How do you use ANS on a calculator for iteration?

Enter the starting value and press =, then type the formula with ANS in place of xn and press = repeatedly.

How do you rearrange f(x)=0 for iteration?

Rewrite it as x=g(x) and use x{n+1}=g(xn).

What equation does the limit L of a convergent iteration satisfy?

L=g(L), which reproduces the original equation.

When does a sequence converge?

When its terms get closer and closer to a single value.

State the change-of-sign test for a root.

If f is continuous and f(a), f(b) have opposite signs, a root lies between a and b.

What must a full change-of-sign answer include?

Both values (or signs), a statement that f is continuous, and a conclusion about the root.

How do you confirm \alpha=3.196 to 3 d.p.?

Show that f(3.1955) and f(3.1965) have opposite signs.

Why keep full calculator values between iterations?

Rounding at each step builds up error; round only at the final answer.

Name two ways an iteration can fail.

The terms diverge, or they oscillate or settle on a different root from the one wanted.

f(x)=x^3+2x-7: give a rearrangement x=g(x).

x=\sqrt[3]{7-2x} (or x=\frac{7-x^3}{2}).

Iterative methodsEdexcel International A Level Maths: Flashcards

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What is a recurrence relation of the form $x_{n+1}=f(x_n)$?

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What is a recurrence relation of the form $x_{n+1}=f(x_n)$ ?: A rule giving each term from the previous one, starting from a value $x_0$ .
How do you use ANS on a calculator for iteration?: Enter the starting value and press =, then type the formula with ANS in place of $x_n$ and press = repeatedly.
How do you rearrange $f(x)=0$ for iteration?: Rewrite it as $x=g(x)$ and use $x_{n+1}=g(x_n)$ .
Rearrange $x^3-4x-3=0$ to $x=\sqrt[3]{\ldots}$ .: $x^3=4x+3$ , so $x=\sqrt[3]{4x+3}$ .
What equation does the limit L of a convergent iteration satisfy?: $L=g(L)$ , which reproduces the original equation.
When does a sequence converge?: When its terms get closer and closer to a single value.
State the change-of-sign test for a root.: If $f$ is continuous and $f(a)$ , $f(b)$ have opposite signs, a root lies between $a$ and $b$ .
What must a full change-of-sign answer include?: Both values (or signs), a statement that f is continuous, and a conclusion about the root.
How do you confirm $\alpha=3.196$ to 3 d.p.?: Show that $f(3.1955)$ and $f(3.1965)$ have opposite signs.
Why keep full calculator values between iterations?: Rounding at each step builds up error; round only at the final answer.
Name two ways an iteration can fail.: The terms diverge, or they oscillate or settle on a different root from the one wanted.
$f(x)=x^3+2x-7$ : give a rearrangement $x=g(x)$ .: $x=\sqrt[3]{7-2x}$ (or $x=\frac{7-x^3}{2}$ ).

Exam questions on Iterative methods

The equation $x^3-4x-3=0$ has a root $\alpha$ with $2<\alpha<3$ . The recurrence relation $x_{n+1}=\sqrt[3]{4x_n+3}$ , with $x_0=2$ , is used to find $\alpha$ .
Find $x_2$ and $x_3$ , giving each to 4 decimal places.2 marks
The equation $f(x)=x^3+2x-7=0$ has a single real root $\alpha$ .
Use the iteration $x_{n+1}=\sqrt[3]{7-2x_n}$ with $x_0=1.5$ to find $x_2$ , giving your answer to 3 decimal places.2 marks
The function $f(x)=\ln x+x-3$ , for $x>0$ , has a root $\alpha$ .
Show that $2<\alpha<2.5$ .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).