Does false position method always converge?

Does the False Position Method Always Work? Let’s Break It Down.

So, you’re trying to find the root of an equation, and you’ve stumbled upon the False Position method, also known as Regula Falsi. Sounds fancy, right? It’s a pretty old-school numerical technique for zeroing in on solutions. But does it always get you there? Well, that’s where things get interesting.

Think of the False Position method like this: you’re trying to find a treasure hidden somewhere between two landmarks. This method, like the Bisection method, starts you off with two points, a and b, that act as your landmarks. The trick? The function’s value at these points, f(a) and f(b), need to have opposite signs. Imagine f(a) is “below sea level” and f(b) is “above sea level.” Somewhere in between, you’ve got to cross “sea level,” which represents the root we’re hunting for.

Now, you draw a straight line connecting your two landmarks. Where that line crosses “sea level” (the x-axis) is your estimated treasure location, c. You calculate c using this formula:

c = a – f(a) * (b – a) / (f(b) – f(a))

Okay, so you’ve found c. Is that the treasure? Maybe, maybe not. You check the value of the function at c, f(c). If f(c) is “below sea level,” then you know the treasure is somewhere between c and b. So, you replace a with c, and start again. If f(c) is “above sea level,” you replace b with c. You keep repeating this process, narrowing down your search until you’re close enough to the real treasure.

So, Here’s the Good News…

Unlike some other methods, like Newton’s method, you don’t need to calculate any complicated derivatives. Even better, the False Position method will find the root, as long as your function is continuous and you start with those two points where the function values have opposite signs. In fact, recent research has even loosened some of the old rules about the function’s derivatives, proving it works for all continuous functions!

…But There’s a Catch

Here’s the thing: while it’s guaranteed to eventually find the root, it can be slower than watching paint dry. I’m talking really slow. This often happens when the function is curvy, especially near the root.

Imagine your treasure map isn’t a straight line, but a winding road with a sharp curve. The straight line approximation of the False Position method might lead you to a point where one of your landmarks just sits there, stubbornly refusing to move. One end of your search interval gets stuck, while the other slowly, painfully crawls towards the solution.

Why Does This Happen?

It all boils down to the straight-line approximation. If the function isn’t close to a straight line, your estimated treasure location, c, might not be a great guess. One of your starting points ends up stagnating, and you’re stuck with a super slow, one-sided search.

What Can You Do About It?

Thankfully, some clever folks have come up with tweaks to the False Position method to speed things up. Algorithms like the Illinois algorithm and the Anderson-Björck algorithm try to prevent that “sticking” problem by adjusting the function value at the stagnant point. Think of it as giving that stubborn landmark a little nudge to get it moving again.

You could also consider the Secant method. It doesn’t guarantee that the root stays bracketed, but it can converge faster. And, if you can easily calculate the derivative of your function, Newton’s method is often even faster.

The Bottom Line

The False Position method will find the root, eventually, if you start with the right conditions. But be warned: it can be slow. Really slow. Keep an eye out for that “sticking” behavior, and don’t be afraid to explore other root-finding methods if you need to speed things up. Ultimately, the best method depends on the specific problem you’re trying to solve.