How do you turn f(x)=0 into an iteration?

Rearrange to x=g(x) and use x{n+1}=g(xn).

If an iteration converges to \alpha, what is true of \alpha?

\alpha=g(\alpha), so \alpha is a root of the original equation.

Where does the Newton-Raphson formula come from?

The tangent to y=f(x) at xn meets the x-axis at x{n+1}.

What does a staircase diagram show?

Monotone convergence, when 0<g'(\alpha)<1.

What does a cobweb diagram show?

Convergence with terms alternating either side of the root, when -1<g'(\alpha)<0.

When does a fixed-point iteration diverge?

When |g'(\alpha)|1, so the error grows at each step.

When does Newton-Raphson fail completely?

When f'(xn)=0: the tangent is horizontal and the formula divides by zero.

Other ways Newton-Raphson can go wrong?

It converges to a different root, or oscillates, when x0 is near a stationary point or far from the root.

How do you know when to stop iterating?

When successive values agree to the required number of decimal places; then check with a sign change.

Iteration and Newton-RaphsonAQA A-Level Maths: Flashcards

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What is an iteration $x_{n+1}=g(x_n)$?

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What is an iteration $x_{n+1}=g(x_n)$ ?: A rule that produces each term from the previous one, starting from $x_0$ .
How do you turn $f(x)=0$ into an iteration?: Rearrange to $x=g(x)$ and use $x_{n+1}=g(x_n)$ .
If an iteration converges to $\alpha$ , what is true of $\alpha$ ?: $\alpha=g(\alpha)$ , so $\alpha$ is a root of the original equation.
State the Newton-Raphson formula.: $x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$
Where does the Newton-Raphson formula come from?: The tangent to $y=f(x)$ at $x_n$ meets the $x$ -axis at $x_{n+1}$ .
Newton-Raphson for $f(x)=x^2-7$ ?: $x_{n+1}=x_n-\frac{x_n^2-7}{2x_n}$
What does a staircase diagram show?: Monotone convergence, when $0<g'(\alpha)<1$ .
What does a cobweb diagram show?: Convergence with terms alternating either side of the root, when $-1<g'(\alpha)<0$ .
When does a fixed-point iteration diverge?: When $|g'(\alpha)|>1$ , so the error grows at each step.
When does Newton-Raphson fail completely?: When $f'(x_n)=0$ : the tangent is horizontal and the formula divides by zero.
Other ways Newton-Raphson can go wrong?: It converges to a different root, or oscillates, when $x_0$ is near a stationary point or far from the root.
How do you know when to stop iterating?: When successive values agree to the required number of decimal places; then check with a sign change.

Exam questions on Iteration and Newton-Raphson

The equation $x^3+x-3=0$ has a root $\alpha$ between 1 and 2. A student uses the iteration $x_{n+1}=\sqrt[3]{3-x_n}$ with $x_0=1$ .
Find $x_2$ and $x_3$ , giving each to 4 decimal places.2 marks
The Newton-Raphson method is used to approximate $\sqrt7$ by solving $f(x)=0$ , where $f(x)=x^2-7$ , with starting value $x_0=3$ .
Find $x_2$ , giving your answer to 4 decimal places.2 marks
The equation $f(x)=0$ , where $f(x)=x^3-2x-5$ , has a root $\alpha$ close to 2. The Newton-Raphson method is used with $x_0=2$ .
Show that the Newton-Raphson iteration is $x_{n+1}=x_n-\frac{x_n^3-2x_n-5}{3x_n^2-2}$ , and hence show that $x_1=2.1$ .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).