Equations obtained in real life by people such as scientists and engineers often are not easy to solve. Algebraic techniques such as transposing terms, factorising and using the quadratic formula are not suitable. Another method is to use numerical methods which are available to approximate an answer or root of the equation to a certain level of accuracy. Two such methods, studied in this course, are the Bisection method and the Newton-Rhapson method.
Roots of an Equation
The root(s) of an equation are the value(s) of the variable, which make the equation true.
The roots of an equation are the solutions of f(x) = 0. This is where the graph of the function f(x) crosses the x-axis.
e.g. The three roots of the equation 3x3 + 3x2 − 3x − 1 = 0 are shown on the diagram below.
When the graph of a function crosses the x-axis its sign changes from positive to negative.
e.g. To show that there is a root between x = -2 and x = -1, evaluate the function at both values.
f(-2) = -7
f(-1) = 2
One value is positive and one value is negative ⇒ there is a root between x = -2 and x = -1.
The Bisection Method
In the graph above for the equation 3x3 + 3x2 − 3x − 1 = 0 there is also a root between x = 0 and x = 1. We can use the bisection method to find the value of this root to a required number of decimal places.
The concept described above is used in this process. By bisecting the interval containing the root (xm), the root can be found more accurately. xm = (x1 + x2 )/2
To approximate the root to 1 decimal place the bisecting process is repeated until x2 − x1 < 0.1 (in the end column).
Each repetition is called an iteration.
|
x1
|
f(x1)
|
x2
|
f(x2)
|
xm
|
f(xm)
|
Notes
|
x2 -x1
|
|
0
|
-1
|
1
|
2
|
0.5
|
-1.375
|
xm replaces x1
|
1
|
|
0.5
|
-1.375
|
1
|
2
|
0.75
|
-0.297
|
xm replaces x1
|
0.5
|
|
0.75
|
-0.297
|
1
|
2
|
0.875
|
0.682
|
xm replaces x2
|
0.25
|
|
0.75
|
-0.297
|
0.875
|
0.682
|
0.8125
|
0.152 |
xm replaces x2
|
0.125
|
|
0.75
|
-0.297
|
0.8125
|
0.152
|
0.78125
|
-0.082
|
0.0625
|
The f(x) values do not have to be to many decimal places it is the sign that is important.
The process is stopped because x2 − x1 < 0.0625, thus the approximate value of the root is0.78125 which is 0.8 to 1 decimal place. This is done because both x1 and x2 now round to 0.8.
The Bisection method is accurate but lengthy. To estimate the root in the example above to 2 decimal places would require several more iterations.
Graphical Representation of the Bisection Method
The graph below shows the intervals used in the example above and illustrates how the root (where the curve crosses the x-axis) is approached. The more iterations taken, the more accurate becomes the approximation.
Look at an interactive spreadsheet (Microsoft Excel) illustrating the bisection method.
(Windows users, right click and "Save target as..." to save the files on your computer.)
