Question

2) Use the Gauss-Seidel method to solve the following system until the percentage relative error is below 0.5% -2x1 + 2x2 – X3 = 25 - 3x1 - 6x2 + 2x3 = -40.5 X1 + x2 + 5x3 = -25.5 a) Record the table-style values. (Iteration, X1, X2, X3, Error X1, Error X2, Error X3). х iteration error X1 x2 x3

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Answer 1

Given -2x, +2xg - X3 = 25 - 3x,- 6x2 + 2x3= - 40.5 x + x + 503 =-25.5 from o -> X, = 2xg - 33-25 २ From => Hg = -32, +223 +40.5 6. 41 From => 23= -25.5-H, -12 5 x=0 X₂ = 0 2,= 0, Tst iteration II iteration, - 12.5 H,= -25 - q = -3xl-12.5) + 2x04 40.5 13 6 Hz = -25.5-1-12.5)-13 5:2 5

0. Given (TV -2x, +2X9 - X3 = 25 + 2x z= - 40.5 X,+ xy + 5x3 = -25.5 - 3x, - 6x2 ili from o => X, = 2X2 - 33-25 २ from 11 » Hq = -30, + 2x3 + 40.5 6 From » Xz= -25.5-8,-42 5 Hence, By Gauss- siedal method – Tst iteration X=0, Ug = 0, X₂ = 0 다. iteration, 2,= -2 -25 -12.5 My = - 3x(-12,5) + 2x0440.5 13 6 Iz = -25.5-1-12.5)-13 -512 5

iteration, = 3.1 X, = 2x13 - (-5.2)-25 3.467 Kg = -3x 3.1 +2x 65.2)+40,5 - 6.4134 Xz = -25.5-3.1-3.467 5 In iteration- -5.826 X, = 2X3.467-(-6.4134)-25 2 7.525 Mg = - 3xl-5,826) + 2x(-6,4134)+40,5 6. -5.4398 Xz = -25.5-1-51826) - 7.525 5 5 iteratim -2.25 % = 2X7.525-(-5:4398) - 25 २ My = -3x(-2,25) + 2% (-5-4398) +40.5. 6.06 6. Az = -25.5 - (~2.25) - 6:06 -5.862

stexotion -3.509 x, = 2x6.06- (-51862) - 25 2 Xg = -3X(-3.509) + 2x (-5.862) +40.5 = 6.5505 6 -5.741 X3= -25.5-(-3.509) - 6.5505 5 yu iteratima - -3.076 X = 2X 6.5505 - (25.747) -25 6.37 Hg = - 3X (-3.076) + 2X (-5.747) + 40.5 6 6.37 Xz = -25.5- (-3,076) – -5.758 5 iteration Hentt, - 3.251 2y = ax 6-37-(-5.758) - 25 2 Xa = -3% (-3.251) + 2x6-5.758) +40.5 6.45 6

W3 = -25.5 - (-3.251) – 6.45 = -5.74 5 Hence, x = -3.25 X = 6,45 a final answer X ₃ = -5.74