Objective:
Lab Report 06
Implementation of
Bisection method and False Position method in coding form to find the value of
unknown in MATLAB.
Theory:
·
Bisection Method:
The bisection method is used to find the roots of a
polynomial equation. It separates the interval and subdivides the interval in
which the root of the equation lies. The principle behind this method is the
intermediate theorem for continuous functions. It works by narrowing the gap
between the positive and negative intervals until it closes in on the correct
answer. This method narrows the gap by taking the average of the positive and negative
intervals. It is a simple method and it is relatively slow. The bisection
method is also known as interval halving method, root-finding method, binary
search method or dichotomy method.
Let us consider a continuous function “f” which is defined on the closed
interval [a, b], is given with f(a) and f(b) of different signs. Then by
intermediate theorem, there exists a point x belongs to (a, b) for which f(x) =
0.
Bisection Method Algorithm:
Follow the below procedure to
get the solution for the continuous function:
i.
Calculate c, the midpoint of the interval, c = a + b/2.
ii.
Calculate the function value at the midpoint, f(c).
iii.
If convergence is satisfactory (that is, c - a is
sufficiently small, or |f(c)| is sufficiently small), return c and stop iterating.
iv.
Examine the sign of f(c) and replace either (a, f(a))
or (b, f(b)) with (c, f(c)) so that there is a zero crossing within the new interval.
Repeat above three steps until
f(t) = 0.
The bisection method is an approximation method to find the roots of the
given equation by repeatedly dividing the interval. This method will divide the
interval until the resulting interval is found, which is extremely small.
MATLAB Code:
>>clc
>>close all
>>clear all
>>xl=input('Enter Lower
Boundary limit: ');
>>xu =input ('Enter Upper
Boundary limit: ');
>>n= input('Enter number
of iteration ');
>>fx1=
(668.06/xl)*(1-exp(-0.146843*xl))-40;
>>fxu= (668.06/xu)
*(1-exp(-0.146843*xu))-40;
>>if fx1*fxu<0
>>disp('Guees is right')
>>else
>>disp('your guess is
wrong')
>>break
>>end
>>for i=1:1:n
>>xr(i)=(xl+xu)/2;
>>fx1=
(668.06/xl)*(1-exp(-0.146843*xl))-40;
>>fxu=
(668.06/xu)*(1-exp(-0.146843*xu))-40;
>>fxr(i)=(668.06/xr(i))*(1-exp(-0.146843*xr(i)))-40;
>>if fx1*fxr (i) <0
>>xu=xr (i);
>>else if fx1*fxr(i)>0
>>xl=xr(i);
>>else
>>xr(i)
>>end
>>end
>>if
i>1
>> e(i)=
((xr(i)-xr(i-1))/xr(i))*100;
>>e(i-1)
>>if norm (e(i))<.02
>>xr(i-1)
>>break
>>end
>>end
>>end Screenshots: Input:
Output:
· False Position Method:
ThisMATLAB program for finding real root of non-linear
equation using Regula Falsi Method with Output. Regula Falsi method is also
known as False Position Method. In this MATLAB program for false position
method, y is nonlinear function, a & b are two initial guesses and e is
tolerable error.
This is one of the iterative methods that give you the
root if the function changes its sign: from positive to negative or from
negative to positive. Despite the fact that bisection is an entirely legitimate
strategy for determining roots, its “brute force” approach is generally
inefficient. False position is based on graphical approach. A shortcoming of
the bisection method is that, in dividing the interval from xl to xu into
equivalent parts, no record is taken of the values of f (xl) and f (xu).
For instance, if f (xl) is very near to zero than f
(xu), it is just like that the root is nearer to xl than to xu (as shown in the
figure below). An alternate method that exploits this graphical understanding
is to join f (xl) and f (xu) by a straight line. The intersection of this line
with the x-axis gives an improved version of the root.
The way that the substitution of a curve by a straight
line gives a “false position” of the root is the actual point of the name,
method of false position, or in Latin, regula falsi method. It is additionally
called the linear interpolation method. Because it takes the same approach
where two points of a function are joined with a straight line.
Difference b/w False Position and Bisection:
The difference between bisection method and
false-position method is that in bisection method, both limits of the interval
have to change. This is not the case for false position method, where one limit
may stay fixed throughout the computation while the other guess converges on
the root.
 |
MATLAB Code:
>>clc
>>clear all
>>close all
>>xl= input('Enter value of
lower boundary limit');
>>xu= input('Enter value of
upper boundary limit');
>>n= input('Enter number of
iteration');
>>fxl=(668.06/xl)*(1-exp(-0.146843*xl))-40;
>>fxu=(668.06/xu)*(1-exp(-0.146843*xu))-40;
>>if fxl*fxu<0
>> disp('Guess is right')
>>else
>> disp('Guess is wrong')
>> break
>>end
>>for i=1:1:n
>> fxl=(668.06/xl)*(1-exp(-0.146843*xl))-40;
>>
fxu=(668.06/xu)*(1-exp(-0.146843*xu))-40;
>>
xr(i)=xu-(((fxu)*(xl-xu))/(fxl-fxu))
>>
fxu=(668.06/xu)*(1-exp(-0.146843*xu))-40;
>>
fxr(i)=(668.06/xr(i))*(1-exp(-0.146843*xr(i)))-40;
>>if fxl*fxr(i)<0
>> xu=xr(i)
>> else if fxl*fxr(i)>0
>> xl=xr(i)
>> else
>> xr(i)
>> end
>> end
>> if i>1
>> e(i)=((xr(i)-xr(i-1))/xr(i))*100;
>> e(i-1)
>> if norm(e(i))<.02
>> xr(i-1)
>> break
>>
end
>> end
>>end
 |
MATLAB Screenshots: Input:
Output:
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