Numerical Analysis By Shanker Rao

34 NUMERICAL ANALYSIS
The above value of x 1
is a closer approximation to the root of f(x) = 0 than x 0
. Similarly if x 2
denotes a better approximation, starting with x 1
, we get
Proceeding in this way we get
x
f x1
= x −
f ′ x
2 1
bg
bg .
1
f x
xn+ 1 = xn
−
f ′ x
b b ng . (12)
The above is a general formula, known as Newton–Raphson formula. Geometrically, Newton’s
method is equivalent to replacing a small arc of the curve y = f(x) by a tangent line drawn to a point
of the curve. For definition sake, let us suppose f ′′ bg x > 0, for a ≤ x ≤ b and f(b) > 0 (see Fig.
2.3) whose x 0
= b, for which f x f ′′ x > .
bg bg
0 0 0
Draw the tangent line to the curve y = f(x) at the point B 0
[x 0
, f(x 0
)].
Let us take the abscissa of the point of intersection of this tangent with the
x-axis, as the first approximation x 1
of the root of c. Again draw a tangent line through B [x 1
, f(x 1
)],
whose abscissa of the intersection point with the x-axis gives us the second approximation x 2
of the root
c and so on. The equation of the tangent at the point B x
[x n
, f(x n
)] [n = 0, 1, 2, …, n] is given by
b ng b ngb ng.
y − f x = f ′ x x − x
Putting y = 0, x = x n + 1'
we get
f x
xn+ 1 = xn
−
f ′ x
Y
b b ng .
B 0
B 1
b
x
B 2
O
fa ( )
a
C
= n
x´
x 2
x 1
x n
X
A
Fig. 2.3