UPSC 2014 Maths Optional Paper 2 Q6b — Step-by-Step Solution

15 marks · Section B

Gauss-Seidel iteration · Numerical Analysis · asked 7× in 13 yrs · Read the full method →

Question

Solve the system of equations

2x_1-x_2=7,\;-x_1+2x_2-x_3=1,\;-x_2+2x_3=1

using Gauss–Seidel iteration method (Perform three iterations).

Standard Gauss-Seidel iteration with most-recent-update rule (vs. Jacobi).

x_1=\dfrac{7+x_2}{2},\quad x_2=\dfrac{1+x_1+x_3}{2},\quad x_3=\dfrac{1+x_2}{2}.

Initial guess: $(x_1,x_2,x_3)=(0,0,0)$ .

Iteration 1:

Iteration 2:

Iteration 3:

Iteration	$x_1$	$x_2$	$x_3$
0	$0.0000$	$0.0000$	$0.0000$
1	$3.5000$	$2.2500$	$1.6250$
2	$4.6250$	$3.6250$	$2.3125$
3	$5.3125$	$4.3125$	$2.65625$

\boxed{\;\text{After 3 iterations: }(x_1,x_2,x_3)=(5.3125,\;4.3125,\;2.65625).\;}