---
title: "Limitations of Newton-Raphson"
subject: "Numerical Method"
chapter: 2
key: "semester_4/numerical_method:limitations-newton-raphson"
source: https://www.bctengineer.com/q/semester_4/numerical_method/limitations-newton-raphson
site: BCT Engineering
---

# Limitations of Newton-Raphson

Numerical Method · Chapter 2: Solutions of Nonlinear Equations

- Asked in: 2070 Bhadra, 4 marks
- Video: [LIMITATIONS OF NEWTON RAPHSON METHOD](https://www.youtube.com/watch?v=AtiH5tZdq5o) — PLAIN AND CIVIL - MATHEMATICS
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/numerical_method/limitations-newton-raphson
- Whole chapter: https://www.bctengineer.com/md/semester_4/numerical_method/chapter-2

## Question

Discuss the limitations of Newton-Raphson method while finding a real root of a non-linear equation.

## Short answer

Newton-Raphson needs an analytic derivative and a good initial guess; it fails when f'(x)=0 or very small, may diverge or converge to a different root, and is only linearly convergent at multiple roots. It does not use a bracketing interval, so convergence is not guaranteed, especially near inflection points or extrema.

## Answer

Newton-Raphson iteration x(n+1) = x(n) − f(x(n)) / f′(x(n)) is fast (quadratic convergence), but it has these limitations:

1. **f′(x) must be known and calculable.** The derivative must be derived analytically and evaluated at each step, which costs extra work for complicated functions.
2. **Division by zero.** If f′(x(n)) = 0 (or very small) at some iterate, the tangent is horizontal (or nearly so) and the next value is undefined or thrown far away.
3. **Needs a good initial guess.** If x0 is far from the root, the method may diverge, oscillate between two values, or converge to a different root.
4. **Convergence not guaranteed.** It converges only if |f(x)·f″(x)| / [f′(x)]² < 1 near the root.
5. **Slow convergence at multiple roots.** If the root is repeated (f′(root) = 0), convergence becomes only linear.
6. **Poor behaviour near an inflection point or a local maximum/minimum** of f, where the tangent leads away from the root.
7. **Cannot use a bracketing interval,** so there is no automatic guarantee that the root stays inside a known interval, unlike bisection.

#### Example of failure
For f(x) = x^(1/3) the iteration gives x(n+1) = −2x(n), so it diverges from every starting value except the root itself.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. One limitation is:
   A. f'(x) must be derived and evaluated each step
   B. It needs no derivative
   C. It needs two guesses
   D. It is linear
   Answer: A. This costs extra work. (Key point: Derivative)

2. If f'(x(n)) = 0:
   A. The method converges faster
   B. The tangent is horizontal and the next value is undefined
   C. Root is found
   D. Nothing changes
   Answer: B. Division by zero. (Key point: Zero derivative)

3. At a multiple root convergence becomes:
   A. Cubic
   B. Quadratic
   C. Only linear
   D. Exact
   Answer: C. f'(root) = 0 slows it. (Key point: Multiple root)

4. For f(x) = x^(1/3), the iteration gives:
   A. Quick convergence
   B. x(n+1)=x(n)/2
   C. Constant sequence
   D. x(n+1) = -2x(n), diverging
   Answer: D. It moves away from the root at 0. (Key point: Failure)

## Related in Numerical Method

- [Newton-Raphson algorithm](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Anewton-raphson-algorithm) — Numerical Method, chapter 2
- [Derive Newton-Raphson formula](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Anewton-raphson-derive) — Numerical Method, chapter 2, asked 2×
- [N-R: sin x = e^-x with limitations](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Anewton-raphson-sinx-exp) — Numerical Method, chapter 2

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