CUMPLE REGLA1 -14 0 -103 0 > 15

5 -12 2 -33 5 > 141 -14 0 -103 14 > 1 NO CUMPLE REGLA1 -3 12 10 12 > 4

1 -14 0 -103 1 > 141 -3 12 10 3 > 135 -12 2 -33 2 > 17

METODO GAUSS JORDAN

0 3 -13 -502 -6 1 444 0 8 4

NO HAY SOLUCION POSIBLE, YA QUE NO CUMPLE LA REGLA

4 0 8 42 -6 1 440 3 -13 -50

1/4 F1 1 0 2 12 -6 1 440 3 -13 -50

1 0 2 1-2 F1+F2 0 -6 -3 42

0 3 -13 -50

1 0 2 1- 1/6 F2 0 1 0.5 -7

0 3 -13 -50

1 0 2 10 1 1 -7

-3 F2+F3 0 0 -14.50 -29

F3↔F1

1 0 2 10 1 1 -7

- 2/29 F3 0 0 1 2

-2 F3+F1 1 0 0 -3-1 F3+F2 0 1 0 -8

0 0 1 2

X1= -3X2= -8.00X3= 2

PRUEBA1. Ecuacion -50 = -502. Ecuacion 44 = 443. Ecuacion 4 = 4

7-537

-1

45

-51-2685

8

-5580

72685

-1

5185

0 4088

-48

CUMPLE REGLA

NO CUMPLE REGLA

NO CUMPLE REGLA

NO HAY SOLUCION POSIBLE, YA QUE NO CUMPLE LA REGLA

1 -3 12 10 15 -12 2 -33 121 -14 0 -103 0

X1= X2= X3= 0

ITERACION 1 ITERACION 2 ITERACION 3X1 #DIV/0! X2 #DIV/0! X3

#DIV/0! #DIV/0!#DIV/0! #DIV/0!#DIV/0! #DIV/0!#DIV/0! #DIV/0!#DIV/0! #DIV/0!#DIV/0! #DIV/0!#DIV/0! #DIV/0!#DIV/0! #DIV/0!#DIV/0! #DIV/0!

ITERACIONESX1 X2 X3 X1 X2 X3

0 0 0 - - -#DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!#DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!#DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!#DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!#DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!#DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!#DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!#DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

> 15> 7 CUMPLE REGLA> 15

#DIV/0!#DIV/0!#DIV/0!#DIV/0!#DIV/0!#DIV/0!#DIV/0!#DIV/0!#DIV/0!#DIV/0!

y= ax+by= ax+b+E E= y-ax-b Error estandar relativo

3. Regresion Polinomica

ajuste los datos a una curva grado 2

xi yi xi^2 xi^3 xi^4 xiyi0 2.1 0 0 0 01 7.7 1 1 1 7.72 13.6 4 8 16 27.23 27.2 9 27 81 81.64 40.9 16 64 256 163.65 61.1 25 125 625 305.5

15 152.6 55 225 979 585.6n=6 m=2

Sistema de ecuaciones en tamaño (m+1)

Y(PROMEDIO)25.4333333y=a0+a1x+a2x^2 X(PROMEDIO 2.5

SV 16171.36116a0+15a1+55a2=152,615a0+55a1+225a2=585,655a0+225a1+979a2=2488,8

6 15 55 152.6 0.821428571415 55 225 585.6 -0.58928571455 225 979 2488.8 0.0892857143

Error estandar relativo S(x/r)=raiz(sr/n-2)Coeficiente de determinacion r^2=Coeficiente de correlacion r

(Yi-Y)^2 (Yi-Y')^2x1^2y1 Y SR SV

0 2.47857143 0.14331633 544.4444447.7 6.69857143 1.00285918 314.471111

54.4 14.64 1.0816 140.027778244.8 26.3028571 0.80486531 3.12111111654.4 41.6871429 0.61959388 239.217778

1527.5 60.7928571 0.09433673 1272.111112488.8 152.6 3.74657143 2513.39333

Y=2,478+2,352X+1860X^2

INVERSAA-1*B

-0.5892857142857 0.08928571 152.6 2.478570.72678571428571 -0.13392857 585.6 2.35929-0.1339285714286 0.02678571 2488.8 1.86071