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Objective: Software Used: Computer Specifications:: To Understand Bi-Section Method. Matlab

The document discusses three root-finding algorithms: 1) Bisection method, which iteratively halves the interval containing a root and converges linearly but is comparatively slow. 2) Regula falsi method, which always converges but can be slower than bisection in some cases. 3) Newton-Raphson method, which uses calculus to converge quadratically but requires calculating derivatives and may not converge for some functions. Matlab code is provided to demonstrate the implementation and output of each method.

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Abhishek Bhalla
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142 views16 pages

Objective: Software Used: Computer Specifications:: To Understand Bi-Section Method. Matlab

The document discusses three root-finding algorithms: 1) Bisection method, which iteratively halves the interval containing a root and converges linearly but is comparatively slow. 2) Regula falsi method, which always converges but can be slower than bisection in some cases. 3) Newton-Raphson method, which uses calculus to converge quadratically but requires calculating derivatives and may not converge for some functions. Matlab code is provided to demonstrate the implementation and output of each method.

Uploaded by

Abhishek Bhalla
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© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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OBJECTIVE: To understand Bi-section method.

SOFTWARE USED: Matlab

COMPUTER SPECIFICATIONS:
FLOWCHART:
MATLAB
CODE:

close all

clear all

clc

f=@(x)(x^3-x-2);

xl=input(‘Enter the first approximation xl:’);

xu=input(‘Enter the second approximation xu:’);

acc=input(‘Enter the value of accuracy:’);

while((f(xl)*f(xu))>0)

xl=input(‘Enter the value of first approximation xl:’);


xu=input(‘Enter the value of second approximation xu:’);

end

while(abs(xu-xl)>acc)

xm=(xl-xu)/2;

if(f(x)*f(xm)<0)

xu=xm;

else

xl=xm;

end

end

fprintf(‘The root of equation is:’,xm);

Plot of Error of Bisection Method:-


OUTPUT/ITERATIONS:
The input for the method is a continuous function f, an interval [a, b], and the function
values f(a) and f(b). The function values are of opposite sign (there is at least one zero
crossing within the interval). Each iteration performs these steps:

1. Calculate c, the midpoint of the interval, c = a + b/2.

2. Calculate the function value at the midpoint, f(c).

3. If convergence is satisfactory (that is, c - a is sufficiently small, or |f(c)| is


sufficiently small), return c and stop iterating.

4. 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.

Iteration an bn cn f(cn)

1 1 2 1.5 −0.125

2 1.5 2 1.75 1.6093750

3 1.5 1.75 1.625 0.6660156

4 1.5 1.625 1.5625 0.2521973

5 1.5 1.5625 1.5312500 0.0591125

6 1.5 1.5312500 1.5156250 −0.0340538

7 1.5156250 1.5312500 1.5234375 0.0122504

8 1.5156250 1.5234375 1.5195313 −0.0109712

9 1.5195313 1.5234375 1.5214844 0.0006222

10 1.5195313 1.5214844 1.5205078 −0.0051789

11 1.5205078 1.5214844 1.5209961 −0.0022794

12 1.5209961 1.5214844 1.5212402 −0.0008289

13 1.5212402 1.5214844 1.5213623 −0.0001034

14 1.5213623 1.5214844 1.5214233 0.0002594

15 1.5213623 1.5214233 1.5213928 0.0000780


After 15 iterations, it becomes apparent that there is a convergence to about 1.521: a root for the
polynomial.

RESULT: The method is guaranteed to converge to a root of f if f is a continuous function on


the interval [a, b] and f(a) and f(b) have opposite signs. The absolute error is halved at each step so
the method converges linearly, which is comparatively slow.
OBJECTIVE: To understand regula falsi method.

SOFTWARE USED: Matlab

COMPUTER SPECIFICATIONS:
PLOT OF ERROR FOR REGULA-FALSI METHOD:
FLOWCHART:
MATLAB CODE:
Clear all

close all

clc;

f=@(x) x^3-2*x-5;

a=2; b=3;

for i=1:10

x0=a; x1=b;

fprintf('\n Hence root lies between (%.4f,%.0f)',a,b)

x2(i)=x0-(x1-x0)/(f(x1)-f(x0))*f(x0);

if f(x2(i))>0

b=x2(i);

else a=x2(i);

end

fprintf('\n Therefore, x2=%.4f \n Here, f(x2)=%.4f',x2(i),f(x2(i)))

p=x2(i);

end

OUTPUT/ITERATIONS:
RESULT:
The fact that regula falsi always converges, and has versions that do well at avoiding
slowdowns, makes it a good choice when speed is needed. However, its rate of
convergence can drop below that of the bisection method.

OBJECTIVE: To understand Newton-Raphson method.


SOFTWARE USED: Matlab

COMPUTER SPECIFICATIONS:
FLOWCHART:

MATLAB CODE:
a=input('Enter the function in the form of variable x:','s');

x(1)=input('Enter Initial Guess:');

error=input('Enter allowed Error:');

f=inline(a)

dif=diff(sym(a));

d=inline(dif);

for i=1:100

x(i+1)=x(i)-((f(x(i))/d(x(i))));

err(i)=abs((x(i+1)-x(i))/x(i));

if err(i)<error

break

end

end

root=x(i)

Sample output:

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