Overview
Newton's Method is a powerful technique for finding roots of equations. Its iterative formula is:
\( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \)
Other related methods include the secant, bisection, and false position methods. This page focuses on Newton's Method and provides an interactive example.
Additional Notes
- Newton's Method can be derived using a Taylor series expansion.
- Its convergence depends on the choice of initial guess and the properties of \( f(x) \).
- The secant method approximates the derivative numerically and requires only function evaluations.
- Bracketing methods like bisection guarantee convergence under certain conditions.
Secant Method
Use if \(f'(x)\) is unavailable.
Iterative formula: \( x_{n+1} = x_n - f(x_n) \cdot \frac{x_n - x_{n-1}}{f(x_n) - f(x_{n-1})} \).
Instructions: Start with two initial guesses and repeat until convergence.
Bisection Method
Use when \(f(a)\) and \(f(b)\) have opposite signs.
Process:
- Calculate midpoint: \( c = \frac{a+b}{2} \).
- If \( f(c) \) has the same sign as \( f(a) \), set \( a = c \); otherwise, set \( b = c \).
- Repeat until the interval \( [a, b] \) is sufficiently small.
Convergence is linear.
False Position (Regula Falsi) Method
Similar to bisection but uses a secant line.
Iterative formula: \( c = b - f(b) \cdot \frac{b-a}{f(b)-f(a)} \).
Instructions: Update the interval by replacing the endpoint that shares the sign with \( f(c)
\).
Order of Convergence
To estimate the order \( p \) of convergence, if you have errors \( e_{n-1}, e_n, e_{n+1} \) from successive iterations, use:
\( p \approx \frac{\ln\left(|e_{n+1}/e_n|\right)}{\ln\left(|e_n/e_{n-1}|\right)} \)
This gives you an idea of how fast the error decreases (e.g., \( p \approx 2 \) for quadratic convergence and \( p \approx 1 \) for linear).