1 Change of sign method
Consider this equation
From the graph we can see point A lies between -2 and -3
When x= -2 given
Y=1.6
When x= -3 given
Y= -28.1
There is a change in sign between – 2 and -3 means that there is a root between the two points.
Decimal research
For function f(x) = , there is a change sign between the interval of -2 and -3 means there is a root.
In this method i am going to use Excel spreadsheet to do decimal search by take increments in of the size 0.1 between the interval -2 and -3 and work out the function value for each.
To work out x I’ll use the formula r+1= r+0.1 given function of f(x)
➔ f(x)+1=
We can see that the function of f(x) is change sign from -2.2 to -2.1 therefore the root must lie between these two points
- I now repeat the process by change the increment from 0.1 to 0.01 between -2.2 and -2.1 given that
To work out x I’ll use the formula r+1= r+0.01 to give the function of x
➔ f(x)+1=
From the table there is a sign change from -2.19 to -2.18 which means this become the new interval in which the root is between.
- The interval had narrow down to -2.18 and -2.19, I’ll then now change the increment size to 0.001 which will zoom in the interval even more
In this case r+1= r+0.001 given that the
f(x) +1 =
There is a change of sign between -2.182 and -2.183 means the root must lies between this intervals.
So my estimate root is:
= -2.1825
The error bounds are ± 0.0005 so the upper bound is -2.1825+0.0005=-2.182
and the lower bound is -2.1825-0.0005= -2.1830 so the root must lie between -2.182 and -2.183
Failure of change of sign method
Change of sign method can not apply to all equation here is an example for which the change of sign fails to find a root. Considering this equation
y
This can be show on the graph:
From the graph we can see there is a real root lie between -2 and -3
when f (-2) =
f (-2) = -0.441
When f (-3) =
f (-3) = -2.596
But there is no change in sign between f (-2) and f (-3) because the graph is merely touches the x axis result in both points of the interval given the same sign this is the case when change in sign method fails to work out the root.
In other cases where this method would fails to work is when there’re roots are very much close together because they are likely to be in the same interval result a missing of change in sign.
And also when a function given a discontinuous curve the whole fraction has an asymptote between some specific intervals that make the change in size think there is a root present but there is no real root present.
Fixed point Iteration using method
Given the equation = 0
This can be rearrange into the form of x=g(x)
= 0
=>
=>