False Position Method (Regula Falsi)

The False Position Method (or Regula Falsi) is a root-finding algorithm that combines features of the Bisection Method and the Secant Method. It approximates the root of a continuous function by drawing a secant line between two initial points with opposite signs.

Formula for False Position Method:

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

Where:

$a$ and $b$ are initial guesses such that $f(a)$ and $f(b)$ have opposite signs.
$c$ is the x-intercept of the secant line joining $(a, f(a))$ and $(b, f(b))$ .

Example

Given Function

f(x) = x^3 - 2x - 5 = 0

Error Margin is 0.001

Step 1: Choose Initial Points

Choose $a = 2$ and $b = 3$ :

f(2) = 2^3 - 2(2) - 5 = -1 \quad (\text{negative})

f(3) = 3^3 - 2(3) - 5 = 16 \quad (\text{positive})

Step 2: Calculate First Approximation

c_1 = \frac{2(16) - 3(-1)}{16 - (-1)} = \frac{32 + 3}{17} = \frac{35}{17} \approx 2.0588

f(2.0588) = (2.0588)^3 - 2(2.0588) - 5 \approx -0.3908

Conclusion

By repeating the False Position formula, the root converges to 2.0945.

False Position Method Solver

PreviousBisection Method

Unit 1•2 of 28

NextIteration Method

False Position Method (Regula Falsi)

Formula for False Position Method:

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

Where:

$a$ and $b$ are initial guesses such that $f(a)$ and $f(b)$ have opposite signs.
$c$ is the x-intercept of the secant line joining $(a, f(a))$ and $(b, f(b))$ .

Example

Given Function

f(x) = x^3 - 2x - 5 = 0

Error Margin is 0.001

Step 1: Choose Initial Points

Choose $a = 2$ and $b = 3$ :

f(2) = 2^3 - 2(2) - 5 = -1 \quad (\text{negative})

f(3) = 3^3 - 2(3) - 5 = 16 \quad (\text{positive})

Step 2: Calculate First Approximation

c_1 = \frac{2(16) - 3(-1)}{16 - (-1)} = \frac{32 + 3}{17} = \frac{35}{17} \approx 2.0588

f(2.0588) = (2.0588)^3 - 2(2.0588) - 5 \approx -0.3908

Conclusion

By repeating the False Position formula, the root converges to 2.0945.

False Position Method Solver

PreviousBisection Method

Unit 1•2 of 28

NextIteration Method

False Position Method (Regula Falsi)

Example

Step 1: Choose Initial Points

Step 2: Calculate First Approximation

Conclusion

False Position Method Solver

NETZ Engine

False Position Method (Regula Falsi)

Example

Step 1: Choose Initial Points

Step 2: Calculate First Approximation

Conclusion

False Position Method Solver