Register  
 
About Us | Help | Sign in
 
   

Revision:Iteration

From The Student Room

TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > Iteration


This is a way of solving equations. An iteration formula might look like the following:

\displaystyle x_{n+1} = 2  + \frac{1}{x_n}.


You are usually given a starting value, which is called x0.

If x0 = 3, substitute 3 into the original equation where it says xn.

This will give you x1.

(This is because if \displaystyle n = 0,\ x_1 = 2 + \frac{1}{x_0} and \displaystyle x_0 = 3).


\displaystyle x_1 = 2 + \frac{1}{3} = 2.333 333 (by substituting in 3).


To find x2, substitute the value you found for x1.

\displaystyle x_2 = 2 + \frac{1}{2.333 333} = 2.428 571


Repeat this until you get an answer to a suitable degree of accuracy. This may be about the 5th value for an answer correct to 3s.f. In this example, x5 = 2.414...


Example

(a) - Show that

\displaystyle x = 1 + \frac{11}{x - 3}

is a rearrangement of the equation:

\displaystyle x^2 - 4x - 8 = 0.


(b) - Use the iterative formula:

\displaystyle x_{n+1} = 1 + \frac{11}{x_n - 3}

together with a starting value of x1 = -2 to obtain a root of the equation:

\displaystyle x^2 - 4x - 8 = 0

accurate to one decimal place.


(a) multiply everything by (x - 3):


\displaystyle x(x - 3) = 1(x - 3) + 11

so \displaystyle x^2 - 3x = x + 8

so \displaystyle x^2 - 4x - 8 = 0


(b) x1 = -2


\displaystyle x_2 = 1 +  \frac{11}{-2 - 3} (substitute -2 into the iteration formula)

\displaystyle = -1.2


\displaystyle x_3 = 1 + \frac{11}{-1.2 - 3} (substitute -1.2 into the above formula)

\displaystyle = -1.619


\displaystyle x_4 = -1.381

\displaystyle x_5 = -1.511

\displaystyle x_6 = -1.439

\displaystyle x_7 = -1.478


therefore, to one decimal place, x = 1.5.


Comments

collapse
Clearing & Results
 
 

Or get advice in our Clearing and Applications forum

collapse Personal Statement - Capitalising first letter of subject?
collapse Give Up Uni Place to Apply for Law????
collapse Should Predicted Grades be Abolished?
collapse Sporting UCAS Points Question.
collapse Wait untill Oct 16th?
 
Recent Threads
 
collapse A2 Maths Core 3&4 textbook help!
started by: Big_C.
replies: 0
last post: 1 Minute Ago
collapse It's like he doesn't want to be around me anymore?
started by: xoJessicaAnn
replies: 16
last post: 1 Minute Ago
collapse Question for straight girls
started by: Germi
replies: 13
last post: 1 Minute Ago
collapse Amir Khan got knocked out....within the first round !!
started by: Jeev
forum: Sport
replies: 143
last post: 2 Minutes Ago
collapse Pretty or natural?
started by: Y'anami!
replies: 34
last post: 2 Minutes Ago