This preview shows pages 1–9. Sign up to view the full content.
1
NUMERICAL ANALYSIS
AAOC C341
CHAPTER 2
Root Finding Methods for
Nonlinear Equations
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document 2
Root Finding Methods
for Nonlinear Equations
±
Bisection Method
±
Secant Method
±
RegulaFalsi Method
±
Newton’s Method
±
Muller’s Method
±
Fixed Point Method
±
Newton’s Method for Multiple Roots
±
System of Nonlinear Equations
Newton’s Method
and
Fixed Point Method
3
Introduction:
±
In scientific and engineering studies, a frequently
occurring problem is
to find the
roots or
zeros of equations of the form f(x) =0.
±
f(x)may be
algebraic
or
transcendental
or a
combination of both.
±
Algebraic functions of the form
P(x)= a
0
+ a
1
x + …
+ a
n
x
n
,
are called
polynomials
.
±
A
nonalgebraic
function
is
called
a
transcendental
function.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document 4
Solution Methods
±
Polynomials up to degree 4 can be solved exactly.
Since finding the root of
f(x)=0
is not always
possible by analytical means, we have to go for
some other techniques or methods.
±
Solution Methods are either
Bracketing:
Bisection Method,
Regula Falsi Method
±
or
Iterative:
Secant Method, Newton's Method, Muller's
Method, FixedPoint Method.
±
Rate of Convergence.
±
Advantages/Disadvantages.
5
The Bisection Method
The Bisection method
is based on the
Intermediate Value Theorem
and
Cantor's
Intersection Theorem.
Intermediate Value Theorem:
Let
f(x)
be a continuous function in [
a,b
]
and let
k
be
any number between
f(a)
and
f(b)
. Then, there exists
a number
c
in (
a, b
) such that
f(c) = k
.
Hence,
an equation
f(x)=0
where
f(x)
is a real
continuous function, has at least one root between
a
and
b
if
f(a) f(b) < 0
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document 6
Bisection method continued …
Cantor's Intersection Theorem:
If a sequence of closed intervals [
a
n
, b
n
] is
such that [
a
n+1
, b
n+1
]
⊂
[
a
n
, b
n
] for all
n
, and
lim
(b
n
a
n
)=0
, then
consists of exactly one point.
Algorithm for Bisection Method
:
Suppose
f(x)
is continuous and we have
two
numbers
a
1
and
b
1
such that
f(a
1
)f(b
1
)<0
, then by
Intermediate Value Theorem
we know that
there is a root for
f(x)
between
a
1
and
b
1
.
[]
n
n
n
b
a
,
0
∩
∞
=
7
Bisection method continued …
Then we have to divide the interval into two parts and
take the middle point suppose that is
c
, if
f(c ) =0
then
we get the root, if not so then by
IVT
the root lies within
a
and
c
or
c
and
b
, depending on whether
f(a
1
)f(c) <0
or
f(c)f(b
1
)<0
. Then we have to take the next interval in
which the root lies and continue our process.
Since we are dividing the interval by half each time, we
are approaching to zero of the function, hence this
algorithm gives us the approximate solution. Now we
have to put some accuracy criteria to stop the process.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document 8
Bisection methodcontinued …
Algorithm for Bisection Method:
Step 1:
Choose
a
and
b
as two initial guesses for
the root such that
f(a) f(b) < 0
.
Then by
IVT
one root
is guaranteed for
f(x)
in [
a,b
].
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 02/23/2011 for the course MATH 122 taught by Professor Ritadubey during the Spring '11 term at Amity University.
 Spring '11
 ritadubey
 Linear Equations, Numerical Analysis, Equations

Click to edit the document details