A-Level Maths (Edexcel 9MA0) · Pure Year 2 · Chapter 10: Numerical methods · Lesson 2

Iteration

112 marks

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Quick questions to start.

A1
Needed today
A sequence is given by $u_{n + 1} = \frac{2}{u _{n}} + 1$ , $u_{1} = 1$ . Find $u_{2}$ , $u_{3}$ and $u_{4}$ .
[2 marks]
Answer
A2
Needed today
Show that $x^{3} - 4 x - 1 = 0$ can be written as $x = 3 4 x + 1$ .
[2 marks]
A3
Common slip
Using $x_{n + 1} = 3 x_{n} + 2$ with $x_{0} = 3$ , find $x_{1}$ and $x_{2}$ to 4 decimal places. Use the full calculator value of $x_{1}$ to find $x_{2}$ .
[2 marks]
Answer
A4
Needed today
Given $f (x) = x^{3} - 3 x - 1$ , show that $f (x) = 0$ has a root in $[1.8, 1.9]$ .
[3 marks]

Example, then your turn

A-Level

Your teacher works through each example with you. Then try the one beside it.

Example 1

Show that $x^{3} + 2 x - 9 = 0$ can be written in the form $x = 3 9 - 2 x$ .

Hint Keep

x^{3}

on one side and move everything else across. Then take the cube root.

B1
Show that $x^{3} - 6 x + 2 = 0$ can be written in the form $x = 6 - \frac{2}{x}$ , for $x > 0$ .
[3 marks]

Example 2

Use $x_{n + 1} = 3 9 - 2 x_{n}$ with $x_{0} = 1.5$ to find $x_{1}$ , $x_{2}$ , $x_{3}$ and $x_{4}$ to 4 decimal places.

Hint Type

1.5

, press =, then type the formula using Ans and press = again and again. Write each value to 4 d.p. but never type the rounded value back in.

B2
Use $x_{n + 1} = 6 - \frac{2}{x _{n}}$ with $x_{0} = 2$ to find $x_{1}$ , $x_{2}$ , $x_{3}$ and $x_{4}$ to 4 decimal places.
[4 marks]
Answer

Example 3

The iterates in the last example approach a root $α$ of $x^{3} + 2 x - 9 = 0$ . Show that $α = 1.76$ to 2 decimal places.

Hint Iterates that look settled do not prove anything. Test the bounds

1.755

and

1.765

f (x) = x^{3} + 2 x - 9

B3
The iterates in B2 approach a root $β$ of $x^{3} - 6 x + 2 = 0$ . Show that $β = 2.262$ to 3 decimal places.
[4 marks]

Example 4

The diagram shows $y = x$ , $y = e^{- x}$ and the first steps of $x_{n + 1} = e^{- x_{n}}$ from $x_{0} = 0.1$ . Find $x_{1}$ , $x_{2}$ , $x_{3}$ . Say whether this is a staircase or a cobweb, and use $g^{'} (x)$ to explain why it converges.

Hint Up to the curve, across to

y = x

, up or down to the curve again. If the steps go round the root it is a cobweb. Then look at the size and the sign of

g^{'} (α)

B4
For the iteration in B2, $g (x) = 6 - \frac{2}{x}$ . Say whether it gives a staircase or a cobweb and how you can tell from the iterates. Find $g^{'} (x)$ and show that $∣ g^{'} (β) ∣ < 1$ , where $β = 2.262$ .
[5 marks]

Example 5

The equation $x^{2} - 5 x + 3 = 0$ has roots $0.697$ and $4.303$ . Use $x_{n + 1} = \frac{x _{n}^{2} + 3}{5}$ with $x_{0} = 4$ to find $x_{1}$ , $x_{2}$ , $x_{3}$ . Explain why this iteration cannot find the root $4.303$ .

Hint Watch where the iterates go. Then work out

g^{'} (x) = \frac{2 x}{5}

at each root.

B5
The equation $x^{3} - 4 x + 1 = 0$ has a root near $1.86$ . Two rearrangements are $x = \frac{x ^{3} + 1}{4}$ and $x = 3 4 x - 1$ . Use $x_{0} = 2$ in each to find $x_{1}$ , $x_{2}$ , $x_{3}$ . Say which one finds the root near $1.86$ and why.
[5 marks]
Answer

Practice

A-Level

C1
Nia uses $x_{n + 1} = 2 + ln x_{n}$ with $x_{0} = 3$ . She finds $x_{3}$ and $x_{4}$ and says that because both round to $3.14$ , the root of $x - 2 - ln x = 0$ is $3.14$ to 2 decimal places. Find $x_{1}$ to $x_{4}$ and show that she is wrong.
[6 marks]

C2
Ben rearranges $x^{2} - 3 x - 5 = 0$ as $x = 3 x - 5$ . He uses it with $x_{0} = 4$ and his calculator gives an error at $x_{4}$ . Find his mistake. Write the correct rearrangement and use it with $x_{0} = 4$ to find $x_{1}$ to $x_{4}$ .
[5 marks]
Answer
C3
The equation $x^{3} - 5 x + 1 = 0$ has three roots. Use $x_{n + 1} = \frac{x _{n}^{3} + 1}{5}$ with $x_{0} = 0.5$ , and $x_{n + 1} = 5 - \frac{1}{x _{n}}$ with $x_{0} = 2$ , to find $x_{1}$ , $x_{2}$ , $x_{3}$ for each. Explain why neither iteration can find the negative root, $- 2.330$ .
[5 marks]
Answer
C4
Kemi wants the root $β = 2.262$ of $x^{3} - 6 x + 2 = 0$ . She uses $x_{n + 1} = \frac{x _{n}^{3} + 2}{6}$ with $x_{0} = 2.3$ . Find $x_{1}$ , $x_{2}$ , $x_{3}$ and use $g^{'} (x)$ to explain what happens.
[5 marks]

Exam-style questions

Edexcel exam style

D1
$f (x) = ln (x + 2) - x + 1$ , $x > - 2$ . The equation $f (x) = 0$ has a root $α$ .
(a)
Show that $α$ lies in $[2.5, 2.6]$ .
[2 marks]
(b)
Show that $f (x) = 0$ can be written as $x = 1 + ln (x + 2)$ . Use $x_{n + 1} = 1 + ln (x_{n} + 2)$ with $x_{0} = 2$ to find $x_{1}$ , $x_{2}$ and $x_{3}$ to 4 decimal places.
[4 marks]
Answer
(c)
Show that $α = 2.505$ to 3 decimal places.
[3 marks]
(d)
Use $g^{'} (x)$ to explain why the iterates in (b) form a staircase.
[2 marks]
Total for D1: 11 marks

D2
Priya writes: "The roots of $x^{2} - 6 x + 4 = 0$ are $3 \pm 5$ . If I start at $x_{0} = 6$ , the iteration $x_{n + 1} = \frac{x _{n}^{2} + 4}{6}$ will find the larger root, $3 + 5$ ."
Priya is wrong.
(a)
Find $x_{1}$ , $x_{2}$ and $x_{3}$ starting from $x_{0} = 6$ , and describe what happens.
[3 marks]
Answer
(b)
Use $g^{'} (x)$ to explain why this iteration can never find $3 + 5$ .
[2 marks]
(c)
Starting from $x_{0} = 1$ , find $x_{1}$ and $x_{2}$ and state which root the iteration finds.
[2 marks]
Answer
Total for D2: 7 marks
D3
The equation $2 cos x = x^{2}$ has one positive root $α$ . Use the iteration $x_{n + 1} = 2 cos x_{n}$ with $x_{0} = 1$ to find $x_{1}$ to $x_{4}$ to 4 decimal places, and describe how the iterates behave. Hence find $α$ to 3 decimal places, showing that your answer is correct to this accuracy. Use radians.
[7 marks]
Answer
D4
The profit $P$ , in thousands of pounds, of a pop-up shop $t$ months after it opens is modelled by $P = 40 - 50 e^{- 0.4 t} - 4 t$ . After about $0.75$ months the shop makes a profit. Later the profit falls to zero again, at $t = T$ .
(a)
Show that $T$ lies between $9$ and $10$ .
[2 marks]
(b)
Show that $P = 0$ can be written as $t = 10 - 12.5 e^{- 0.4 t}$ .
[2 marks]
(c)
Use $t_{n + 1} = 10 - 12.5 e^{- 0.4 t_{n}}$ with $t_{0} = 9$ to find $t_{1}$ , $t_{2}$ and $t_{3}$ to 4 decimal places.
[3 marks]
Answer
(d)
Find $T$ to 2 decimal places, showing that your answer is correct. Give the time in months.
[3 marks]
Answer
Total for D4: 10 marks
D5
A sequence is defined by $x_{n + 1} = 4 - x_{n} + 5$ , $x_{0} = 1$ .
(a)
Find $x_{1}$ , $x_{2}$ , $x_{3}$ and $x_{4}$ to 4 decimal places.
[3 marks]
Answer
(b)
The sequence converges to $α$ . Show that $α^{2} - 9 α + 11 = 0$ and hence find the exact value of $α$ .
[4 marks]
Answer
(c)
Explain whether the iteration gives a staircase or a cobweb diagram.
[2 marks]
Total for D5: 9 marks
D6
The equation $e^{x} - 5 x + 2 = 0$ has two roots, $α \approx 0.88$ and $β \approx 2.19$ .
(a)
Show that the equation can be written as $x = ln (5 x - 2)$ .
[2 marks]
(b)
Use $x_{n + 1} = ln (5 x_{n} - 2)$ with $x_{0} = 2$ to find $x_{1}$ to $x_{4}$ to 4 decimal places. State which root the iterates approach.
[3 marks]
Answer
(c)
Starting at $x_{0} = 0.8$ , find $x_{1}$ and $x_{2}$ . Explain why this iteration cannot be used to find $α$ .
[3 marks]
Total for D6: 8 marks

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
The equation $x^{3} - 2 x - 5 = 0$ has one real root $α$ . An iteration is $x_{n + 1} = x_{n} - λ (x_{n}^{3} - 2 x_{n} - 5)$ , where $λ$ is a positive constant. Find the range of values of $λ$ for which iterates that start close to $α$ converge to it, and the value of $λ$ that gives the fastest convergence, to 3 significant figures. Then use $λ = 0.1$ and $x_{0} = 2$ to find $x_{3}$ to 4 decimal places.
[9 marks]
Hint 1 · What to try
Here $g (x) = x - λ (x^{3} - 2 x - 5)$ , so $g^{'} (x) = 1 - λ (3 x^{2} - 2)$ .
Hint 2 · The first line
Find $α$ to 4 decimal places first, by a sign-change search. Then it converges when $- 1 < g^{'} (α) < 1$ .
Hint 3 · The full method
It is fastest when $g^{'} (α) = 0$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can rearrange f(x) = 0 into the form x = g(x) and use the iterative formula with a starting value.
I can find iterates to a stated accuracy using the full calculator value each time.
I can draw a staircase or cobweb diagram and say which one an iteration gives.
I can use \|g'(α)\| < 1 to explain why an iteration converges, diverges or finds another root.

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