Numerical Methods (Problems and solutions) by m.k.jain and s.r.k.lyengar and r.k.jain pdf

INTRODUCTION
We consider the methods for determining the roots of the equation
f   (x ) = 0 (1.1)
which may be given explicitly as a polynomial of degree  n in  x  or  f   (x ) may be defined
implicitly as a transcendental function. A transcendental equation (1.1) may have no root,
a finite or an infinite number of real and / or complex roots while a polynomial equation (1.1)
has exactly  n (real and / or complex) roots. If the function  f  (x ) changes sign in any one of
the intervals [ x * –  ε ,  x*], [ x *,  x* +  ε], then  x* defines an approximation to the root of  f   (x )
with accuracy  ε. This is known as  intermediate value  theorem. Hence, if the interval [ a,  b]
containing  x * and  ξ  where  ξ is the exact root of (1.1), is sufficiently small, then
|  x * –  ξ  | ≤   b – a
can be used as a measure of the error.
There are two types of methods that can be used to find the roots of the equation (1.1).
( i ) Direct methods  : These methods give the exact value of the roots (in the absence of
round off errors) in a finite number of steps. These methods determine all the roots at
the same time.
( ii ) Iterative methods  : These methods are based on the idea of successive approximations.
Starting with one or more initial approximations to the root, we obtain a sequence of
iterates {x
k

} which in the limit converges to the root. These methods determine one or
two roots at a time.
Definition 1.1 A sequence of iterates {x
k
} is said to converge to the root ξ  if
lim
k →∞
 | x
k
 – ξ  | = 0.
If x
k
,  x
k –1
, ... , x
k–m+1
 are m approximates to a root, then we write an iteration method in
the form
 x
k + 1
 = φ( x
k
,  x
k–1
, ... ,  x
k –m +1)
(1.2)
where we have written the equation (1.1) in the equivalent form
x  = φ ( x ).
The function φ  is called the iteration function. For m = 1, we get the one-point iteration