Método de Jacobi

Margen de error de 1x10-12

5x1 +2x2 -3.5x3 -4x4 =4.25-2x1 +4x2 -10x3 -x4 =4-x1 +2x2 -10x3 +x4 =-22x1 -x2 +x3 -4x4 =11.5

x1=2.000000000000449 e1=4.394262731465383x10-13

x2=3.9999999999972684 e2=1.2423395645563986x10-13

x3=0.49999999999937844 e3=7.325251516485889x10-13

x4=3.0000000000003455 e4=1.780797731498546x10-13

Número de iteraciones=65

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8x1 -4x2 +3x3 -4x4 =12-2x1 +13x2 -3x3 +x4 =-10.88x1 +0.5x2 -9x3 +x4 =-6.05-4.37x1 +8x2 -3x3 +4x4 =-0.71

x1=3.0000000000000044 e1=1.8015218946250847x10-13

x2=0.09999999999991321 e2=8.082423619278154x10-13

x3=4.000000000000404 e3=3.457234498682388x10-13

x4=5.899999999999873 e4=1.0040932304067733x10-13

Número de iteraciones=74

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1.5x1 +0.2x2 +x3 -x4 =7.62x1 +3x2 -x3 +x4 =16x1 -x2 +3x3 -2x4 =5x1 -x2 +3x3 -2x4 =10

x1=1.999999999999786 e1=9.503509090792357x10-14

x2=2.9999999999999734 e2=3.172277255695642x10-13

x3=3.9999999999996714 e3=4.2787995369057047x10-13

x4=1.0000000000002425 e4=1.5409895581793437x10-13

P á g i n a | 1

Número de iteraciones=38

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4x1 -3x2 +2x3 -x4 =-0.9x1 +3x2 -2x3 -x4 =1.22x1 -x2 +8x3 +3x4 =8.12x1 +4x2 -0.5x3 +5x4 =10.2

x1=0.4999999999999445 e1=9.570122472269912x10-13

x2=1.0000000000003721 e2=5.169198402652805x10-13

x3=0.5999999999999662 e3=2.1057230033724987x10-13

x4=1.1000000000003385 e4=5.349256391374108x10-14

Número de iteraciones=171

**************************************************

6x1 -0.3x2 +x3 -2x4 =6.053x1 +4x2 -x3 -3x4 =-10.84x1 -5x2 +8x3 -x4 =4.6x1 -2x2 -3x3 +10x4 =-0.2

x1=1.000000000000044 e1=2.5535129566377477x10-14

x2=0.500000000000206 e2=2.384759056893854x10-13

x3=0.4000000000001984 e3=2.8657631823122636x10-13

x4=0.10000000000015201 e4=8.788803018675536x10-13

Número de iteraciones=63

P á g i n a | 2

Método de Gauss-Seidel

Margen de error de 1x10-12

5x1 +2x2 -3.5x3 -4x4 =4.25-2x1 +4x2 -10x3 -x4 =4-x1 +2x2 -10x3 +x4 =-22x1 -x2 +x3 -4x4 =11.5

x1= 1.9999999999993423 e1= 6.961098364402021x10-13

x2= 3.999999999999233 e2= 9.392486788330625x10-14

x3= 0.4999999999998176 e3= 2.0372592501879054x10-13

x4= 2.9999999999998175 e4= 2.5490720645395144x10-13

Número de iteraciones=44

**************************************************

8x1 -4x2 +3x3 -4x4 =12-2x1 +13x2 -3x3 +x4 =-10.88x1 +0.5x2 -9x3 +x4 =-6.05-4.37x1 +8x2 -3x3 +4x4 =-0.71

x1= 3.000000000000105 e1= 3.256654205567012x10-14

x2= 0.10000000000004719 e2= 3.5929592634414675x10-13

x3= 4.00000000000014 e3= 3.1974423109203385x10-14

x4= 5.900000000000125 e4= 4.666700171305644x10-14

Número de iteraciones=34

**************************************************

1.5x1 +0.2x2 +x3 -x4 =7.62x1 +3x2 -x3 +x4 =16x1 -x2 +3x3 -2x4 =5x1 -x2 +3x3 -2x4 =10

x1= 2.000000000000157 e1= 3.9224179460003696x10-13

x2= 2.999999999999979 e2= 3.508304757815519x10-14

x3= 4.000000000000006 e3= 8.10462807976363x10-15

x4= 0.9999999999999668 e4= 1.659783421814664x10-13

P á g i n a | 3

Número de iteraciones=20

**************************************************

4x1 -3x2 +2x3 -x4 =-0.9x1 +3x2 -2x3 -x4 =1.22x1 -x2 +8x3 +3x4 =8.12x1 +4x2 -0.5x3 +5x4 =10.2

x1= 0.49999999999990474 e1= 4.16333634234513x10-14

x2= 1.0000000000001223 e2= 2.523536934972672x10-13

x3= 0.5999999999999863 e3= 1.3248661427193837x10-13

x4= 1.0999999999999386 e4= 1.8348958716078612x10-13

Número de iteraciones=41

**************************************************

6x1 -0.3x2 +x3 -2x4 =6.053x1 +4x2 -x3 -3x4 =-10.84x1 -5x2 +8x3 -x4 =4.6x1 -2x2 -3x3 +10x4 =-0.2

x1= 0.9999999999999836 e1= 2.4535928844216363x10-14

x2= 0.4999999999999015 e2= 2.9398705692079936x10-13

x3= 0.3999999999999349 e3= 2.4272250875871433x10-13

x4= 0.09999999999996242 e4= 5.608014053139929x10-13

Número de iteraciones=32

P á g i n a | 4

Método Iterativo Secuencial

Margen de error de 1x10-12

f1(x,y)=x2+y2-4x-6y+11=0

f2(x,y)=x2+y2-6x-8y+21=0

f1(x,y)=0 x=g1(x,y) x=(x2+y2-6y+11)/4f2(x,y)=0 x=g2(x,y) y=(x2+y2-6x+21)/8

Iteración xi yi |xi+1-xi|190 0.999999999998229 3.999999999998188 8.445759532951282x10-13

**************************************************

f1(x,y)=7x3-10x-y-1=0

f2(x,y)=8y3-11y+x-1=0

f1(x,y)=0 x=g1(x,y) x=(7x3-y-1)/10f2(x,y)=0 x=g2(x,y) y=(8y3+x-1)/11

Iteración xi yi |xi+1-xi|12 -0.09053305397079696 -0.09986367086961603 9.97339195916772x10-13

**************************************************

f1(x,y)=x2+y2+xy-21=0

f2(x,y)=x+y-1=0

f1(x,y)=0 x=g1(x,y) x=√− y2−xy+21f2(x,y)=0 x=g2(x,y) y=-x+1

Iteración xi yi |xi+1-xi|50 4.999999999999826 -3.9999999999996554 4.964980920523348x10-13

**************************************************

P á g i n a | 5

f1(x,y)=-x+y-1=0

f2(x,y)=x2+y2-5=0

f1(x,y)=0 x=g1(x,y) x=y-1f2(x,y)=0 x=g2(x,y) y=√−x2+5

Iteración xi yi |xi+1-xi|80 0.9999999999995324 2.0 6.61322943597606x10-13

**************************************************

f1(x,y)=x2+y-9=0

f2(x,y)=x-y+3=0

f1(x,y)=0 x=g1(x,y) x=√− y+9f2(x,y)=0 x=g2(x,y) y=x+3

Iteración xi yi |xi+1-xi|43 1.9999999999997595 5.000000000000418 8.541012330017849x10-13

**************************************************

f1(x,y)=x2-8x-y+18=0

f2(x,y)=-x2+6x-y-4=0

f1(x,y)=0 x=g1(x,y) x=(x2-y+18)/8f2(x,y)=0 x=g2(x,y) y=-x2+6x-4

Iteración xi yi |xi+1-xi|36 2.381966011250066 4.618033988749936 4.395074426442159x10-13

P á g i n a | 6

Método de Newton-Raphson

Margen de error de 1x10-12

f1(x,y)=x2+y2-4x-6y+11=0

f2(x,y)=x2+y2-6x-8y+21=0

∂ f 1∂x

=2 x−4∂ f 2∂x

=2 x−6

∂ f 1∂ y

=2 y−6∂ f 2∂ y

=2 y−8

Iteración xi yi |xi+1-xi|9 3.000000000000001 2.0000000000000004 3.774758283725532x10-15

**************************************************

f1(x,y)=7x3-10x-y-1=0

f2(x,y)=8y3-11y+x-1=0

∂ f 1∂x

=21x2−10∂ f 2∂x

=1

∂ f 1∂ y

=−1∂ f 2∂ y

=24 y2−11

Iteración xi yi |xi+1-xi|4 -0.09053305397081238 -0.09986367086952219 3.867996145289936x10-15

**************************************************

f1(x,y)=x2-2y-2=0

f2(x,y)=-y2+x-1=0

P á g i n a | 7

∂ f 1∂x

=2 x∂ f 2∂x

=1

∂ f 1∂ y

=−2∂ f 2∂ y

=−2 y

Iteración xi yi |xi+1-xi|7 1.1303954347672789 -0.3611030805286474 1.1102230246251565x10-13

**************************************************

f1(x,y)=x2+3x-y+2=0

f2(x,y)=2x-y+3=0

∂ f 1∂x

=2 x+3∂ f 2∂x

=2

∂ f 1∂ y

=−1∂ f 2∂ y

=−1

Iteración xi yi |xi+1-xi|6 0.6180339887498952 4.236067977499791 2.1051889619536875x10-13

**************************************************

f1(x,y)=x2-8x-y+18=0

f2(x,y)=-x2+6x-y-4=0

∂ f 1∂x

=2 x−8∂ f 2∂x

=−2 x+6

∂ f 1∂ y

=−1∂ f 2∂ y

=−1

P á g i n a | 8

Iteración xi yi |xi+1-xi|7 2.381966011250105 4.618033988749895 8.881784197001252x10-16

P á g i n a | 9