Question

20x+2y+6z=28,
X+20y+9z=-23,
2x-7y-20z=-57.
In gauss seidal method

Answer 1

using the gauss seidal method, values of x=1.88, y=2.09, and z= -2.305.

Solution:

First, check for the convergence of approximations,

20 > 2 + 6

20 > 1 + 9

20 > 2 + 7

Hence, the given system of equations is strongly diagonally dominant, which ensures the convergence of approximations. Let us take the initial approximation, x(0) = 0, y(0) = 0 and z(0) = 0.

First Iteration:

x(1) = 1/20[28 – 2 × 0 – 6 × 0 ] = 1.400

y(1) = 1/20[ 23 - 1.400 – 9 × 0 ] = 1.0800

z(1) = 1/20[ - 57 – 2 × 1.400 + 7 × 1.0800] = - 2.612

Second Iteration:

x(2) = 1/20[28 – 2 × 1.0800 – 6 × - 2.612 ] = 2.0756

y(2) = 1/20[ 23 - 2.0756 – 9 × (- 2.612) ] = 2.22162

z(2) = 1/20[ -57 – 2 × 2.0756 + 7 × 2.22162] = - 2.279

Third Iteration:

x(3) = 1/20[28 – 2 ×2.22162 – 6 × (- 2.279) ] = 1.8618

y(3) = 1/20[ 23 - 1.8618 – 9 × (- 2.279) ] = 2.0829

z(3) = 1/20[ -57 – 2 ×1.8618 + 7 × 2.0829] = - 2.3071

Fourth Iteration:

x(4) = 1/20[28 – 2 ×2.0829 – 6 × (- 2.3071) ] = 1.88

y(4) = 1/20[ 23 - 1.88 – 9 × (- 2.3071) ] = 2.09

z(4) = 1/20[ -57 – 2 ×1.88 + 7 × 2.09] = - 2.305

Fifth Iteration

x(5) = 1/20[28 – 2 ×2.09 – 6 × (- 2.305) ] = 1.88

y(5) = 1/20[ 23 - 1.88 – 9 × (- 2.305) ] = 2.09

z(5) = 1/20[ -57 – 2 ×1.88 + 7 × 2.09] = - 2.305

∴ the values of x, y, and z in the fourth and fifth iterations are similar.

So, the value of x =1.88, y= 2.09, and z= - 2.305

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