8x-3y+2z=20, 4x+11y-z=33, 6x+3y+12z=35. Solve the equations in 5 steps by Guass-Jacobi method.
Answers
Answer:
8x - 3y + 2z = 20
4x - 11y -z = 33
6x + 3y + 12z = 35
Rewrite 1st equation as
8x = 20 + 3y - 2z
x = 1/8 ( 20 + 3y - 2z ) etc as an iterative process to find better and better values for x
Then we have:
x k+1 = 1/8 ( 20 + 3y k - 2z k )
y k+1 = 1/-11 ( 33 - 4x k+1 + z k )
z k+1 = 1/12 ( 35 - 6 x k+1 - 3 y k+1 )
Initial gauss ( x, y, z ) = ( 0, 0, 0 )
1st Approximation
x₁ = 1/8 ( 20 + 3 (0) - 2 ( 0 ) )
= 1/8 ( 20 )
= 2.5
y₁ = 1/-11 ( 33 - 4 ( 2.5 ) + ( 0 ) )
= 1/-11 ( 23 )
= - 2.90909
z₁ = 1/12 ( 35 - 6 ( 2.5 ) - 3 ( - 2.90909 ) )
= 1/12 ( 35 - 15 + 6.72727
= 1/12 ( 26.272727 )
= 2.189394
2nd Approximation
x₂ = 1/8 ( 20 + 3 ( - 2.090909 ) - 2 ( 2.189394) )
= 1/8 ( 20 - 6.272727 - 4.378788 )
= 1/8 ( 9.348485 )
= 1.168561
y₂ = 1/-11 ( 33 - 4 ( 1.168561) + ( 2.189394 ) )
= 1/-11 ( 33 - 4.674244 + 2.189394 )
= 1/-11 ( 30.5155152 )
= - 2.774105
z₂ = 1/12 ( 35 - 6 ( 1.168561 ) - 3 ( - 2.774105 ) )
= 1/12 ( 35 - 7.011366 + 8.322315 )
= 1/12 ( 36.31095 )
= 3.025913
3rd Approximation
x₃ = 1/8 ( 20 + 3 ( -2.774105 ) - 2 ( 3.025913 ) )
= 1/8 ( 5.625861 )
= 0.703233
y₃ = 1/-11 ( 33 - 4 ( 0.703233 ) + ( 3.025913 ) )
= 1/-11 ( 33.212982 )
= - 3. 019362
z₃ = 1/12 ( 35 - 6 ( 0.703233 ) - 3 ( - 3. 019362 ) )
= 1/12 ( 39.83869 )
= 3.319891