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Thursday, August 30, 2012
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
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