1 Feb 1999

umber Of Roots Of An Equation

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.

1. If y = f(x) is easy to sketch, the number of x-intercepts is the number of roots.
2. Otherwise, rewrite the equation as g(x) = h(x) where the graphs of y = g(x) and y = h(x) are easy to sketch.�� Then the number of intersection points between the two graphs is the number of roots of the equation.

xistence Of Roots In An Interval

 If� the graph of y = f(x) is continuous in [a, b] (ie, there is no break in the graph from a to b), and f(a) and f(b) are of opposite sign, then the equation f(x) = 0 has a root in the interval (a, b).

inear Interpolation

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:
 af(b) - bf(a) c =� ����� f(b) - f(a)

ewton-Raphson Method

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:
 f(x1) x2 = x1 - ��� f �(x1)

nder-/Over- Estimation

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.

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