Change of Sign Method - Mathematical Essay
In order to find the roots of an equation that cannot be solved
algebraically, I can use numerical methods to do this instead. One of
these methods is the change of sign method. From looking at a graph of
my equation I can find two integers that my root lies between, then
from there, using spreadsheets, I can use the change of sign method to
discover where the root lies to five decimal places.
I have chosen to try to solve the equation: 5x3-7x+1=0
First, I drew the graph of y=5x3-7x+1 in autograph to find where the
roots roughly lie by looking where the graph cuts the x-axis.
Y=5x3-7x+1
[IMAGE]
From this graph I can see that there are three roots, in the intervals
[-2, -1], [0, 1] and [1, 2]. Looking at the root in the interval [1,
2], we bisect this interval and find the midpoint, 1.5.
f(1.5) = 7.375, so f(1.5)> 0. Since f(1) <0, the root is in [1,
1.5].
Now I am going to take the midpoint of this second interval, 1.25.
f(1.25) = 2.016, so f(1.25)> 0. Since f(1) <0, the root is in [1,
1.25].
The midpoint of this reduced interval is 1.125.
f(1.125) = 0.244, so f(1.125)> 0. Since f(1) <0, the root is in [1,
1.125].
The method then continues in this manner until the required degree of
accuracy is achieved. However, this takes a long time to converge the
root, and can be solved more rapidly using a spreadsheet.
To put my data into a spreadsheet, I first needed to design one that
achieved the desired purpose: finding the roots of the equation using
the change of sign method. Below, the formula which I typed into my
spreadsheet are shown:
Change of sign method
y=5x^3-7x+1
a
b
(a+b)/2
f(a)
f(b)
f((a+b)/2)
1
2
=(A5+B5)/2
=5*A5^3-7*A5+1
=5*B5^3-7*B5+1
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