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Ryan Guttridge

Rearrangement Method
Method and Example
This method will find an interval in which the root lies
...
Below are the roots of the
equation f(x) = 0
...
When x0 = 0, the iterative formula produces
the following values:
5

The function g(x) converges on the graph y=x at
0
...


Ryan Guttridge

This convergence can be demonstrated graphically, in the form of a staircase (see below):
y

y= g(x)

y=x

x

x0

x1

At the point x0, when a vertical line is drawn, when it touches the curve g(x), it travels
parallel to the x axis until it touches the line y=x
...


The repetition of this process is shown in greater detail on the graph below:
y
y=x

y= g(x)

x

x1

x2 x3 x4 x5

Ryan Guttridge
There are instances however, when a rearrangement of the equation: x5-5x+3=0 fails to find
the root, one such instance is where g(x) = 5√5x-3
...
When x0 = 0, the iterative formula
produces the following values:

The function g(x) converges on the graph y=x at 1
...


This divergence can be demonstrated graphically, in the form of a staircase (see below):
y
y= g(x)

Root that this rearrangement of
the equation is converging on

x7 x2 x1

x0

Desired root
x
y=x

A particular arrangement will only converge on the root if -1< g’(x) <1 for values of x close
to the root
...


Ryan Guttridge
y
y= g(x)
At this point, the gradient of g(x)>1, as
it is steeper than the line y=x that has
gradient 1
...


x

y=x
The method succeeded in finding the desired root in the original equation because the gradient
was between -1< g’(x) <1, this can be identified because at points close to the root, the line is
not as steep as the line y=x, nor is it roughly perpendicular to it (as steep as y = -x)

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