2.2. Bisection Method

The Bisection method is a simple and reliable method used for finding a root of a continuous function when the root is bracketed between two values. It is particularly useful when we know that the function changes sign between two values, i.e., f(a)⋅f(b)<0.

The Bisection method is a numerical technique for locating the roots of a function—these roots are the points where the function's value equals zero. To use this method, we first need two values, a and b, such that the function takes on opposite signs at these points. This implies that there is at least one root between these two points due to the Intermediate Value Theorem. The main advantage of the bisection method is its simplicity and reliability.

Imagine you are trying to find the depth of a swimming pool by testing various points. If you know that one end of the pool is shallow (where the water is fewer feet) and the other end is deep (where the water is several feet), you can safely assume that somewhere in between, the transition from shallow to deep occurs. The Bisection method helps pinpoint exactly where that transition happens.

1. Start with an interval [a,b] such that f(a)⋅f(b)<0 (i.e., the function has different signs at the endpoints).
2. Compute the midpoint c=a+b/2.
3. Check the sign of f(c):
   • If f(a)⋅f(c)<0, the root lies between a and c, so set b=c.
   • If f(b)⋅f(c)<0, the root lies between c and b, so set a=c.
4. Repeat the process until the interval is sufficiently small, i.e., |b−a| is less than a specified tolerance.

To find the root using the Bisection method, you proceed through a series of steps:

1. Define an interval [a,b]. You must ensure that the function has different signs at these endpoints.
2. Calculate the midpoint c of this interval.
3. Determine where the root lies by checking the signs of the function at the midpoint versus the endpoints:
   • If the sign of f(c) matches f(a), it means the root is between c and b; hence, we can drop the left part and set b to c.
   • If the sign matches f(b), then we update a to c.
4. You continue narrowing down the interval until it is smaller than a pre-defined tolerance level, which defines how close you want to be to the actual root.

Think of using a searchlight in a dark room to find an object that you know is somewhere between two walls. You first shine your light across the entire space between the walls. You notice the light reflects off the object at a specific point. Each time, you deduce which wall to keep searching in the dark room, progressively narrowing down until you find the exact spot of the object.

Advantages:

• Simple to implement.
• Always converges if the function is continuous and the initial interval is chosen correctly.

Disadvantages:

• Slow convergence.
• Requires an initial bracketing of the root.

The Bisection method boasts advantages such as ease of implementation, making it a favored choice for beginners in numerical methods. Additionally, it guarantees convergence as long as the function remains continuous within the selected interval. However, it has notable disadvantages; notably, its convergence can be slow, meaning that while it will get to the answer, it takes a significant number of iterations. Moreover, you must have an initial range where you believe the root lies, which could limit its practical applicability in some scenarios.

Imagine following a treasure map. The simplicity of using the map makes it approachable and easy to follow, ensuring eventually you’ll find the treasure if you stick to it. However, you might find yourself wandering for much longer than you anticipated before finding the treasure hidden beneath an 'X' mark—you may also need to be careful to select the right area to start your search, or else you are wasting your time!

For f(x)=x²−4 (the root is x=2):

• Initial interval: a=1, b=3.
• Midpoint: c=(1+3)/2=2.
• Since f(2)=0, the root is found.

As an application of the Bisection method, consider the function f(x)=x²−4, where we know the root is 2. We start with a=1 and b=3, both of which yield signs indicating different intervals (f(1)=−3 and f(3)=5). We compute the midpoint, c, which equals 2.5. From testing the midpoint, we see that f(2)=0, confirming that 2 is indeed the root of the function. This simple step confirms that our method effectively finds the root quickly.

Picture a light switch in a room. You know the switch will turn the light on with a flick (indicating a change in state, analogous to crossing zero with the function). You start testing the switch from a distance, and at one point, you discover it positively (the light turns on) at switch 'C'. You've bracketed your search, and the moment that switch responds suggests you've successfully identified where to act.

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