Proof of Quadratic Convergence for Newton's Method

Q: What does 'twice continuously differentiable' imply for the function $ f $?

It means the function $ f $ and its first and second derivatives, $ f' $ and $ f'' $, are continuous. This smoothness is essential for Taylor expansions and for the existence of the second derivative in the error analysis.

Q: Why is the condition $ f'(x^*) \neq 0 $ important?

If $ f'(x^*) = 0 $, the tangent line at $ x^* $ would be horizontal, and the method would involve division by zero, rendering it undefined at the root itself. This condition ensures $ x^* $ is a simple root.

Q: Can we illustrate the error propagation using a Taylor expansion?

Yes, by expanding $ f(x_k) $ around $ x^* $ using a second-order Taylor expansion, we can show that the difference $ x_{k+1} - x^* $ is proportional to $ (x_k - x^*)^2 $, demonstrating quadratic convergence.

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The Formal Theorem

Let

f: \mathbb{R} \to \mathbb{R}

be a twice continuously differentiable function such that

f(x^*) = 0

and

f'(x^*) \neq 0

. If

x_0

is sufficiently close to

x^*

, then the sequence

\{x_k\}_{k=0}^{\infty}

generated by Newton's method,

x_{k+1} = x_k - \frac{f(x_k)}{f'(x_k)}

, converges to

x^*

quadratically, meaning there exists a constant

C > 0

such that

|x_{k+1} - x^*| \leq C |x_k - x^*|^2

for all

k \geq 0

Analytical Intuition.

Imagine a treasure hunter,

x_k

, searching for a hidden treasure, the root

x^*

. They can't see the treasure directly, but they can feel the slope of the terrain,

f'(x_k)

, and the altitude,

f(x_k)

. Newton's method is like taking a giant leap in the direction that promises the fastest descent towards sea level (the root). The 'quadratic convergence' is the astonishing part: this method doesn't just inch closer; it *doubles down* on its accuracy with each step. The closer you are to the treasure, the more precisely your next leap will land, often reducing the error by a factor of its square. It's like a warp drive for root-finding, exponentially shrinking the distance to the target.

CAUTION

Institutional Warning.

The assumption of 'sufficiently close' to $x^*$ is crucial. If the initial guess $x_0$ is too far, the tangent line might lead to divergence or convergence to a different root.

Academic Inquiries.

What does 'twice continuously differentiable' imply for the function $f$ ?

It means the function $f$ and its first and second derivatives, $f'$ and $f''$ , are continuous. This smoothness is essential for Taylor expansions and for the existence of the second derivative in the error analysis.

Why is the condition $f'(x^*) \neq 0$ important?

If $f'(x^*) = 0$ , the tangent line at $x^*$ would be horizontal, and the method would involve division by zero, rendering it undefined at the root itself. This condition ensures $x^*$ is a simple root.

How is 'sufficiently close' to $x^*$ quantified in a practical sense?

While the proof establishes existence, practical quantification often involves analyzing the magnitude of $|x_0 - x^*|$ relative to the bounds of $f''$ and the inverse of $f'$ near $x^*$ . Often, an iterative application of the error bound gives a practical range.

Can we illustrate the error propagation using a Taylor expansion?

Yes, by expanding $f(x_k)$ around $x^*$ using a second-order Taylor expansion, we can show that the difference $x_{k+1} - x^*$ is proportional to $(x_k - x^*)^2$ , demonstrating quadratic convergence.

Standardized References.

Definitive Institutional SourceDennis, J. E., & Schnabel, R. E. (1996). *Numerical Methods for Unconstrained Optimization and Nonlinear Equations*. SIAM.

Intermediate

Weierstrass Extreme Value Theorem: Guaranteeing Existence of Optima

Exploring the cinematic intuition of Weierstrass Extreme Value Theorem: Guaranteeing Existence of Optima.

Intermediate

Local Optima are Global Optima for Convex Functions

Exploring the cinematic intuition of Local Optima are Global Optima for Convex Functions.

Intermediate

Hessian Matrix and Second-Order Optimality Conditions

Exploring the cinematic intuition of Hessian Matrix and Second-Order Optimality Conditions.

Intermediate

Jensen's Inequality for Convex Functions

Exploring the cinematic intuition of Jensen's Inequality for Convex Functions.

Institutional Citation

Reference this proof in your academic research or publications.

NICEFA Visual Mathematics. (2026). Proof of Quadratic Convergence for Newton's Method: Visual Proof & Intuition. Retrieved from https://nicefa.org/library/fundamentals-of-optimization/proof-of-quadratic-convergence-for-newton-s-method

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The Formal Theorem

Analytical Intuition.

Institutional Warning.

Academic Inquiries.

What does 'twice continuously differentiable' imply for the function f f f?

Why is the condition f′(x∗)≠0 f'(x^*) \neq 0 f′(x∗)=0 important?

How is 'sufficiently close' to x∗ x^* x∗ quantified in a practical sense?

Can we illustrate the error propagation using a Taylor expansion?

Standardized References.

Related Proofs Cluster.

Weierstrass Extreme Value Theorem: Guaranteeing Existence of Optima

Local Optima are Global Optima for Convex Functions

Hessian Matrix and Second-Order Optimality Conditions

Jensen's Inequality for Convex Functions

Institutional Citation

Dominate the Logic.

What does 'twice continuously differentiable' imply for the function $f$ ?

Why is the condition $f'(x^*) \neq 0$ important?

How is 'sufficiently close' to $x^*$ quantified in a practical sense?