ANALISIS NUMERICO

Método de Jacobi

Ejercicio 1:

17X1 - 2X2 - 3X3 = 500
-5X1 + 21X2 - 2X3 = 200
-5X1 - 5X2 + 22X3 = 30

Para K=0

X10=0 ; X20=0 ; X30=0

Primera iteración k+1=1

X11=500+2(0)+3(0) =500 = 29.412  Máx
17 17

X21=200+5(0)+2(0) = 200 = 9.524
17 17

X31=30+5(0)+5(0) = 30 = 1.364
17 17

|X10-X11| = 29.412  Máx
|X20-X21| = 9.524|X30-X31| = 1.364

E. Relativo = Máx |Xi0-Xi1| = ℮k
℮k=29.412

E. Absoluto = ℮k = 29.412 = 1 = 100%
Máx |Xi1| 29.412


Segunda iteración k+1=2X12=500+2(9.524)+3(1.364) = 523.14 = 30.773  Máx
17 17

X22=200+5(29.412)+2(1.364) = 349.788 = 16.657
17 17

X32=30+5(29.412)+5(9.524) = 224.68 = 10.21317 17

|X11-X12| = 1.361
|X21-X22| = 7.133
|X31-X32| = 8.849  Máx

E. Relativo = Máx |Xi1-Xi2| = ℮k
℮k=8.849

E. Absoluto = ℮k = 8.849 = 0.288 = 28.8%Máx |Xi2| 30.773


Tercera iteración k+1=3

X13=500+2(16.657)+3(10.213) = 563.953 = 33.174  Máx
17 17

X23=200+5(30.773)+2(10.213) =374.291 = 17.823
17 17

X33=30+5(30.773)+5(16.657) = 267.15 = 12.143
17 17

|X12-X13| = 2.401  Máx
|X22-X23| = 1.166
|X32-X33| = 1.93

E.Relativo = Máx |Xi2-Xi3| = ℮k
℮k=2.401

E. Absoluto = ℮k = 2.401 = 0.072 = 7.2%
Máx |Xi3| 33.174




Cuarta iteración k+1=4

X14=500+2(17.823)+3(12.143) =572.075 = 33.652  Máx
17 17

X24=200+5(33.174)+2(12.143) = 390.156 = 18.579
17 17

X34=30+5(33.174)+5(17.823) = 284.985 = 12.95417 17

|X13-X14| = 0.478
|X23-X24| = 0.756
|X33-X34| = 0.811  Máx

E. Relativo = Máx |Xi3-Xi4| = ℮k
℮k=0.811

E. Absoluto = ℮k = 0.811 = 0.024 = 2.4%...