UPSC 2019 Maths Optional Paper 2 Q7b — Step-by-Step Solution

15 marks · Section B

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

Question

Apply Gauss-Seidel iteration method to solve the following system of equations:

\begin{aligned}2x+y-2z&=17,\\ 3x+20y-z&=-18,\\ 2x-3y+20z&=25,\end{aligned}

correct to three decimal places.

Technique

Gauss–Seidel: solve each row for its diagonal variable and substitute the most recent values immediately; iterate to 3-dp stability.

Solution

Setup. Solve each equation for its “diagonal” unknown. Equation 2 ( $20y$ ) and equation 3 ( $20z$ ) are strongly diagonally dominant; equation 1 supplies $x$ :

x=\frac{17-y+2z}{2},\qquad y=\frac{-18-3x+z}{20},\qquad z=\frac{25-2x+3y}{20}.

Gauss–Seidel uses each freshly updated value immediately. Start $x=y=z=0$ .

Step 1 — Iterations

$n$	$x$	$y$	$z$
1	8.5000	−2.1750	0.0738
2	9.6613	−2.3455	−0.0680
3	9.6048	−2.3441	−0.0621
4	9.6100	−2.3446	−0.0627
5	9.6096	−2.3446	−0.0626
6	9.6096	−2.3446	−0.0627

Sample of iteration 1: $x=\frac{17-0+0}{2}=8.5$ ; $y=\frac{-18-3(8.5)+0}{20}=\frac{-43.5}{20}=-2.175$ ; $z=\frac{25-2(8.5)+3(-2.175)}{20}=\frac{25-17-6.525}{20}=0.07375$ .

Successive values agree to three decimals from iteration 5 onward.

Answer

\boxed{\;x\approx 9.610,\qquad y\approx -2.345,\qquad z\approx -0.063.\;}