Given the system of equations x1 22 3x3 3 3x1 x2 x3 3

Given the system of equations: x-1 - 2_2 + 3x_3 = 3 3x_1 + x_2 + x_3 = 3 x_1 - 3x_2 - x_3 = 9 Solve variables using Gauss-Seidel method. Show equations used for iterations. Make five iterations. Show all of your work.

Solution

Step : 1

x 0 = 3/1- (-2 x 0)/1- (3 x 0)/1 = 3
x 1 = 3/1- (3 x 0)/1- (1 x 0)/1 = 3
x 2 = 9/-1- (1 x 0)/-1- (-3 x 0)/-1 = -9

Step : 2

x 0 = 3/1- (-2 x 3)/1- (3 x -9)/1 = 36
x 1 = 3/1- (3 x 3)/1- (1 x -9)/1 = 3
x 2 = 9/-1- (1 x 3)/-1- (-3 x 3)/-1 = -15

Step : 3

x 0 = 3/1- (-2 x 3)/1- (3 x -15)/1 = 54
x 1 = 3/1- (3 x 36)/1- (1 x -15)/1 = -90
x 2 = 9/-1- (1 x 36)/-1- (-3 x 3)/-1 = 18

Step : 4

x 0 = 3/1- (-2 x -90)/1- (3 x 18)/1 = -231
x 1 = 3/1- (3 x 54)/1- (1 x 18)/1 = -177
x 2 = 9/-1- (1 x 54)/-1- (-3 x -90)/-1 = 315

Step : 5

x 0 = 3/1- (-2 x -177)/1- (3 x 315)/1 = -1296
x 1 = 3/1- (3 x -231)/1- (1 x 315)/1 = 381
x 2 = 9/-1- (1 x -231)/-1- (-3 x -177)/-1 = 291

Step : 6

x 0 = 3/1- (-2 x 381)/1- (3 x 291)/1 = -108
x 1 = 3/1- (3 x -1296)/1- (1 x 291)/1 = 3600
x 2 = 9/-1- (1 x -1296)/-1- (-3 x 381)/-1 = -2448

Step : 7

x 0 = 3/1- (-2 x 3600)/1- (3 x -2448)/1 = 14547
x 1 = 3/1- (3 x -108)/1- (1 x -2448)/1 = 2775
x 2 = 9/-1- (1 x -108)/-1- (-3 x 3600)/-1 = -10917

Step : 8

x 0 = 3/1- (-2 x 2775)/1- (3 x -10917)/1 = 38304
x 1 = 3/1- (3 x 14547)/1- (1 x -10917)/1 = -32721
x 2 = 9/-1- (1 x 14547)/-1- (-3 x 2775)/-1 = 6213

Step : 9

x 0 = 3/1- (-2 x -32721)/1- (3 x 6213)/1 = -84078
x 1 = 3/1- (3 x 38304)/1- (1 x 6213)/1 = -121122
x 2 = 9/-1- (1 x 38304)/-1- (-3 x -32721)/-1 = 136458

Step : 10

x 0 = 3/1- (-2 x -121122)/1- (3 x 136458)/1 = -651615
x 1 = 3/1- (3 x -84078)/1- (1 x 136458)/1 = 115779
x 2 = 9/-1- (1 x -84078)/-1- (-3 x -121122)/-1 = 279279

Step : 11

x 0 = 3/1- (-2 x 115779)/1- (3 x 279279)/1 = -606276
x 1 = 3/1- (3 x -651615)/1- (1 x 279279)/1 = 1675569
x 2 = 9/-1- (1 x -651615)/-1- (-3 x 115779)/-1 = -998961

Step : 12

x 0 = 3/1- (-2 x 1675569)/1- (3 x -998961)/1 = 6348024
x 1 = 3/1- (3 x -606276)/1- (1 x -998961)/1 = 2817792
x 2 = 9/-1- (1 x -606276)/-1- (-3 x 1675569)/-1 = -5632992

Step : 13

x 0 = 3/1- (-2 x 2817792)/1- (3 x -5632992)/1 = 22534563
x 1 = 3/1- (3 x 6348024)/1- (1 x -5632992)/1 = -13411077
x 2 = 9/-1- (1 x 6348024)/-1- (-3 x 2817792)/-1 = -2105361

Step : 14

x 0 = 3/1- (-2 x -13411077)/1- (3 x -2105361)/1 = -20506068
x 1 = 3/1- (3 x 22534563)/1- (1 x -2105361)/1 = -65498325
x 2 = 9/-1- (1 x 22534563)/-1- (-3 x -13411077)/-1 = 62767785

Step : 15

x 0 = 3/1- (-2 x -65498325)/1- (3 x 62767785)/1 = -319300002
x 1 = 3/1- (3 x -20506068)/1- (1 x 62767785)/1 = -1249578
x 2 = 9/-1- (1 x -20506068)/-1- (-3 x -65498325)/-1 = 175988898

Step : 16

x 0 = 3/1- (-2 x -1249578)/1- (3 x 175988898)/1 = -530465847
x 1 = 3/1- (3 x -319300002)/1- (1 x 175988898)/1 = 781911111
x 2 = 9/-1- (1 x -319300002)/-1- (-3 x -1249578)/-1 = -315551277

Step : 17

x 0 = 3/1- (-2 x 781911111)/1- (3 x -315551277)/1 = 2510476056
x 1 = 3/1- (3 x -530465847)/1- (1 x -315551277)/1 = 1906948821
x 2 = 9/-1- (1 x -530465847)/-1- (-3 x 781911111)/-1 = -2876199189

Step : 18

x 0 = 3/1- (-2 x 1906948821)/1- (3 x -2876199189)/1 = 12442495212
x 1 = 3/1- (3 x 2510476056)/1- (1 x -2876199189)/1 = -4655228976
x 2 = 9/-1- (1 x 2510476056)/-1- (-3 x 1906948821)/-1 = -3210370416

Step : 19

x 0 = 3/1- (-2 x -4655228976)/1- (3 x -3210370416)/1 = 320653299
x 1 = 3/1- (3 x 12442495212)/1- (1 x -3210370416)/1 = -34117115217
x 2 = 9/-1- (1 x 12442495212)/-1- (-3 x -4655228976)/-1 = 26408182131

Step : 20

x 0 = 3/1- (-2 x -34117115217)/1- (3 x 26408182131)/1 = -147458776824
x 1 = 3/1- (3 x 320653299)/1- (1 x 26408182131)/1 = -27370142025
x 2 = 9/-1- (1 x 320653299)/-1- (-3 x -34117115217)/-1 = 102671998941

Step : 21

x 0 = 3/1- (-2 x -27370142025)/1- (3 x 102671998941)/1 = -362756280870
x 1 = 3/1- (3 x -147458776824)/1- (1 x 102671998941)/1 = 339704331534
x 2 = 9/-1- (1 x -147458776824)/-1- (-3 x -27370142025)/-1 = -65348350758

Step : 22

x 0 = 3/1- (-2 x 339704331534)/1- (3 x -65348350758)/1 = 875453715345
x 1 = 3/1- (3 x -362756280870)/1- (1 x -65348350758)/1 = 1153617193371
x 2 = 9/-1- (1 x -362756280870)/-1- (-3 x 339704331534)/-1 = -1381869275481

Step : 23

x 0 = 3/1- (-2 x 1153617193371)/1- (3 x -1381869275481)/1 = 6452842213188
x 1 = 3/1- (3 x 875453715345)/1- (1 x -1381869275481)/1 = -1244491870551
x 2 = 9/-1- (1 x 875453715345)/-1- (-3 x 1153617193371)/-1 = -2585397864777

Step : 24

x 0 = 3/1- (-2 x -1244491870551)/1- (3 x -2585397864777)/1 = 5267209853232
x 1 = 3/1- (3 x 6452842213188)/1- (1 x -2585397864777)/1 = -16773128774784
x 2 = 9/-1- (1 x 6452842213188)/-1- (-3 x -1244491870551)/-1 = 10186317824832

Step : 25

x 0 = 3/1- (-2 x -16773128774784)/1- (3 x 10186317824832)/1 = -64105211024061
x 1 = 3/1- (3 x 5267209853232)/1- (1 x 10186317824832)/1 = -25987947384525
x 2 = 9/-1- (1 x 5267209853232)/-1- (-3 x -16773128774784)/-1 = 55586596177575

Step : 26

x 0 = 3/1- (-2 x -25987947384525)/1- (3 x 55586596177575)/1 = -218735683301772
x 1 = 3/1- (3 x -64105211024061)/1- (1 x 55586596177575)/1 = 136729036894611.02
x 2 = 9/-1- (1 x -64105211024061)/-1- (-3 x -25987947384525)/-1 = 13858631129505

Step : 27

x 0 = 3/1- (-2 x 136729036894611.02)/1- (3 x 13858631129505)/1 = 231882180400710.03
x 1 = 3/1- (3 x -218735683301772)/1- (1 x 13858631129505)/1 = 642348418775814
x 2 = 9/-1- (1 x -218735683301772)/-1- (-3 x 136729036894611.02)/-1 = -628922793985614

Step : 28

x 0 = 3/1- (-2 x 642348418775814)/1- (3 x -628922793985614)/1 = 3171465219508473
x 1 = 3/1- (3 x 231882180400710.03)/1- (1 x -628922793985614)/1 = -66723747216513.125
x 2 = 9/-1- (1 x 231882180400710.03)/-1- (-3 x 642348418775814)/-1 = -1695163075926741

Step : 29

x 0 = 3/1- (-2 x -66723747216513.125)/1- (3 x -1695163075926741)/1 = 4952041733347200
x 1 = 3/1- (3 x 3171465219508473)/1- (1 x -1695163075926741)/1 = -7819232582598676
x 2 = 9/-1- (1 x 3171465219508473)/-1- (-3 x -66723747216513.125)/-1 = 3371636461158003.5

Step : 30

x 0 = 3/1- (-2 x -7819232582598676)/1- (3 x 3371636461158003.5)/1 = -25753374548671356
x 1 = 3/1- (3 x 4952041733347200)/1- (1 x 3371636461158003.5)/1 = -18227761661199600
x 2 = 9/-1- (1 x 4952041733347200)/-1- (-3 x -7819232582598676)/-1 = 28409739481143220

Step : 31

x 0 = 3/1- (-2 x -18227761661199600)/1- (3 x 28409739481143220)/1 = -121684741765828860
x 1 = 3/1- (3 x -25753374548671356)/1- (1 x 28409739481143220)/1 = 48850384164870850
x 2 = 9/-1- (1 x -25753374548671356)/-1- (-3 x -18227761661199600)/-1 = 28929910434927436

Step : 32

x 0 = 3/1- (-2 x 48850384164870850)/1- (3 x 28929910434927436)/1 = 10911037024959396
x 1 = 3/1- (3 x -121684741765828860)/1- (1 x 28929910434927436)/1 = 336124314862559170
x 2 = 9/-1- (1 x -121684741765828860)/-1- (-3 x 48850384164870850)/-1 = -268235894260441400

Step : 33

x 0 = 3/1- (-2 x 336124314862559170)/1- (3 x -268235894260441400)/1 = 1476956312506442800
x 1 = 3/1- (3 x 10911037024959396)/1- (1 x -268235894260441400)/1 = 235502783185563230
x 2 = 9/-1- (1 x 10911037024959396)/-1- (-3 x 336124314862559170)/-1 = -997461907562718100

Step : 34

x 0 = 3/1- (-2 x 235502783185563230)/1- (3 x -997461907562718100)/1 = 3463391289059280400
x 1 = 3/1- (3 x 1476956312506442800)/1- (1 x -997461907562718100)/1 = -3433407029956610000
x 2 = 9/-1- (1 x 1476956312506442800)/-1- (-3 x 235502783185563230)/-1 = 770447962949753100

Step : 35

x 0 = 3/1- (-2 x -3433407029956610000)/1- (3 x 770447962949753100)/1 = -9178157948762480000
x 1 = 3/1- (3 x 3463391289059280400)/1- (1 x 770447962949753100)/1 = -11160621830127593000
x 2 = 9/-1- (1 x 3463391289059280400)/-1- (-3 x -3433407029956610000)/-1 = 13763612378929111000

Step : 36

x 0 = 3/1- (-2 x -11160621830127593000)/1- (3 x 13763612378929111000)/1 = -63612080797042524000
x 1 = 3/1- (3 x -9178157948762480000)/1- (1 x 13763612378929111000)/1 = 13770861467358327000
x 2 = 9/-1- (1 x -9178157948762480000)/-1- (-3 x -11160621830127593000)/-1 = 24303707541620300000

Step : 37

x 0 = 3/1- (-2 x 13770861467358327000)/1- (3 x 24303707541620300000)/1 = -45369399690144250000
x 1 = 3/1- (3 x -63612080797042524000)/1- (1 x 24303707541620300000)/1 = 166532534849507260000
x 2 = 9/-1- (1 x -63612080797042524000)/-1- (-3 x 13770861467358327000)/-1 = -104924665199117500000

Step : 38

x 0 = 3/1- (-2 x 166532534849507260000)/1- (3 x -104924665199117500000)/1 = 647839065296367000000
x 1 = 3/1- (3 x -45369399690144250000)/1- (1 x -104924665199117500000)/1 = 241032864269550260000
x 2 = 9/-1- (1 x -45369399690144250000)/-1- (-3 x 166532534849507260000)/-1 = -544967004238666000000

Step : 39

x 0 = 3/1- (-2 x 241032864269550260000)/1- (3 x -544967004238666000000)/1 = 2.1169667412550985e+21
x 1 = 3/1- (3 x 647839065296367000000)/1- (1 x -544967004238666000000)/1 = -1.398550191650435e+21
x 2 = 9/-1- (1 x 647839065296367000000)/-1- (-3 x 241032864269550260000)/-1 = -75259527512283800000

Step : 40

x 0 = 3/1- (-2 x -1.398550191650435e+21)/1- (3 x -75259527512283800000)/1 = -2.5713218007640184e+21
x 1 = 3/1- (3 x 2.1169667412550985e+21)/1- (1 x -75259527512283800000)/1 = -6.275640696253011e+21
x 2 = 9/-1- (1 x 2.1169667412550985e+21)/-1- (-3 x -1.398550191650435e+21)/-1 = 6.312617316206403e+21