How does Newton's method converge?

Newton's method typically converges with exceptional speed, most commonly exhibiting quadratic convergence when certain conditions are met, meaning the number of accurate decimal places roughly doubles with each iteration.

Understanding Newton's Method Convergence

Newton's method is an iterative algorithm used to find successively better approximations to the roots (or zeroes) of a real-valued function. It works by starting with an initial guess and then repeatedly improving it by moving along the tangent line of the function at the current point until it intersects the x-axis.

The Core Idea: Tangent Lines

At each step, Newton's method uses the first derivative of the function to estimate where the function crosses the x-axis.
The iterative formula for Newton's method is:

• x_{n+1} = x_n - f(x_n)/f'(x_n)
  • Where x_n is the current approximation, f(x_n) is the function's value, and f'(x_n) is its derivative at x_n.

Rates of Convergence

The "rate of convergence" describes how quickly an iterative method approaches its solution.

Convergence Type	Description	Speed	Conditions (General)
Linear	Error reduces by a constant factor in each step.	Relatively slow, but consistent.	Common for simple fixed-point iteration.
Quadratic	Error is squared in each step; number of correct digits roughly doubles.	Extremely fast and efficient.	Requires stronger conditions on the function and initial guess.
Cubic	Error is cubed in each step; number of correct digits roughly triples.	Even faster, but rarer for basic methods.	Even stronger conditions, often for higher-order methods.

Quadratic Convergence: The Ideal Scenario

Newton's method's hallmark is its quadratic convergence, making it one of the most efficient root-finding algorithms. This rapid convergence occurs under specific conditions:

1. Simple Root: The function f(x) must have a simple root p, meaning f(p) = 0 but f'(p) ≠ 0. If f'(p) were zero, the tangent line would be horizontal, causing issues.
2. Sufficient Smoothness: The function f(x) must be at least twice continuously differentiable (i.e., f'(x) and f''(x) exist and are continuous) in an interval containing the root.
3. Good Initial Guess: The initial guess x_0 must be sufficiently close to the actual root p. If x_0 is too far, the method might diverge or converge to a different root.

These conditions ensure that the iterative process behaves predictably. In the context of fixed-point iteration, where Newton's method can be viewed as x_{n+1} = g(x_n) with g(x) = x - f(x)/f'(x), quadratic convergence is guaranteed when g'(p) = 0 at the fixed point p. Furthermore, if the second derivative of this iteration function, g''(x), is bounded by some M (i.e., g''(x) < M) on an open interval around the root, these conditions are sufficient to ensure the quadratic convergence of the sequence generated by Newton's method. For simple roots, g'(p) for Newton's method naturally evaluates to zero.

Linear Convergence: When Things Slow Down

While usually quadratic, Newton's method can converge linearly in certain situations, typically when the root is multiple (i.e., f(p) = 0 and f'(p) = 0, also known as a repeated root). In such cases, the tangent line approximation is less effective, and the convergence rate degrades from quadratic to linear. This means the number of correct digits will increase steadily, but not double with each step.

Factors Influencing Convergence

Several factors dictate whether Newton's method will converge and at what rate:

1. Proximity of Initial Guess: The closer the initial guess x_0 is to the true root p, the higher the likelihood of convergence and the faster it typically converges quadratically.
2. Nature of the Function's Derivative:
   • If f'(p) = 0 (multiple root), convergence will be linear or problematic.
   • If f'(x) is very small near the root, the division f(x_n)/f'(x_n) can lead to large steps, potentially overshooting the root or causing divergence.
3. Presence of Inflection Points or Local Extrema: If f'(x) approaches zero or changes sign near the initial guess or subsequent iterations, the method can jump to a distant point, oscillate, or diverge.

Practical Insights and Considerations

• Speed Advantage: When it converges, Newton's method is remarkably fast, making it highly preferred for applications requiring rapid root finding, such as in scientific computing and engineering.
• Sensitivity to Initial Guess: Its main drawback is its sensitivity to the initial guess. A poor starting point can lead to divergence or convergence to an unexpected root.
• Requires Derivative: Unlike some other methods (like the Bisection Method or Secant Method), Newton's method requires the computation of the function's derivative, which isn't always analytically available or easy to compute.
• Applications: It's widely used in solving non-linear equations, optimization problems (finding extrema by setting the derivative to zero), and in numerical analysis as a core algorithm.

Newton's method is a powerful tool due to its rapid quadratic convergence, provided the initial conditions and function properties are favorable.