10.3PureStretch

The Newton-Raphson Method

Newton-Raphson is a fast iterative method that uses the tangent to a curve to home in on a root. Each step follows the tangent line down to the x-axis to get a better estimate.

26 min Video by Zeeshan Zamurred Numerical Methods

Edexcel A level Maths: 10.3 Newton Raphson MethodWatch the full walkthrough before the notes below.

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What you'll be able to do

State the Newton-Raphson formula
Apply it to find roots
Understand the tangent interpretation
Recognise when it fails

The formula

Starting from $x_{0}$ , the next estimate is $x_{n + 1} = x_{n} - \frac{f ( x _{n} )}{f ^{'} ( x _{n} )}$ . Geometrically, you follow the tangent at $x_{n}$ down to where it crosses the $x$ -axis.

x_{n + 1} = x_{n} - \frac{f ( x _{n} )}{f ^{'} ( x _{n} )}

Newton-Raphson iteration.

f (1.5) = 0.25

f^{'} (x) = 2 x

f^{'} (1.5) = 3

x_{1} = 1.5 - \frac{0.25}{3} \approx 1.4167

Answer

x_{1} \approx 1.4167

(approaching √2).

Tip — Newton-Raphson usually converges much faster than fixed-point iteration.

When it fails

It fails or diverges if $f^{'} (x_{n}) = 0$ (the tangent is horizontal — division by zero) or if the starting value is too far from the root.

Formula recap

x_{n + 1} = x_{n} - \frac{f ( x _{n} )}{f ^{'} ( x _{n} )}

Newton-Raphson.

Common mistakes to avoid

Using f(x) instead of f′(x) in the denominator.

Divide by the derivative f′(xₙ).

Ignoring the failure when f′(xₙ) = 0.

A horizontal tangent makes the method break down.

Key takeaways

xₙ₊₁ = xₙ − f(xₙ)/f′(xₙ).
Follows the tangent to the x-axis.
Fails if f′(xₙ) = 0 or x₀ is poor.

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