|
Given an equation f(x) = 0, a number a such that f(a) = 0 is called a root of the equation.� In other words, a root is a solution of the equation.
To find the number of roots of an equation, usually graphical method is used.
If�
then the equation f(x) = 0 has a root in the interval (a, b). |
Theory:��Suppose we know that there is a root of the equation f(x) = 0 in the interval (a, b), where b - a is small.� Then in the ideal situation, the chord joining the points P(a, f(a)) and Q(b, f(b)) will be close to the curve y = f(x).� In this case, the x-intercept, c, of the chord PQ will be close to the root of the equation.
Formula: |
|
Theory:� Suppose we know that a root of f(x) = 0 is close to x = x1.� Then in the ideal situation, the tangent to the curve at x = x1 will be close to y = f(x) in the surrounding of x = x1.
Therefore, we can use the x-intercept, x2, of
the tangent to be an approximation to the root of f(x) = 0.
Formula: |
|
Whether linear interpolation or Newton-Raphson method will give an under- or over- estimation depends greatly on the shape of the curve near the root.
By computing the signs (ie + or -) of� f �(x) and� f �(x) on an interval, we can deduce the general shape of the curve y = f(x) on the interval.� The following table shows all the four cases.
� | f �(x) > 0 | f �(x) < 0 |
f �(x) > 0 | ||
f �(x) < 0 |
Just remember the following results and the table will become easy.
�
f �(x) > 0� � | the curve is increasing |
f �(x) < 0� � | the curve is decreasing |
f �(x) > 0� � | the curve concaves upwards |
f �(x) < 0� � | the curve concaves downwards |
Let's look at the case when f �(x) > 0 and f �(x) > 0.
It is clear from the above diagram that linear interpolation (red chord) produces an under-estimation whereas the Newton-Raphson method (blue tangent) produces an overestimation.
Similarly, you may deduce the results for the other 3 cases.