Notesale: Turn your study into money

Document Preview

Extracts from the notes are below, to see the PDF you'll receive please use the links above

---

Ryan Guttridge

Change of sign method – Decimal Search
Method and Example
This method will find an interval in which the root lies
...
It is possible to use
the change of sign method on a table in order to indentify and specify with greater accuracy
the interval in which the root lies
...

They are [-2,-1] [0, 1] and [1, 2]
...


The symbol

represents a change in sign
...
1
...

As is evident from the table; there is a sign change, and
therefore a root; in the interval [1
...
2]
...
1, 1
...
Proving the method
to be successful thus far
...
1, 1
...
1, 1
...
01
...

As seen from the table, there is a sign change and therefore a root in
the interval [1
...
16]
...
15, 1
...
Once again, proving the method
to be successful thus far
...
15, 1
...
15, 1
...
001, which is again shown in the table below
...
159, 1
...

When the graph of y= x9-5x+2 is magnified we can see that the
root is in the interval [1
...
160]
...

y

y=x9-5x+2
x

This root lies
in the interval
[1
...
160]
In this particular instance the root is to be found to an accuracy of
±0
...
p
...
159,
1
...
However, to be able to find an actual value for the root
to 3 decimal places, the method must be repeated a final time to
determine the final decimal place i
...
whether the root is 1
...
160 to 3 d
...

It can be seen from the table that there is a sign change, and thus a
root in the interval [1
...
1599]
...
1598, 1
...
Showing, as this is the final
stage that this method has proved successful in this instance
...


This root lies in
the interval
[1
...
1599]

The interval has now been narrowed down to the acceptable level of accuracy (3 d
...
) – so the
root is 1
...
p
...
1598, 1
...

This can be confirmed by using the change of sign method with the error bounds, the lower
bound of the root is 1
...
1605, using the change of sign method,
f(1
...
00927, and f(1
...
01524, this change of sign showing that the root does
lie between the lower bound of 1
...
1605
...

Consider the equation: 3
...
249x2-15
...

This extensive integer search has only identified one change of sign in
the interval [1, 2]
...


Ryan Guttridge
However, the graph of: y=3
...
249x2-15
...

y

y=3
...
249x2-15
...

x

Repeated root that
was missed because
f(x) does not change
sign between -3 and
-2 and thus it is not
detected in the
change of sign
method
...


So when the curve touches the x axis the method fails to find all of the roots
...
88x2+11
...
94
The table shows a search for a change of sign in integer intervals
...

The interval [2, 3] will be broken down into increments of 0
...


From the table it is evident that there is a change of sign between 2
...
6, so now to continue to narrow down the search in the interval

Ryan Guttridge
[2
...
6] to find the root
...

However, on an enlarged picture of the graph…
y
y=x3-5
...
21x-6
...


Root identified in the interval
[2
...
6]

So when there are several roots very close together it is easy to miss a pair of them
...
This failure is due to there
being two roots very close together, as can be seen in more detail on the graph below
...
This results in neither root being identified due to there being no change of sign
evident and thus the interval is not examined more closely – hence the failure of the change of
sign method in this instance

---