Bisection Method Root Finding For F(x) = 4x² - 3 In [0, 1]

The bisection method is a powerful numerical technique for approximating the roots of a continuous function. It's a root-finding algorithm that repeatedly bisects an interval and then selects the subinterval where a root must lie for further processing. This method is based on the Intermediate Value Theorem, which states that if a continuous function f(x) changes sign over an interval [a, b], then there exists at least one root within that interval.

Understanding the Bisection Method

At its core, the bisection method is an iterative process. It begins with an interval [a, b] where the function f(x) has opposite signs at the endpoints, i.e., f(a) * f(b) < 0. This ensures that a root lies within the interval. The method then proceeds as follows:

1. Calculate the midpoint: Find the midpoint of the interval, denoted as c = (a + b) / 2.
2. Evaluate the function at the midpoint: Compute f(c).
3. Determine the subinterval:
   • If f(c) = 0, then c is the root, and the process terminates.
   • If f(a) * f(c) < 0, then a root lies in the interval [a, c]. Set b = c.
   • If f(b) * f(c) < 0, then a root lies in the interval [c, b]. Set a = c.
4. Repeat: Go back to step 1 with the new interval [a, b].

The process continues until the interval becomes sufficiently small or a desired level of accuracy is achieved. The bisection method is guaranteed to converge to a root if the initial interval contains a root and the function is continuous within that interval. However, it's a relatively slow method compared to other root-finding algorithms, such as Newton's method, but its simplicity and guaranteed convergence make it a valuable tool.

Advantages of the Bisection Method:

• Guaranteed Convergence: The method is guaranteed to converge to a root if the initial interval contains a root and the function is continuous within that interval. This is a significant advantage over other methods that may not always converge.
• Simplicity: The algorithm is straightforward and easy to implement. It only involves basic arithmetic operations and function evaluations.
• Robustness: The method is not sensitive to the shape of the function, unlike methods that rely on derivatives.

Disadvantages of the Bisection Method:

• Slow Convergence: The bisection method has a linear rate of convergence, meaning that the interval size is halved in each iteration. This can be slow compared to other methods with faster convergence rates.
• Requires Initial Interval: The method requires an initial interval [a, b] where the function changes sign. Finding such an interval may not always be easy.
• Cannot Find Complex Roots: The bisection method is designed to find real roots. It cannot be used to find complex roots.

Applying the Bisection Method to f(x) = 4x² - 3 = 0 in the Interval [0, 1]

Let's apply the bisection method to find the root of the equation f(x) = 4x² - 3 = 0 in the interval [0, 1]. This is a classic example that demonstrates the method's iterative nature. We will perform four iterations to approximate the root. This step-by-step process shows how the method narrows down the interval containing the root with each iteration.

Initial Interval: [a, b] = [0, 1]

First, we need to verify that there is a root within the interval [0, 1]. We do this by evaluating the function at the endpoints:

• f(0) = 4(0)² - 3 = -3
• f(1) = 4(1)² - 3 = 1

Since f(0) is negative and f(1) is positive, there is a root in the interval [0, 1] according to the Intermediate Value Theorem.

Iteration 1:

1. Calculate the midpoint: c = (0 + 1) / 2 = 0.5
2. Evaluate the function at the midpoint: f(0.5) = 4(0.5)² - 3 = -2
3. Determine the subinterval: Since f(0) is negative and f(0.5) is negative, the root does not lie in [0, 0.5]. Since f(0.5) is negative and f(1) is positive, the root lies in the interval [0.5, 1].
4. Update the interval: [a, b] = [0.5, 1]

Iteration 2:

1. Calculate the midpoint: c = (0.5 + 1) / 2 = 0.75
2. Evaluate the function at the midpoint: f(0.75) = 4(0.75)² - 3 = -0.75
3. Determine the subinterval: Since f(0.5) is negative and f(0.75) is negative, the root does not lie in [0.5, 0.75]. Since f(0.75) is negative and f(1) is positive, the root lies in the interval [0.75, 1].
4. Update the interval: [a, b] = [0.75, 1]

Iteration 3:

1. Calculate the midpoint: c = (0.75 + 1) / 2 = 0.875
2. Evaluate the function at the midpoint: f(0.875) = 4(0.875)² - 3 = 0.0625
3. Determine the subinterval: Since f(0.75) is negative and f(0.875) is positive, the root lies in the interval [0.75, 0.875].
4. Update the interval: [a, b] = [0.75, 0.875]

Iteration 4:

1. Calculate the midpoint: c = (0.75 + 0.875) / 2 = 0.8125
2. Evaluate the function at the midpoint: f(0.8125) = 4(0.8125)² - 3 = -0.359375
3. Determine the subinterval: Since f(0.8125) is negative and f(0.875) is positive, the root lies in the interval [0.8125, 0.875].
4. Update the interval: [a, b] = [0.8125, 0.875]

Approximate Root:

After four iterations, the interval containing the root is [0.8125, 0.875]. We can take the midpoint of this interval as an approximation of the root: (0.8125 + 0.875) / 2 = 0.84375. Thus, after four iterations, the bisection method gives us an approximate root of 0.84375.

Summary of Iterations:

Calculating the Exact Root:

The exact root can be calculated analytically by solving the quadratic equation 4x² - 3 = 0. Adding 3 to both sides and then dividing by 4 gives x² = 3/4. Taking the square root of both sides gives x = ±√(3/4) = ±√3 / 2. Since we are looking for the root in the interval [0, 1], the positive root is the one we want. The exact root is x = √3 / 2 ≈ 0.866025.

Comparing Approximate and Exact Root:

Our approximation after four iterations is 0.84375, while the exact root is approximately 0.866025. The bisection method has brought us reasonably close to the actual root after only four iterations. However, there is still a difference, and further iterations would be needed to improve the accuracy. The error can be reduced by continuing the iterations, each time halving the interval and converging closer to the true root. The number of iterations required depends on the desired level of accuracy.

Error Analysis:

The error after n iterations of the bisection method can be estimated by the formula |bₙ - aₙ| / 2, where [aₙ, bₙ] is the interval after n iterations. In our case, after 4 iterations, the interval is [0.8125, 0.875], so the error is approximately (0.875 - 0.8125) / 2 = 0.03125. This error represents the maximum possible difference between our approximation and the true root within the interval.

Practical Applications of the Bisection Method:

• Root Finding in Engineering and Science: The bisection method is widely used in engineering and scientific calculations to find roots of equations that model physical phenomena. This includes solving for equilibrium points, critical values, and other parameters.
• Optimization Algorithms: The method can be incorporated into optimization algorithms to find the minimum or maximum of a function. By finding the roots of the derivative of the function, we can identify potential extrema.
• Computer Graphics and Simulations: The bisection method is used in computer graphics to solve equations related to ray tracing, collision detection, and other simulations.
• Financial Modeling: In finance, the bisection method can be used to find the internal rate of return (IRR) of an investment or to price financial instruments.
• Numerical Analysis: As a fundamental algorithm in numerical analysis, the bisection method serves as a basis for understanding more advanced root-finding techniques.

Conclusion

The bisection method is a reliable and straightforward technique for approximating roots of continuous functions. While its convergence is relatively slow compared to other methods, its guaranteed convergence and ease of implementation make it a valuable tool in various fields. By repeatedly bisecting an interval and selecting the subinterval containing a root, the method efficiently narrows down the search space, providing increasingly accurate approximations. In our example, four iterations of the bisection method applied to the equation 4x² - 3 = 0 in the interval [0, 1] yielded an approximate root of 0.84375, which is reasonably close to the exact root of approximately 0.866025. Further iterations would improve the accuracy, showcasing the iterative power of the bisection method.