What is the formula of the regula falsi method?

The Regula Falsi method, also known as the False Position Method, is an iterative numerical technique used to find the real roots of a continuous function. It combines aspects of the bisection method and the secant method to converge quickly on a root.

Understanding the Regula Falsi Method

The core idea behind the Regula Falsi method is to approximate the function f(x) within an interval [a, b] with a straight line (a secant line) connecting the points (a, f(a)) and (b, f(b)). The point where this line intersects the x-axis is taken as the new approximation for the root.

This method requires two initial points, a and b, such that a < b and the function values at these points have opposite signs, meaning f(a) * f(b) < 0. This condition ensures that at least one root lies within the interval [a, b].

The Core Formula of Regula Falsi

The formula to calculate the new approximate root, often denoted as c or x_0, derived from the intersection of the secant line with the x-axis, is:

$c = a - \frac{f(a) \cdot (b - a)}{f(b) - f(a)}$

An equivalent form of the formula is:

$c = \frac{a \cdot f(b) - b \cdot f(a)}{f(b) - f(a)}$

Where:

• a and b are the current interval endpoints.
• f(a) and f(b) are the function values at a and b, respectively.
• c is the new approximation for the root.

How the Method Works (Iterative Steps)

The Regula Falsi method is an iterative process that refines the interval containing the root. Here are the steps:

1. Initial Interval Selection: Choose two initial points, a and b, such that a < b and f(a) and f(b) have opposite signs (i.e., f(a) * f(b) < 0). This guarantees a root exists within the interval [a, b].

2. Calculate the New Approximation: Use the Regula Falsi formula to calculate c:
   $c = a - \frac{f(a) \cdot (b - a)}{f(b) - f(a)}$

3. Evaluate the Function at c: Calculate f(c).

4. Check for Root and Update Interval:

   • Root Found: If f(c) = 0, then c is the exact root, and the process terminates.
   • Update Interval: If f(c) is not zero, determine which sub-interval [a, c] or [c, b] contains the root by checking the sign of f(c) relative to f(a) and f(b):
     • If f(a) * f(c) < 0 (meaning f(a) and f(c) have opposite signs), the root lies in the interval [a, c]. For the next iteration, the new interval becomes [a, c] by setting b = c.
     • If f(b) * f(c) < 0 (meaning f(b) and f(c) have opposite signs), the root lies in the interval [c, b]. For the next iteration, the new interval becomes [c, b] by setting a = c.

5. Repeat: Steps 2-4 are repeated until f(c) is sufficiently close to zero (within a predefined tolerance) or the maximum number of iterations is reached.

This method is generally more efficient than the bisection method because it uses the values of the function to guide the next approximation, leading to faster convergence for well-behaved functions. However, it can sometimes converge slowly or experience issues if the function is not sufficiently linear or has multiple roots close together.

For further exploration of root-finding algorithms, you can refer to resources on numerical methods.