Metodos Numericos Gauss y Gauss Jordan
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Transcript of Metodos Numericos Gauss y Gauss Jordan
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
PRESENTADO POR:SHARON GONZALEZGLORIA JIMENEZ
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
1. Regresion lineal (minimos cuadrados) Coeficiente de determinacionCoeficiente de correlacion
2. Sumatoria cuadrada de los residuos Sr=Si
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