**Course Title:** MATLAB Programming: Applications in Engineering, Data Science, and Simulation **Section Title:** Numerical Computation and Linear Algebra **Topic:** Root-finding methods: Bisection, Newton's method, etc. **Overview** In this topic, we will explore various root-finding methods used to solve non-linear equations. These methods are essential in numerical computation and have numerous applications in engineering, data science, and simulation. By the end of this topic, you will understand the different approaches to finding roots and learn how to implement them using MATLAB. **What is Root-finding?** Root-finding involves finding the values of a variable (x) that makes a given function (f(x)) equal to zero. In other words, we want to solve the equation: f(x) = 0 This can be acheived by using different methods, and the choice of method depends on the complexity of the function and the desired level of accuracy. **Bisection Method** The bisection method is a simple yet powerful approach to root-finding. This method works by repeatedly dividing the interval in which the root lies and selecting the new interval based on the midpoint. Here's how the bisection method works: 1. Define the interval `[a, b]` within which the root lies. 2. Calculate the midpoint `c = (a + b) / 2` 3. If `f(c)` is zero or close to it, the midpoint `c` is the approximate root. 4. If `f(c)` is not zero, check if the function value changes sign between `a` and `c`, or between `c` and `b`. This determines which interval contains the root. **MATLAB Implementation: Bisection Method** To implement the bisection method in MATLAB, you can use the following code: ```matlab function x_root = bisection_method(f, a, b, tol) if f(a) * f(b) >= 0 error('Root not within the given interval') end while (b - a) / 2 > tol c = (a + b) / 2; if f(c) == 0 x_root = c; break elseif f(c) * f(a) < 0 b = c; else a = c; end end x_root = (a + b) / 2; end ``` **Newton's Method** Newton's method is a widely used root-finding method that starts with an initial guess for the root and iteratively updates the guess using the formula: x(n+1) = x(n) - f(x(n)) / f'(x(n)) where `x(n)` is the `n-th` estimate of the root. Here's how Newton's method works: 1. Choose an initial guess `x0` for the root. 2. Calculate the function value `f(x0)` and the derivative `f'(x0)`. 3. Update the estimate using the formula: `x(n+1) = x(n) - f(x(n)) / f'(x(n))`. 4. Repeat step 2-3 until convergence. **MATLAB Implementation: Newton's Method** To implement Newton's method in MATLAB, you can use the following code: ```matlab function x_root = newton_method(f, df, x0, tol) x = x0; while true f_x = f(x); df_x = df(x); x_new = x - f_x / df_x; if abs(x_new - x) < tol x_root = x_new; break end x = x_new; end end ``` **Comparison of Root-finding Methods** The bisection method is often slower than Newton's method, but it is guaranteed to converge to the root if the interval is properly defined. On the other hand, Newton's method can be faster if the initial guess is close to the root. | Method | Pros | Cons | | --- | --- | --- | | Bisection | Guaranteed convergence, easy to implement | Can be slower | | Newton's Method | Can be faster, accurate | Requires derivative, sensitive to initial guess | **Conclusion** In this topic, we covered the basics of root-finding methods and discussed two popular approaches: bisection and Newton's method. We also implemented these methods in MATLAB, providing a foundation for further study and practice. **Key Takeaways:** 1. Root-finding methods are used to solve non-linear equations. 2. The bisection method is simple and guaranteed to converge, but can be slower. 3. Newton's method can be faster, but requires the derivative and an initial guess. 4. MATLAB provides an excellent platform for implementing and visualizing root-finding methods. **Read the official MATLAB documentation on optimizing numerical computations** [https://de.mathworks.com/help/matlab/numerical-computation.html](https://de.mathworks.com/help/matlab/numerical-computation.html) **Do you have any questions about this topic?** Please comment below or ask for help on any aspects that you did not understand in this topic. Next Topic: Working with Polynomials in MATLAB (coming soon!) Please provide feedback below on this topic content before leaving the page.

function x_root = bisection_method(f, a, b, tol)
    if f(a) * f(b) >= 0
        error('Root not within the given interval')
    end

    while (b - a) / 2 > tol
        c = (a + b) / 2;
        if f(c) == 0
            x_root = c;
            break
        elseif f(c) * f(a) < 0
            b = c;
        else
            a = c;
        end
    end

    x_root = (a + b) / 2;
end

function x_root = newton_method(f, df, x0, tol)
    x = x0;
    while true
        f_x = f(x);
        df_x = df(x);
        x_new = x - f_x / df_x;
        if abs(x_new - x) < tol
            x_root = x_new;
            break
        end
        x = x_new;
    end
end