Root Finding Techniques
Root finding solves \( f(x) = 0 \), critical in numerical analysis for complex equations. At MathMultiverse, we explore methods like Newton’s, Bisection, and Secant with clear examples, visualizations, and applications.
Newton’s Method
Uses derivative to refine guesses:
\[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \]
Error estimate:
\[ e_{n+1} \approx \frac{f''(x_n)}{2 f'(x_n)} e_n^2 \]
Quadratic convergence, but requires \( f'(x_n) \neq 0 \).
Other Methods
Bisection
Halves interval \([a, b]\) where \( f(a) \cdot f(b) < 0 \):
\[ m = \frac{a + b}{2} \]
Secant
Derivative-free:
\[ x_{n+1} = x_n - f(x_n) \frac{x_n - x_{n-1}}{f(x_n) - f(x_{n-1})} \]
False Position
Interpolates:
\[ x_{n+1} = \frac{a f(b) - b f(a)}{f(b) - f(a)} \]
Fixed-Point
Iterates \( x = g(x) \):
\[ x_{n+1} = g(x_n) \]
Examples
Newton’s Method: \( x^2 - 2 = 0 \)
Initial guess \( x_0 = 1 \):
\[ x_1 = 1.5, \quad x_2 \approx 1.4167, \quad x_3 \approx 1.4142 \]
Approximates \( \sqrt{2} \).
Newton’s Method: \( e^x - 2x - 1 = 0 \)
Initial guess \( x_0 = 1 \):
\[ x_1 = 0, \quad x_2 = 0 \]
Bisection: \( x^3 - x - 2 = 0 \)
Interval \([1, 2]\):
\[ m_1 = 1.5, \quad m_2 = 1.25, \quad m_3 = 1.375 \]
Approaches \( \approx 1.3247 \).
Visualizations
Newton’s Method: \( x^2 - 2 \)
Applications
- Engineering: Solving \( x^3 - x - 2 = 0 \) for structural design, root \( \approx 1.3247 \).
- Physics: Newton’s method for \( e^x - 2x - 1 = 0 \), root at \( x = 0 \).
- Finance: Secant method for IRR calculations in cash flow equations.
- Optimization: Finding critical points of nonlinear functions.