Numerical Analysis By Shanker Rao

20 NUMERICAL ANALYSIS
Theorem 2.1 If a function f (x) assumes values of opposite sign at the end points of interval
(a, b), i.e., f (a) f (b) < 0 then the interval will contain at least one root of the equation f (x) = 0,
in other words, there will be at least one number c ∈ (a, b) such that f (c) = 0. Throughout our
discussion in this chapter we assume that
1. f (x) is continuous and continuously differentiable up to sufficient number of times.
2. f (x) = 0 has no multiple root, that is, if c is a real root f (x) = 0 then f (c) = 0 and
f ′( x) < 0 f ′( x) > 0 in (a, b), (see Fig. 2.1).
Y
x = c
fb ( )
X´
O
x = a
x = b
X
fa ( )
y = f( x)
Y´
Fig. 2.1
2.2 GRAPHICAL SOLUTION OF EQUATIONS
The real root of the equation
f (x) = 0, refer (1)
can be determined approximately as the abscissas of the points of intersection of the graph of the
function y = f (x) with the x-axis. If f (x) is simple, we shall draw the graph of y = f (x) with respect
to a rectangular axis X´OX and Y´OY. The points at which the graph meets the x-axis are the location
of the roots of (1). If f (x) is not simple we replace equation (1) by an equivalent equation say
φ( x) = ψ( x),
where the functions φ( x ) and ψ ( x ) are simpler than f(x). Then we construct the
graphs of y =φ( x) and y =ψ( x). Then the x-coordinate of the point of intersection of the graphs
gives the crude approximation of the real roots of the equation (1).
Example 2.1 Solve the equation x log 10
x = 1, graphically.
Solution The given equation
x log 10
x = 1
can be written as
log 10
x =
1 x