Newton-Raphson
Step 1 The Formula
The Newton-Raphson method uses the derivative to find the root of a differentiable function \( f(x) \).
Iterative Formula
Key Idea: We follow the tangent line at \( x_n \) to where it intersects the x-axis.
Geometric Interpretation
Unlike Bisection (which chops intervals) or Secant (which cuts through two points), Newton’s method uses the slope (tangent) at a single point to “slide” down to the root.
Step 2 Convergence Criteria
Required Condition
Derived from fixed point convergence theory \( |g'(x)| < 1 \).
If \( f'(x) \approx 0 \) (flat tangent), the term \( \frac{f(x)}{f'(x)} \) blows up, causing the method to fail or shoot off to infinity.
Converges fastest when \( f'(x_0) \) is large (steep tangent), directing the next point sharply towards the root.
Step 3 Interactive Simulator
Simulate finding the root of \( f(x) = x^2 – 2 \) (Root: \( \sqrt{2} \approx 1.414 \)).
Step 4 Complex Roots
A Unique Superpower
Unlike bracketing methods (Bisection, Regula Falsi) which rely on sign changes on a real number line, Newton’s method works perfectly in the Complex Plane.
- Can solve equations like \( 2x^2 + 1 = 0 \).
- Requires performing arithmetic with complex numbers \( (a + bi) \).
- Generates fractals (Newton Fractals) based on which root a starting point converges to.
Step 5 Order of Convergence
Simple Root (m=1)
Quadratic (2)Conditions: \( f(\alpha)=0, f'(\alpha) \ne 0 \)
The number of correct digits roughly doubles with every iteration. Extremely fast.
Multiple Root (m > 1)
Linear (1)Conditions: \( f(\alpha)=0, f'(\alpha) = 0 \)
The curve touches the axis tangentially. Convergence slows down significantly.
Quadratic vs Linear Speed
Step 6 Modified Newton-Raphson
If we know the multiplicity \( m \) of a root beforehand, we can restore the lightning-fast quadratic convergence.