What does x{n+1} represent geometrically?

The x-intercept of the tangent to y=f(x) at x=xn.

Newton-Raphson for f(x)=x^2-7?

x{n+1}=\frac12\left(xn+\frac{7}{xn}\right)

When does Newton-Raphson fail completely?

When f'(xn)=0: the tangent is horizontal and has no x-intercept.

Why is a small f'(xn) a problem?

The tangent is nearly horizontal, so x{n+1} is far from the root.

What can happen with f(x)=x^3-2x+2 and x0=0?

The iterates cycle 0,1,0,1,\ldots and never converge.

How do you choose a good x0?

Use a sign change to locate the root, then start close to it where f' is not small.

Is Newton-Raphson a recurrence relation x{n+1}=g(xn)?

Yes, with g(x)=x-\frac{f(x)}{f'(x)}.

Newton-Raphson can converge to a different root. Why?

The start value may lie nearer to (or in the basin of) another root.

Only when f(xn)=0, i.e. xn is already a root.

Newton-Raphson methodEdexcel A-Level Maths: Flashcards

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State the Newton-Raphson formula.: $x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$
What does $x_{n+1}$ represent geometrically?: The $x$ -intercept of the tangent to $y=f(x)$ at $x=x_n$ .
Newton-Raphson for $f(x)=x^2-7$ ?: $x_{n+1}=\frac12\left(x_n+\frac{7}{x_n}\right)$
$f(x)=x^3-2x-5$ with $x_0=2$ : find $x_1$ .: $2-\frac{-1}{10}=2.1$
$f'(x)$ for $f(x)=\mathrm{e}^x-3x$ ?: $\mathrm{e}^x-3$
When does Newton-Raphson fail completely?: When $f'(x_n)=0$ : the tangent is horizontal and has no $x$ -intercept.
Why is a small $f'(x_n)$ a problem?: The tangent is nearly horizontal, so $x_{n+1}$ is far from the root.
What can happen with $f(x)=x^3-2x+2$ and $x_0=0$ ?: The iterates cycle $0,1,0,1,\ldots$ and never converge.
How do you choose a good $x_0$ ?: Use a sign change to locate the root, then start close to it where $f'$ is not small.
Is Newton-Raphson a recurrence relation $x_{n+1}=g(x_n)$ ?: Yes, with $g(x)=x-\frac{f(x)}{f'(x)}$ .
Newton-Raphson can converge to a different root. Why?: The start value may lie nearer to (or in the basin of) another root.
When does $x_{n+1}=x_n$ ?: Only when $f(x_n)=0$ , i.e. $x_n$ is already a root.

Exam questions on Newton-Raphson method

The equation $f(x)=0$ , where $f(x)=x^3-2x-5$ , has a root $\alpha$ near $x=2$ . The Newton-Raphson method is used with $x_0=2$ .
Find $x_2$ , giving your answer to 4 decimal places.2 marks
Let $f(x)=\mathrm{e}^x-3x$ . The equation $f(x)=0$ has a root $\alpha$ close to $0.6$ , and the Newton-Raphson method is used with $x_0=0.5$ .
Given that $x_1=0.610060\ldots$ , find $x_2$ to 4 decimal places.2 marks
The Newton-Raphson method is used to find $\sqrt7$ , as the positive root of $x^2-7=0$ , starting from $x_0=3$ .
Show that the Newton-Raphson formula for this equation simplifies to $x_{n+1}=\frac12\left(x_n+\frac{7}{x_n}\right)$ , and find $x_1$ as an exact fraction.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).