Change of sign method.

Authors Avatar

Melanie Sawyer 7436


Change of sign method

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.

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:

From this spread sheet I can find the upper and lower bounds of the intervals so that I can narrow down the accuracy of the root of the equation.  Shown above are only the first few rows of the spreadsheet, to find the roots to a high enough accuracy, the formulas are filled down for more rows.  If I look at the numbers that are present in the spreadsheet once the equation has been entered, we can find the root to a give number of decimal places.

Error bounds need to be established for the roots.  To make sure that the final answer that I have is correct I can have a look at the error bounds.  For my answer here I can see that to five decimal places, my root between 1 and 2 is 1.10401.  So I can use the change of sign method to check my answer.

Join now!

I can take my final answer one degree of accuracy further and look at the change of sign of f(x) so when:

f(1.104005) = -7.9268 < 0  

f(1.104015) = 3.3557 > 0

Therefore because my answer lies between the positive and negative, it must be accurate.

However, there are some instances where this method does not succeed and we are unable to find the root of the equation using the change of sign method.  For example if there are three roots between two integers, the spreadsheets can only find one of those roots and so ...

This is a preview of the whole essay