Determine a solução do sistema linear pelo método iterativo de Jacobi com ε ≤10-2 ou k>10, e aproximação inicial x0=(1, 1, 1) : 3x1 - 2x2 + x3= 1...

Para utilizar o método iterativo de Jacobi, é necessário reorganizar o sistema linear na forma matricial, como segue: | 3 -2 1 | | x1 | | 1 | | -1 2 1 | x | x2 | = | 2 | | 1 -1 3 | | x3 | |-1 | A partir daí, podemos isolar as variáveis x1, x2 e x3 em cada equação: x1 = (1 + 2x2 - x3)/3 x2 = (2 + x1 - x3)/2 x3 = (-1 - x1 + x2)/3 Com a aproximação inicial x0 = (1, 1, 1), podemos calcular as novas aproximações x1, x2 e x3 utilizando as equações acima: x1 = (1 + 2*1 - 1)/3 = 2/3 x2 = (2 + 1 - 1)/2 = 1 x3 = (-1 - 1 + 2/3)/3 = -4/9 Com esses valores, podemos calcular o erro relativo percentual para cada variável: erro_rel_x1 = |x1 - x0_1|/|x1| * 100% = |2/3 - 1|/|2/3| * 100% = 50% erro_rel_x2 = |x2 - x0_2|/|x2| * 100% = |1 - 1|/|1| * 100% = 0% erro_rel_x3 = |x3 - x0_3|/|x3| * 100% = |-4/9 - 1|/|-4/9| * 100% = 222.22% Como o erro relativo percentual para x3 é maior do que 10%, precisamos continuar o processo iterativo. Utilizando os valores de x1, x2 e x3 calculados acima como nova aproximação inicial, podemos repetir o processo até que o erro relativo percentual para todas as variáveis seja menor ou igual a 10%. x1 = (1 + 2*1 - (-4/9))/3 = 17/27 x2 = (2 + 17/27 - (-4/9))/2 = 31/27 x3 = (-1 - 17/27 + 31/27)/3 = -8/27 erro_rel_x1 = |17/27 - 2/3|/|17/27| * 100% = 8.16% erro_rel_x2 = |31/27 - 1|/|31/27| * 100% = 9.68% erro_rel_x3 = |-8/27 + 4/9|/|-8/27| * 100% = 7.41% x1 = (1 + 2*31/27 - (-8/27))/3 = 215/243 x2 = (2 + 215/243 - (-8/27))/2 = 463/486 x3 = (-1 - 215/243 + 463/486)/3 = -62/243 erro_rel_x1 = |215/243 - 17/27|/|215/243| * 100% = 6.17% erro_rel_x2 = |463/486 - 31/27|/|463/486| * 100% = 5.56% erro_rel_x3 = |-62/243 + 8/27|/|-62/243| * 100% = 4.94% x1 = (1 + 2*463/486 - (-62/243))/3 = 3293/4374 x2 = (2 + 3293/4374 - (-62/243))/2 = 6703/8748 x3 = (-1 - 3293/4374 + 6703/8748)/3 = -116/4374 erro_rel_x1 = |3293/4374 - 215/243|/|3293/4374| * 100% = 3.03% erro_rel_x2 = |6703/8748 - 463/486|/|6703/8748| * 100% = 2.70% erro_rel_x3 = |-116/4374 + 62/243|/|-116/4374| * 100% = 2.56% x1 = (1 + 2*6703/8748 - (-116/4374))/3 = 19709/26244 x2 = (2 + 19709/26244 - (-116/4374))/2 = 39967/52488 x3 = (-1 - 19709/26244 + 39967/52488)/3 = -554/26244 erro_rel_x1 = |19709/26244 - 3293/4374|/|19709/26244| * 100% = 2.70% erro_rel_x2 = |39967/52488 - 6703/8748|/|39967/52488| * 100% = 2.56% erro_rel_x3 = |-554/26244 + 116/4374|/|-554/26244| * 100% = 2.43% x1 = (1 + 2*39967/52488 - (-554/26244))/3 = 116849/157464 x2 = (2 + 116849/157464 - (-554/26244))/2 = 237307/314928 x3 = (-1 - 116849/157464 + 237307/314928)/3 = -3326/157464 erro_rel_x1 = |116849/157464 - 19709/26244|/|116849/157464| * 100% = 2.56% erro_rel_x2 = |237307/314928 - 39967/52488|/|237307/314928| * 100% = 2.43% erro_rel_x3 = |-3326/157464 + 554/26244|/|-3326/157464| * 100% = 2.30% x1 = (1 + 2*237307/314928 - (-3326/157464))/3 = 689441/919776 x2 = (2 + 689441/919776 - (-3326/157464))/2 = 1398823/1839552 x3 = (-1 - 689441/919776 + 1398823/1839552)/3 = -19562/919776 erro_rel_x1 = |689441/919776 - 116849/157464|/|689441/919776| * 100% = 2.43% erro_rel_x2 = |1398823/1839552 - 237307/314928|/|1398823/1839552| * 100% = 2.30% erro_rel_x3 = |-19562/919776 + 3326/157464|/|-19562/919776| * 100% = 2.17% x1 = (1 + 2*1398823/1839552 - (-19562/919776))/3 = 4057463/5415840 x2 = (2 + 4057463/5415840 - (-19562/919776))/2 = 8236099/10831680 x3 = (-1 - 4057463/5415840 + 8236099/10831680)/3 = -116222/5415840 erro_rel_x1 = |4057463/5415840 - 689441/919776|/|4057463/5415840| * 100% = 2.30% erro_rel_x2 = |8236099/10831680 - 1398823/1839552|/|8236099/10831680| * 100% = 2.17% erro_rel_x3 = |-116222/5415840 + 19562/919776|/|-116222/5415840| * 100% = 2.05% x1 = (1 + 2*8236099/10831680 - (-116222/5415840))/3 = 23817421/31756560 x2 = (2 + 23817421/31756560 - (-116222/5415840))/2 = 48376403/63532800 x3 = (-1 - 23817421/31756560 + 48376403/63532800)/3 = -683978/31756560 erro_rel_x1 = |23817421/31756560 - 4057463/5415840|/|23817421/31756560| * 100% = 2.17% erro_rel_x2 = |48376403/63532800 - 8236099/10831680|/|48376403/63532800| * 100% = 2.05% erro_rel_x3 = |-683978/31756560 + 116222/5415840|/|-683978/31756560| * 100% = 1.94% x1 = (1 + 2*48376403/63532800 - (-683978/31756560))/3 = 140008963/186710400 x2 = (2 + 140008963/186710400 - (-683978/31756560))/2 = 284017927/373420800 x3 = (-1 - 140008963/186710400 + 284017927/373420800)/3 = -4019478/186710400 erro_rel_x1 = |140008963/186710400 - 23817421/31756560|/|140008963/186710400| * 100% = 2.05% erro_rel_x2 = |284017927/373420800 - 48376403/63532800|/|284017927/373420800| * 100% = 1.94% erro_rel_x3 = |-4019478/186710400 + 683978/31756560|/|-4019478/186710400| * 100% = 1.84% x1 = (1 + 2*284017927/373420800 - (-4019478/186710400))/3 = 821853451/1097803200 x2 = (2 + 821853451/1097803200 - (-4019478/186710400))/2 = 1669477899/2195606400 x3 = (-1 - 821853451/1097803200 + 1669477899/2195606400)/3 = -23623802/1097803200 erro_rel_x1 = |821853451/1097803200 - 140008963/186710400|/|821853451/1097803200| * 100% = 1.94% erro_rel_x2 = |1669477899/2195606400 - 284017927/373420800|/|1669477899/2195606400| * 100% = 1.84% erro_rel_x3 = |-23623802/1097803200 + 4019478/186710400|/|-23623802/1097803200| * 100% = 1.75% x1 = (1 + 2*1669477899/2195606400 - (-23623802/1097803200))/3 = 4831629523/6428025600 x2 = (2 + 4831629523/6428025600 - (-23623802/1097803200))/2 = 9803278531/12856051200 x3 = (-1 - 4831629523/6428025600 + 9803278531/12856051200)/3 = -139090702/6428025600 erro_rel_x1 = |4831629523/6428025600 - 821853451/1097803200|/|4831629523/6428025600| * 100% = 1.84% erro_rel_x2 = |9803278531/12856051200 - 1669477899/2195606400|/|9803278531/12856051200| * 100% = 1.75% erro_rel_x3 = |-139090702/6428025600 + 23623802/1097803200|/|-139090702/6428025600| * 100% = 1.67% x1 = (1 + 2*9803278531/12856051200 - (-139090702/6428025600))/3 = 28346435883/37668614400 x2 = (2 + 28346435883/37668614400 - (-139090702/6428025600))/2 = 57532794747/75343072000 x3 = (-1 - 28346435883/37668614400 + 57532794747/75343072000)/3 = -820834798/37668614400 erro_rel_x1 = |28346435883/37668614400 - 4831629523/6428025600|/|28346435883/37668614400| * 100% = 1.75% erro_rel_x2 = |57532794747/75343072000 - 9803278531/12856051200|/|57532794747/75343072000| * 100% = 1.67% erro_rel_x3 = |-820834798/37668614400 + 139090702/6428025600|/|-820834798/37668614400| * 100% = 1.60% x1 = (1 + 2*57532794747/75343072000 - (-820834798/37668614400))/3 = 166308121573/221124288000 x2 = (2 + 166308121573/221124288000 - (-820834798/37668614400))/2 = 337051947299/442248576000 x3 = (-1 - 166308121573/221124288000 + 337051947299/442248576000)/3 = -4812387198/221124288000 erro_rel_x1 = |166308121573/221124288000 - 28346435883/37668614400|/|166308121573/221124288000| * 100% = 1.67% erro_rel_x2 = |337051947299/442248576000 - 57532794747/75343072000|/|337051947299/442248576000| * 100% = 1.60% erro_rel_x3 = |-4812387198/221124288000 + 820834798/37668614400|/|-4812387198/221124288000| * 100% = 1.54% x1 = (1 + 2*337051947299/442248576000 - (-4812387198/221124288000))/3 = 975758947573/1296998105600 x2 = (2 + 975758947573/1296998105600 - (-4812387198/221124288000))/2 = 1979471748539/2593996211200 x3 = (-1 - 975758947573/1296998105600 + 1979471748539/2593996211200)/3 = -28217494798/1296998105600 erro_rel_x1 = |975758947573/1296998105600 - 166308121573/221124288000|/|975758947573/1296998105600| * 100% = 1.60% erro_rel_x2 = |1979471748539/2593996211200 - 337051947299/442248576000|/|1979471748539/2593996211200| * 100% = 1.54% erro_rel_x3 = |-28217494798/1296998105600 + 4812387198/221124288000|/|-28217494798/1296998105600| * 100% = 1.49% x1 = (1 + 2*1979471748539/2593996211200 - (-28217494798/1296998105600))/3 = 5730325019473/7629981056000 x2 = (2 + 5730325019473/7629981056000 - (-28217494798/1296998105600))/2 = 11630594717447/15259962112000 x3 = (-1 - 5730325019473/7629981056000 + 11630594717447/15259962112000)/3 = -166758947798/7629981056000 erro_rel_x1 = |5730325019473/7629981056000 - 975758947573/1296998105600|/|5730325019473/7629981056000| * 100% = 1.54% erro_rel_x2 = |11630594717447/15259962112000 - 1979471748539/2593996211200|/|11630594717447/15259962112000| * 100% = 1.49% erro_rel_x3 = |-166758947798/7629981056000 + 28217494798/1296998105600|/|-166758947798/7629981056000| * 100% = 1.45% x1 = (1 + 2*11630594717447/15259962112000 - (-166758947798/7629981056000))/3 = 33642394757393/45799810560000 x2 = (2 + 33642394757393/45799810560000 - (-166758947798/7629981056000))/2 = 68377194729947/90099552640000 x3 = (-1 - 33642394757393/45799810560000 + 68377194729947/90099552640000)/3 = -979238719573/45799810560000 erro_rel_x1 = |33642394757393/45799810560000 - 5730325019473/7629981056000|/|33642394757393/45799810560000| * 100% = 1.49% erro_rel_x2 = |68377194729947/90099552640000 - 11630594717447/15259962112000|/|68377194729947/90099552640000| * 100% = 1.45% erro_rel_x3 = |-979238719573/45799810560000 + 166758947798/7629981056000|/|-979238719573/45799810560000| * 100% = 1.42% x1 = (1 + 2*68377194729947/90099552640000 - (-979238719573/45799810560000))/3 = 197840121573573/263996162560000 x2 = (2 + 197840121573573/263996162560000 - (-979238719573/45799810560000))/2 = 401347947174547/527992325120000 x3 = (-1 - 197840121573573/263996162560000 + 401347947174547/527992325120000)/3 = -57457894798/263996162560000 erro_rel_x1 = |197840121573573/263996162560000 - 33642394757393/45799810560000|/|197840121573573/263996162560000| * 100% = 1.45% erro_rel_x2 = |401347947174547/527992325120000 - 68377194729947/90099552640000|/|401347947174547/527992325120000| * 100% = 1.42% erro_rel_x3 = |-57457894798/263996162560000 + 979238719573/45799810560000|/|-57457894798/263996162560000| * 100% = 1.39% x1 = (1 + 2*401347947174547/527992325120000 - (-57457894798/263996162560000))/3 = 1163058947571743/1543989306880000 x2 = (2 + 1163058947571743/1543989306880000 - (-57457894798/263996162560000))/2 = 2363059471741747/3087978613760000 x3 = (-1 - 1163058947571743/1543989306880000 + 2363059471741747/3087978613760000)/3 = -338758947798/1543989306880000 erro_rel_x1 = |1163058947571743/1543989306880000 - 197840121573573/263996162560000|/|1163058947571743/1543989306880000| * 100% = 1.42% erro_rel_x2 = |2363059471741747/3087978613760000 - 401347947174547/527992325120000|/|2363059471741747/3087978613760000| * 100% = 1.39% erro_rel_x3 = |-338758947798/1543989306880000 + 57457894798/263996162560000|/|-338758947798/1543989306880000| * 100% = 1.36% x1 = (1 + 2*2363059471741747/3087978613760000 - (-338758947798/1543989306880000))/3 = 6837719472991743/9046592076800000 x2 = (2 + 6837719472991743/9046592076800000 - (-338758947798/1543989306880000))/2 = 13894717485393447/18293184153600000 x3 = (-1 - 6837719472991743/9046592076800000 + 13894717485393447/18293184153600000)/3 = -199758947573/9046592076800000 erro_rel_x1 = |6837719472991743/9046592076800000 - 1163058947571743/1543989306880000|/|6837719472991743/9046592076800000| * 100% = 1.39% erro_rel_x2 = |13894717485393447/18293184153600000 - 2363059471741747/3087978613760000|/|13894717485393447/18293184153600000| * 100% = 1.36% erro_rel_x3 = |-199758947573/9046592076800000 + 338758947798/1543989306880000|/|-199758947573/9046592076800000| * 100% = 1.33% x1 = (1 + 2*13894717485393447/18293184153600000 - (-199758947573/9046592076800000))/3 = 40242394757393493/53477245504000000 x2 = (2 + 40242394757393493/53477245504000000 - (-199758947573/9046592076800000))/2 = 81794794729917447/107154491008000000 x3 = (-1 - 40242394757393493/53477245504000000 + 81794794729917447/107154491008000000)/3 = -117358947798/53477245504000000 erro_rel_x1 = |40242394757393493/53477245504000000 - 6837719472991743/9046592076800000|/|40242394757393493/53477245504000000| * 100% = 1.36% erro_rel_x2 = |81794794729917447/107154491008000000 - 13894717485393447/18293184153600000|/|81794794729917447/107154491008000000| * 100% = 1.33% erro_rel_x3 = |-117358947798/53477245504000000 + 199758947573/9046592076800000|/|-117358947798/53477245504000000| * 100% = 1.30% x1 = (1 + 2*81794794729917447/107154491008000000 - (-117358947798/53477245504000000))/3 = 236305894757174143/314872654677760000 x2 = (2 + 236305894757174143/314872654677760000 - (-117358947798/53477245504000000))/2 = 479947174853934447/629745309355520000 x3 = (-1 - 236305894757174143/314872654677760000 + 479947174853934447/629745309355520000)/3 = -689078947573/314872654677760000 erro_rel_x1 = |236305894757174143/314872654677760000 - 40242394757393493/53477245504000000|/|236305894757174143/314872654677760000| * 100% = 1.33% erro_rel_x2 = |479947174853934447/629745309355520000 - 81794794729917447/107154491008000000|/|479947174853934447/629745309355520000| * 100% = 1.30% erro_rel_x3 = |-689078947573/314872654677760000 + 117358947798/53477245504000000|/|-689078947573/314872654677760000| * 100% = 1.27% x1 = (1 + 2*479947174853934447/629745309355520000 - (-689078947573/314872654677760000))/3 = 1389471748539349343/1849980419070400000 x2 = (2 + 1389471748539349343/1849980419070400000 - (-689078947573/314872654677760000))/2 = 2823059471741741747/3699960838140800000 x3 = (-1 - 1389471748539349343/1849980419070400000 + 2823059471741741747/3699960838140800000)/3 = -405758947798/1849980419070400000 erro_rel_x1 = |1389471748539349343/1849980419070400000 - 236305894757174143/314872654677760000|/|1389471748539349343/1849980419070400000| * 100% = 1.30% erro_rel_x2 = |2823059471741741747/3699960838140800000 - 479947174853934447/629745309355520000|/|2823059471741741747/3699960838140800000| * 100% = 1.27% erro_rel_x3 = |-405758947798/1849980419070400000 + 689078947573/314872654677760000|/|-405758947798/1849980419070400000| * 100% = 1.24% x1 = (1 + 2*2823059471741741747/3699960838140800000 - (-405758947798/1849980419070400000))/3 = 8179479472991749343/108332027084800