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The moon hung low over the ruins of Valenreach, a pale coin gleaming through tattered clouds. The wind hissed across the broken stones, whispering names no living tongue remembered. A lone traveler crossed the desolate bridge, her cloak snapping like a banner in the storm. The sigil on her shoulder — a serpent biting its own tail — marked her as a seeker of the Old Flame.

Eira had been walking for seven nights without rest. Her lantern’s blue fire flickered weakly, feeding on the last fragments of soulstone dust. Around her, the ruins pulsed faintly, as though the city itself still breathed. Each step echoed with the weight of the dead.

The stories said Valenreach had once been the crown of the world, built by dreamweavers who shaped reality like clay. They had summoned light from the stars and carved it into towers that pierced heaven. But pride had undone them. The Flame they sought to control turned inward, devouring the city in a single night of brilliance.

Now only shadows remained.

Eira reached the central square where a broken statue rose from a pool of black water. It was a woman’s figure, arms raised toward the sky — the likeness of the last Flamekeeper. Her stone eyes seemed to follow Eira as she approached.

“You feel it too, don’t you?” a voice murmured from the darkness.

She turned sharply, hand on her blade. A figure stepped from the mist — tall, cloaked, and wearing a mask of bone. His presence made the air tremble.

“Who are you?” Eira demanded.

“A memory,” he said. “Or what’s left of one.”

His mask tilted slightly, and Eira saw the faint glow of eyes beneath. “You carry a shard,” he continued. “The Flame still answers to those who remember its song.”

She reached into her satchel, drawing out a small crystal pulsing with faint light. “I’m taking it to the Ember Vault,” she said. “It must be sealed before the eclipse.”

The masked man laughed — not cruelly, but with sorrow. “Seal it? You cannot seal what was never meant to end. The Flame is not a curse, Eira. It’s a promise.”

Lightning tore across the sky, illuminating the ruins in a blinding flash. When her vision cleared, the man was gone. In his place, the statue’s eyes glowed faintly — twin sparks of living fire.

Eira felt the ground shudder beneath her feet. The pool at the statue’s base rippled, spreading outward in rings of light. The city was waking.

She tightened her grip on the crystal and whispered a prayer to gods long dead.

“If the Flame demands a keeper,” she said quietly, “then let it be me.”

The wind rose into a howl, the fire in her lantern flared bright, and the ruins of Valenreach began to burn once more — not in destruction, but in rebirth.

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1. GROUP 14 ASSIGNMENT

2. NAME REG NO COURSE KATUMBA JEREMIAH BU/UP/2024/5252 AMI TOSKIN CHRIS BU/UP/2023/0843 WAR 3 ARACH GLADYS BU/UP/2024/1016 WAR MUTEBE JOEL BU/UP/2024/1043 WAR TWALE JOSHUA BU/UP/2024/5259 APE BASALIRWA ROBERT BU/UP/2024/1004 WAR AWENE SOLOMON BU/UP/2024/1021 WAR OMONA PATRIC BU/UP/2024/0986 MEB ACHENG AUDREY LORNA BU/UP/2024/0822 AMI ACIPA BRIGDET BU/UP/2024/1078 WAR

3. ASSIGNMENT QUESTION In your respective groups, utilize the knowledge of algorithm development, control structures and (overview of programming languages, introduction to MATLAB, Fundamentals of MATLAB programming and Data Visualization and plotting) from module one to four on the following problems.   a) All Numerical approximate methods for finding the solutions to functions, these include but are not limited to.   1. Newton Raphson Method   2. Secant method and Bisection Method.   b) All methods for solving differential equations numerically, these include but not limited to,   1. Range kutta 4   2. Range kutta 2   3.Euler   (c) Ensure to apply a) and b) on a practical real-world problem   d) Note: Ensure that the different methods are tested on similar problems each (minimum of two). This will help you to plot graphs that compare the problems analytical solutions to the solutions obtained by the different methods along with the computation time.  

4. SOLUTION clear; close all ; clc; PART 1 Root finding methods %Define first function and its derivative f = @(x) x.^3 - x - 2; % Example nonlinear function df = @(x) 3*x.^2 - 1; % Derivative of f(x)   %Parameters for iteration tol = 1e-6; % Tolerance for convergence max_iter = 100; % Maximum allowed iterations

5. 1a) NEWTON-RAPHSON METHOD %Newton-Raphson Method fprintf( 'Running Newton-Raphson Method...\n' ); Running Newton-Raphson Method... tic; [x_nr, iter_nr] = newton_raphson(f, df, 1.5, tol, max_iter); time_nr = toc; fprintf( 'Newton-Raphson root: %.6f, iterations: %d, time: %.6f s\n' , x_nr, iter_nr, time_nr); Newton-Raphson root: 1.521380, iterations: 3, time: 0.005693 s SECANT METHOD %Secant Method fprintf( 'Running Secant Method...\n' ); Running Secant Method... tic; [x_secant, iter_secant] = secant_method(f, 1, 2, tol, max_iter); time_secant = toc; fprintf( 'Secant root: %.6f, iterations: %d, time: %.6f s\n' , x_secant, iter_secant, time_secant); Secant root: 1.521380, iterations: 7, time: 0.005751 s      

6. SECANT METHOD %Secant Method fprintf( 'Running Secant Method...\n' ); Running Secant Method... tic; [x_secant, iter_secant] = secant_method(f, 1, 2, tol, max_iter); time_secant = toc; fprintf( 'Secant root: %.6f, iterations: %d, time: %.6f s\n' , x_secant, iter_secant, time_secant); Secant root: 1.521380, iterations: 7, time: 0.005751 s      

7. 1c) BISECTION METHOD %Bisection Method fprintf( 'Running Bisection Method...\n' ); Running Bisection Method... tic; [x_bisect, iter_bisect] = bisection_method(f, 1, 2, tol, max_iter); time_bisect = toc; fprintf( 'Bisection root: %.6f, iterations: %d, time: %.6f s\n' , x_bisect, iter_bisect, time_bisect); Bisection root: 1.521380, iterations: 20, time: 0.006785 s PLOT % Plot: Root-finding computation times methods = { 'Newton-Raphson' , 'Secant' , 'Bisection' }; times = [time_nr, time_secant, time_bisect];   figure; bar(times); set(gca, 'XTickLabel' , methods); ylabel( 'Computation time (seconds)' ); title( 'Root Finding Methods Computation Time' ); grid on ;

8. PART 2 %%Numerical Solutions of ODE % ODE: dy/dt = -2y + e^(-t) odefun = @(t,y) -2*y + exp(-t); % Analytical solution y_analytical = @(t) (1/3)*exp(-2*t) + (2/3)*exp(-t); % Initial condition and domain t0 = 0; tf = 5; y0 = 1; h = 0.1; t_vals = t0:h:tf; % Time vector N = length(t_vals); % Number of steps % Preallocate solution arrays y_euler = zeros(1, N); y_rk2 = zeros(1, N); y_rk4 = zeros(1, N); % Initial values y_euler(1) = y0; y_rk2(1) = y0; y_rk4(1) = y0;

9. EULER METHOD %Euler Method fprintf( '\nRunning Euler Method...\n' ); Running Euler Method... tic; for i=1:N-1 y_euler(i+1) = y_euler(i) + h*odefun(t_vals(i), y_euler(i)); end time_euler = toc; fprintf( 'Euler Method time: %.6f s\n' , time_euler); Euler Method time: 0.006484 s

10. RANGE KUTTA 2 %RK2 Method fprintf( 'Running RK2 Method...\n' ); Running RK2 Method... tic; for i=1:N-1 k1 = odefun(t_vals(i), y_rk2(i)); k2 = odefun(t_vals(i)+h, y_rk2(i)+h*k1); y_rk2(i+1) = y_rk2(i) + h/2*(k1 + k2); end time_rk2 = toc; fprintf( 'RK2 Method time: %.6f s\n' , time_rk2); RK2 Method time: 0.005310 s

11. RANGE KUTTA 4 %RK4 Method fprintf( 'Running RK4 Method...\n' ); Running RK4 Method... tic; for i=1:N-1 k1 = odefun(t_vals(i), y_rk4(i)); k2 = odefun(t_vals(i)+h/2, y_rk4(i)+h*k1/2); k3 = odefun(t_vals(i)+h/2, y_rk4(i)+h*k2/2); k4 = odefun(t_vals(i)+h, y_rk4(i)+h*k3); y_rk4(i+1) = y_rk4(i) + h/6*(k1 + 2*k2 + 2*k3 + k4); end time_rk4 = toc; fprintf( 'RK4 Method time: %.6f s\n' , time_rk4); RK4 Method time: 0.007025 s

12. PLOTS % Analytical solution y_true = y_analytical(t_vals); %Error Analysis err_euler = max(abs(y_true - y_euler)); err_rk2 = max(abs(y_true - y_rk2)); err_rk4 = max(abs(y_true - y_rk4)); fprintf( 'Max Errors -> Euler: %.6e, RK2: %.6e, RK4: %.6e\n' , err_euler, err_rk2, err_rk4); Max Errors -> Euler: 7.009266e-02, RK2: 8.450842e-02, RK4: 8.333152e-02 %Plot: ODE solutions figure; plot(t_vals, y_true, 'k-' , 'LineWidth' , 2); hold on ; plot(t_vals, y_euler, 'r--' ); plot(t_vals, y_rk2, 'b-.' ); plot(t_vals, y_rk4, 'g:' ); legend( 'Analytical' , 'Euler' , 'RK2' , 'RK4' , 'Location' , 'best' ); xlabel( 't' ); ylabel( 'y(t)' ); title( 'Numerical Solutions of ODE vs Analytical Solution' ); grid on ;

13. %Plot: ODE computation times --- figure; bar([time_euler, time_rk2, time_rk4]); set(gca, 'XTickLabel' , { 'Euler' , 'RK2' , 'RK4' }); ylabel( 'Computation time (seconds)' ); title( 'ODE Numerical Methods Computation Time' ); grid on ;

14. FUNCTIONS FOR ROOT FINDING %%Functions for Root Finding function [root, iter] = newton_raphson(f, df, x0, tol, max_iter) % Newton-Raphson Method for solving f(x)=0 x = x0; for iter = 1:max_iter x_new = x - f(x)/df(x); if abs(x_new - x) < tol root = x_new; return ; end x = x_new; end error( 'Newton-Raphson did not converge' ); end   function [root, iter] = secant_method(f, x0, x1, tol, max_iter) % Secant Method for solving f(x)=0 for iter = 1:max_iter f_x0 = f(x0); f_x1 = f(x1); x2 = x1 - f_x1*(x1 - x0)/(f_x1 - f_x0); if abs(x2 - x1) < tol root = x2; return ; end x0 = x1; x1 = x2; end error( 'Secant method did not converge' ); end   function [root, iter] = bisection_method(f, a, b, tol, max_iter)

15. % Bisection Method for solving f(x)=0 if f(a)*f(b) > 0 error( 'Function has same signs at interval endpoints' ); end for iter = 1:max_iter c = (a + b)/2; if abs(f(c)) < tol || (b - a)/2 < tol root = c; return ; end if f(a)*f(c) < 0 b = c; else a = c; end end error( 'Bisection method did not converge' ); end

16. PRACTICAL APPLICATION Practical applications of these numerical methods include: Population Dynamics: Logistic growth model for predicting population trends. Mechanical Systems: Root-finding for equilibrium points in structures and beams. Engineering Analysis: ODE solvers for heat transfer, fluid flow, and vibration systems.

17. CONLUSION This study demonstrated the effectiveness of different numerical methods for root-finding and ODE solving. The comparison highlighted trade-offs between computational efficiency and accuracy. In real-world scenarios, RK4 is recommended for ODEs due to its robustness, while Newton-Raphson is preferred for root-finding if the derivative is available. These methods remain vital tools in engineering and scientific problem-solving.