1. Let's state the problem: We want to find numerical solutions of algebraic equations using iteration methods. 2. One common iterative method is the Fixed Point Iteration, where we rewrite the equation $f(x)=0$ as $x=g(x)$ and then use the iteration formula: $$x_{n+1} = g(x_n)$$ 3. Important rules for convergence: - The function $g(x)$ should be continuous in the interval considered. - The absolute value of the derivative $|g'(x)|$ should be less than 1 near the root to ensure convergence. 4. Example: Suppose we want to solve $x^3 + x -1=0$. Rewrite as $x = 1 - x^3$, so $g(x) = 1 - x^3$. 5. Start with an initial guess, say $x_0=0.5$. Calculate $x_1 = g(x_0) = 1 - (0.5)^3 = 1 - 0.125 = 0.875$. 6. Next iteration: $x_2 = g(x_1) = 1 - (0.875)^3 = 1 - 0.6699 = 0.3301$. 7. Continue iterating until $|x_{n+1} - x_n|$ is less than a desired tolerance. 8. This method approximates the root numerically by successive iterations. 9. Other iterative methods include Newton-Raphson and Secant methods, which often converge faster but require derivatives or multiple initial guesses. In summary, iteration methods provide a practical way to numerically solve algebraic equations by repeatedly applying a function until the solution stabilizes.