Part A: Nonlinear Equations

1. Graphical Method

Consider equation

$$f(x) = x^3 - 2x - 5 = 0$$

a) Is equation linear or nonlinear? Justify.
b) Determine two intervals where root may exist using sign changes.
c) Sketch graph of $f(x)$ and estimate root graphically.

2. Bisection Method

Consider

$$f(x) = x^3 - 7 = 0$$

a) Show that root exists in interval $[1,2]$.
b) Why is interval $[1,2]$ suitable for Bisection Method?
c) Determine root using Bisection Method with $\mathrm{TOL} = 0.005$.

3. Fixed Point Iteration Method

Given

$$x^3 + x - 6 = 0$$

a) Write two possible fixed-point forms of equation.
b) Choose suitable initial guess.
c) Find root using Fixed Point Iteration up to error less than $0.005$.

4. Secant Method

Consider

$$f(x) = x^3 - 4x - 9 = 0$$

a) Show that root lies between $2$ and $3$.
b) Select appropriate initial approximations $x_0$ and $x_1$.
c) Apply Secant Method for five iterations and present results in table.

5. Newton-Raphson Method

Consider

$$f(x) = x^3 - 2x^2 + 3 = 0$$

a) Write stopping criterion used in Newton-Raphson Method.
b) Derive Newton-Raphson iteration formula for this equation.
c) Find root after six iterations.

Part B: System of Linear Equations

6. Gaussian Elimination Method

Consider system:

$$3x + y - z = 4$$ $$2x - 4y + 5z = -3$$ $$x + 2y + 3z = 10$$

a) Write augmented matrix.
b) Convert matrix into upper triangular form.
c) Determine values of $x$, $y$, and $z$ using Gaussian Elimination.

7. Gaussian Elimination with Partial Pivoting

Consider:

$$0.2x + 3y + z = 8$$ $$4x + y - z = 7$$ $$2x + 5y + 2z = 10$$

a) Explain why partial pivoting is required in this problem.
b) Perform necessary row interchanges.
c) Solve system using Gaussian Elimination with Partial Pivoting.

8. Gauss-Seidel Method

Given:

$$10x + y - z = 11$$ $$2x + 10y + z = 12$$ $$-2x + 3y + 10z = 14$$

a) Rearrange equations into iterative form.
b) Verify whether system is diagonally dominant.
c) Starting with $x_0 = y_0 = z_0 = 0$, perform four iterations of Gauss-Seidel Method.

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