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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 Numerical Methods
Edexcel A level Maths: 10.3 Newton Raphson MethodCurated by Kepler, not produced by us. Watch it 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
1

The formula

Starting from x0x_0, the next estimate is xn+1=xn−f(xn)f′(xn)x_{n+1} = x_n - \dfrac{f(x_n)}{f'(x_n)}. Geometrically, you follow the tangent at xnx_n down to where it crosses the xx-axis.

xn+1=xn−f(xn)f′(xn)x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}
Newton-Raphson iteration.
Example 1Apply one step to f(x)=x2−2f(x) = x^2 - 2 with x0=1.5x_0 = 1.5.
1f(1.5)=0.25f(1.5) = 0.25, f′(x)=2xf'(x) = 2x, f′(1.5)=3f'(1.5) = 3.
2x1=1.5−0.253≈1.4167x_1 = 1.5 - \tfrac{0.25}{3} \approx 1.4167.
Answer
x1≈1.4167x_1 \approx 1.4167 (approaching √2).

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

2

When it fails

It fails or diverges if f′(xn)=0f'(x_n) = 0 (the tangent is horizontal — division by zero) or if the starting value is too far from the root.

Formula recap

xn+1=xn−f(xn)f′(xn)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.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. The Newton-Raphson formula is:

    • xn−f(xn)f′(xn)x_n - \frac{f(x_n)}{f'(x_n)}
    • xn−f′(xn)f(xn)x_n - \frac{f'(x_n)}{f(x_n)}
    • xn+f(xn)x_n + f(x_n)
    • xn−f(xn)x_n - f(x_n)
    Show answer and explanation

    Answer: xn−f(xn)f′(xn)x_n - \frac{f(x_n)}{f'(x_n)}

    Standard.

  2. Geometrically each step follows the:

    • tangent to the x-axis
    • normal
    • chord
    • asymptote
    Show answer and explanation

    Answer: tangent to the x-axis

    Tangent line.

  3. The method fails when:

    • f′(xn)=0f'(x_n) = 0
    • f(xn)=0f(x_n) = 0
    • xn>0x_n > 0
    • never
    Show answer and explanation

    Answer: f′(xn)=0f'(x_n) = 0

    Horizontal tangent.

  4. For f(x)=x2−2f(x) = x^2 - 2, x0=1.5x_0 = 1.5, x1≈x_1 \approx

    • 1.41671.4167
    • 1.51.5
    • 1.251.25
    • 22
    Show answer and explanation

    Answer: 1.41671.4167

    1.5 − 0.25/3.

  5. Compared to fixed-point iteration, Newton-Raphson usually:

    • converges faster
    • converges slower
    • never converges
    • is the same
    Show answer and explanation

    Answer: converges faster

    Quadratic convergence.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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Related lessons

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Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.