a) Using method of Successive bisection method Or Newton Raphson's method, find the root of the equation x^2 + x 4 = 0 in (1,2) correct to three places of decimals. [4] b) Use Gauss Seidel method to…

a) Finding the root of $x^2+x-4=0$ in (1,2) using the Successive Bisection Method

Step 1: Check the interval

Let $f(x)=x^2+x-4$. We evaluate $f(x)$ at the endpoints of the interval $[1,2]$:

• $f(1)=(1{)}^2+1-4=1+1-4=-2$ (negative)
• $f(2)=(2{)}^2+2-4=4+2-4=2$ (positive)

Since $f(1)$ and $f(2)$ have opposite signs, by the Intermediate Value Theorem, there exists a root in $(1,2)$.

Step 2: Apply the Bisection Method

The bisection formula is: $$x_{n+1}=\frac{a_n+b_n}{2}$$ where $a_n$ and $b_n$ are the current interval endpoints.

We iterate until the interval length $|b_n-a_n|<0.001$ (for 3 decimal places).

At this point, the interval length is $1.564453125-1.5625=0.001953125<0.001$, so we stop.

The approximate root is the midpoint of the final interval: $$\text{Root}\approx \frac{1.5625+1.564453125}{2}=1.5634765625$$ Rounding to three decimal places, the root is: 1.563

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b) Solving the system using the Gauss-Seidel Method

The system is: $$\{\begin{array}{l}3x+4y+8z=7(1)\\ x+20y+z=-18(2)\\ 25x+y-5z=19(3)\end{array}$$

Step 1: Rewrite in Gauss-Seidel form

We solve each equation for the variable with the largest coefficient in its row:

From (1): $$3x=7-4y-8z⟹x=\frac{7-4y-8z}{3}$$

From (2): $$20y=-18-x-z⟹y=\frac{-18-x-z}{20}$$

From (3): $$-5z=19-25x-y⟹z=\frac{25x+y-19}{5}$$

Step 2: Apply Gauss-Seidel iterations

We start with an initial guess $(x_0,y_0,z_0)=(0,0,0)$ and iterate until convergence (changes in $x,y,z$ are very small).

The values stabilize at: $$x\approx 1.6933,y\approx -0.9811,z\approx 0.5233$$

Rounding to three decimal places, the solution is: $x=1.693$, $y=-0.981$, $z=0.523$