Looking at the new content for UK GCSE Mathematics a completely new entry on the specification is “find approximate solutions to equations numerically using iteration”.

The topic of iteration, is it same as it used to be called TRIAL & IMPROVEMENT METHOD? If yes, then why its name is different. Gee I have never read any alternate name to Trial and improvement method that is being called Iteration method. Please reply. Regards

We used to have test only the Trial and Improvement method at GCSE level, If you look at the file I have linked to below on page 13 you will see x^3+x=14 solved by Trial and Improvement. Iterative Techniques refers to other techniques, look at pages 17 and 18 where the same equation is solved far more efficiently using a rearrangement method. https://colleenyoung.files.wordpress.com/2016/03/solution-of-equations-graphical-iteration1.pdf

One can still have some fun !
A classic iterative scheme for root(A) is
Xn+1 = (1/2)(Xn + A/Xn)
This is very quick.
BUT looking at the steady state solution X = (1/2)(X + A/X) and rearranging it:
X = A/X
we get …oh dear….
and if we really want to do an iterative solution
Xn+1 = A/Xn
and X1 = 2, then X2 is 2.5, X3 is 2, X4 is 2.5 and so on…..

I totally agree re the fun! Have already told my students more than they need for the exam! Your cobweb diagram is exactly what we do at Advanced Level for the rearrangement methods. Sometimes a ‘staircase’ to convergence.

I am very interested in the iterative methods stuff generally, but I have quibbles about the Xn+1 = f(Xn) approach, which seems to be quite popular. It looks as though it is a rabbit-out-of-a-hat treatment, particularly when the “tests” are mostly simple arithmetic.
The basic idea is that Yn = f(Xn), and then Yn is converted to Xn+1 around the line y = x
So we get the f(Xn), the Yn+1, the Xn+1 and (repeat) the f(Xn+1).
Regrettably the f’s show convergence on only half the possibilities.
A diagram is attached:

Only have my phone at the moment. Can’t see a diagram. It is as I mentioned previously though, at GCSE level, (the reason for these posts because the subject is new at this level) there is no discussion of when methods will or will not converge, that is done at Advanced Level later.

Interesting to see them being taught at this level. Not all iterative methods operate the same way. The Newton method applies under a certain set of circumstances, and this one operates based on the fixed-point theorem — and is similar to what one would do in order to find the root of cosx=x.

Absolutely Robert; we teach this at Advanced level too, looking at change of sign methods, rearrangements of equations and Newton-Raphson. At GCSE there is no requirement to discuss conditions for convergence for example.

The topic of iteration, is it same as it used to be called TRIAL & IMPROVEMENT METHOD? If yes, then why its name is different. Gee I have never read any alternate name to Trial and improvement method that is being called Iteration method. Please reply. Regards

We used to have test only the Trial and Improvement method at GCSE level, If you look at the file I have linked to below on page 13 you will see x^3+x=14 solved by Trial and Improvement. Iterative Techniques refers to other techniques, look at pages 17 and 18 where the same equation is solved far more efficiently using a rearrangement method. https://colleenyoung.files.wordpress.com/2016/03/solution-of-equations-graphical-iteration1.pdf

One can still have some fun !

A classic iterative scheme for root(A) is

Xn+1 = (1/2)(Xn + A/Xn)

This is very quick.

BUT looking at the steady state solution X = (1/2)(X + A/X) and rearranging it:

X = A/X

we get …oh dear….

and if we really want to do an iterative solution

Xn+1 = A/Xn

and X1 = 2, then X2 is 2.5, X3 is 2, X4 is 2.5 and so on…..

finally:

https://howardat58.files.wordpress.com/2016/07/angle-45-to-45.jpg

I totally agree re the fun! Have already told my students more than they need for the exam! Your cobweb diagram is exactly what we do at Advanced Level for the rearrangement methods. Sometimes a ‘staircase’ to convergence.

Well I thought it would be attached.

Must try again !

(on the next comment 1)

A diagram is attached:

I am very interested in the iterative methods stuff generally, but I have quibbles about the Xn+1 = f(Xn) approach, which seems to be quite popular. It looks as though it is a rabbit-out-of-a-hat treatment, particularly when the “tests” are mostly simple arithmetic.

The basic idea is that Yn = f(Xn), and then Yn is converted to Xn+1 around the line y = x

So we get the f(Xn), the Yn+1, the Xn+1 and (repeat) the f(Xn+1).

Regrettably the f’s show convergence on only half the possibilities.

A diagram is attached:

Only have my phone at the moment. Can’t see a diagram. It is as I mentioned previously though, at GCSE level, (the reason for these posts because the subject is new at this level) there is no discussion of when methods will or will not converge, that is done at Advanced Level later.

Interesting to see them being taught at this level. Not all iterative methods operate the same way. The Newton method applies under a certain set of circumstances, and this one operates based on the fixed-point theorem — and is similar to what one would do in order to find the root of cosx=x.

Absolutely Robert; we teach this at Advanced level too, looking at change of sign methods, rearrangements of equations and Newton-Raphson. At GCSE there is no requirement to discuss conditions for convergence for example.