Understanding the Regula Falsi (False Position) Method

What is the Regula Falsi Method?

The Regula Falsi Method, also known as the False Position Method, is a powerful numerical technique for finding the roots of a function. It combines the reliability of the bisection method with the efficiency of the secant method, making it a versatile tool in numerical analysis.

Unlike the Secant Method, the Regula Falsi Method always keeps the root bracketed between two points where the function changes sign, ensuring convergence while maintaining reasonable efficiency.

Combining Bisection and Secant Methods

The genius of the Regula Falsi Method lies in how it combines the best aspects of both the Bisection and Secant methods:

From Bisection Method:

• Bracketing Property: Like bisection, it maintains a bracket around the root, ensuring convergence.
• Sign Change: Uses the sign change principle to determine which interval contains the root.
• Guaranteed Convergence: Inherits the guaranteed convergence property of bisection.

From Secant Method:

• Linear Interpolation: Uses the secant line between points to estimate the root.
• Faster Convergence: Generally converges faster than pure bisection due to linear interpolation.
• Better Approximation: Uses function values to make more informed guesses.

The Hybrid Approach

In each iteration, the method:

1. Uses linear interpolation (like Secant Method) to find the next approximation
2. Maintains bracketing interval (like Bisection Method) to ensure convergence
3. Updates the interval based on sign change, preserving the root within bounds

This formula represents the hybrid nature:

• Numerator $(b-a)$: Similar to bisection's interval
• Denominator $f(b)-f(a)$: Secant method's slope calculation
• Whole expression: Secant's linear interpolation with bisection's bracketing

How Does the Regula Falsi Method Work?

The method begins with two points $a$ and $b$ where $f(a)$ and $f(b)$ have opposite signs. The next approximation is calculated using:

This formula represents the x-intercept of the line connecting the points $(a, f(a))$ and $(b, f(b))$. After each iteration, the method updates either point $a$ or $b$, while maintaining the sign change condition to ensure the root lies between the two points.

Example of the Regula Falsi Method

Let's consider the function $f(x) = x^2 - 4$, which has a root at $x = 2$. We will use the initial points $a = 1$ and $b = 3$ where:

• $f(a) = f(1) = 1^2 - 4 = -3$
• $f(b) = f(3) = 3^2 - 4 = 5$

Since $f(a)$ and $f(b)$ have opposite signs, we proceed with the Regula Falsi method:

Now, $f(c) = f(1.75) = 1.75^2 - 4 = -0.9375$, which has the opposite sign to $f(b) = 5$. Thus, we replace $a$ with $c$, and the process is repeated until the desired accuracy is achieved.

Next Iteration

In the next iteration, using $a = 1.75$ and $b = 3$, the formula is applied again:

The process continues until the difference between successive values of $c$ becomes small enough to meet a predefined tolerance (e.g., 0.001).

Conclusion

By iterating the Regula Falsi method, we eventually converge on the root $x \approx 2$, demonstrating how the method efficiently approximates the root by maintaining a sign change condition.

Steps of the Regula Falsi Method

1. Initial Bracket:
   • Find two points $a$ and $b$ where $f(a)f(b) < 0$
   • This ensures the root lies between these points
2. Calculate Next Approximation:
   $$c = b - f(b) \frac{b - a}{f(b) - f(a)}$$
3. Update Interval:
   • If $f(c)f(a) < 0$: Set $b = c$
   • If $f(c)f(b) < 0$: Set $a = c$
4. Check Convergence:
   • Check if $|f(c)| < \epsilon$ where $\epsilon$ is the tolerance
   • Or if $|b - a| < \delta$ where $\delta$ is the interval tolerance
5. Iterate:
   • Repeat steps 2-4 until convergence or maximum iterations reached

Basic Example

Let's solve $f(x) = x^3 - x - 2 = 0$ using the Regula Falsi Method.

Step-by-Step Solution:

1. Initial Bracket:
   • Let's try $a = 1$ and $b = 2$
   • $f(1) = 1^3 - 1 - 2 = -2$
   • $f(2) = 2^3 - 2 - 2 = 4$
   • Since $f(1)f(2) < 0$, the root lies in [1, 2]
2. First Iteration:
   $$c_1 = 2 - 4\frac{2-1}{4-(-2)} = 2 - \frac{4}{6} = 1.333$$

   $f(1.333) \approx -0.370$

3. Update:
   • Since $f(1.333)f(2) < 0$, new interval is [1.333, 2]
4. Second Iteration:
   $$c_2 = 2 - 4\frac{2-1.333}{4-(-0.370)} \approx 1.442$$

   $f(1.442) \approx 0.089$

Application Problems with Solutions

Problem 1: Engineering Design

Problem Statement: A cylindrical tank with height h meters and radius r meters needs to hold 100 cubic meters of liquid. The radius is fixed at 3 meters. Find the height needed such that $\pi r^2h = 100$.

Solution:

1. Setup the equation:
   • $f(h) = \pi(3)^2h - 100 = 0$
   • $f(h) = 9\pi h - 100$
2. Initial Bracket:
   • Try h = 3: $f(3) = 27\pi - 100 \approx -15.2$
   • Try h = 4: $f(4) = 36\pi - 100 \approx 13.1$
3. First Iteration:
   $$h_1 = 4 - (13.1)\frac{4-3}{13.1-(-15.2)} = 3.54$$
4. Result: The tank should be approximately 3.53 meters high.

Problem 2: Financial Application

Problem Statement: Find the interest rate (r) that makes an investment of $10,000 grow to $15,000 in 5 years using compound interest: $10000(1+r)^5 = 15000$

Solution:

1. Setup the equation:
   • $f(r) = 10000(1+r)^5 - 15000 = 0$
2. Initial Bracket:
   • Try r = 0.05: $f(0.05) = -2,721.65$
   • Try r = 0.10: $f(0.10) = 1,105.10$
3. Iterations lead to: r ≈ 0.0835 or 8.35%

Advantages and Limitations

Advantages:

• Guaranteed convergence for continuous functions
• No need to calculate derivatives
• More efficient than the bisection method
• Always maintains bracketing of the root

Limitations:

• Generally slower convergence than Newton-Raphson method
• Requires initial bracketing interval
• Can be slower than Secant method in some cases
• May exhibit one-sided convergence

Practical Applications

The Regula Falsi Method is particularly useful in:

• Engineering: Design optimization and system analysis
• Physics: Finding equilibrium points and solving force balance equations
• Economics: Finding break-even points and market equilibrium
• Chemistry: Solving reaction equilibrium equations

Conclusion

The Regula Falsi Method represents a balanced approach to root-finding, combining the reliability of the bisection method with the efficiency of linear interpolation. Its guaranteed convergence and straightforward implementation make it a valuable tool in numerical analysis, particularly when reliability is prioritized over maximum speed.