MATLAB : Numerical Differention and Integration

AHSANULLAH UNIVERSITY OF
SCIENCE AND TECHNOLOGY
EEE 3110
NUMERICAL TECHNIQUE LABORATORY

PRESENTED BY :
AINUL ISLAM (ID 12-02-05-065)
SHADMAN SAKIB AOYON (ID 12-02-05-060)
MIRZA SHAMIM ARMAN (ID 12-02-05-072)
TAUHIDUR RAHMAN AKIF (ID 12-02-05-062)
KAZI WALIDA AFROZE (ID 12-02-05-079)

CONTENTS:
• INTRODUCTION TO NUMERICAL DIFFERENTIATION
• FORWARD DIFFERENCE FORMULA
• CENTRAL DIFFERENCE FORMULA
• RICHARDSON’S EXTRAPOLATION
• INTRODUCTION TO NUMERICAL INTEGRATION
• TRAPEZOIDAL RULE
• SIMPSON’S RULE

Numerical Differentiation :
Numerical differentiation deals with the following problem : we are given the function
y = f (x) and wish to obtain one of its derivatives at the point x = xk. The term “given”
means that we either have an algorithm for computing the function, or possess a
set of discrete data points (xi , yi ), i = 1, 2, . . . , n. In either case, we have access to a
finite number of (x, y) data pairs from which to compute the derivative. If you suspect
by now that numerical differentiation is related to interpolation, you are right—one
means of finding the derivative is to approximate the function locally by a polynomial
and then differentiate it. An equally effective tool is the Taylor series expansion of
f (x) about the point xk. The latter has the advantage of providing us with information
about the error involved in the approximation.
Numerical differentiation is not a particularly accurate process. It suffers from
a conflict between round off errors (due to limited machine precision) and errors
inherent in interpolation. For this reason, a derivative of a function can never be
computed with the same precision as the function itself

Taylor series expansion can be used to generate high-
accuracy formulas for derivatives by using linear algebra
to combine the expansion around several points.
We will discuss three categories of formula:
• Forward difference formula
•Centered difference formula
•Richardson Extrapolation formula

EXAMPLE :
Given that f(x)=(sin x) and h=.001 find the first derivative f’(x)
using forward difference formula in MATLAB :
Code :-
clear all;
close all;
clc;
h=.001;
x=0:(.001):2*pi;
f=inline(‘sin (x)’)
q=f(x)
d=f(x+h)
a=((d-q)/h)
plot(x,a)
d
a

Richardson Extrapolation
• As with integration, the Richardson extrapolation can be used to combine two lower-accuracy
estimates of the derivative to produce a higher-accuracy estimate.
• For the cases where there are two O(h2
) estimates and the interval is halved (h2=h1/2), an
improved O(h4
) estimate may be formed using:
• For the cases where there are two O(h4
) estimates and the interval is halved (h2=h1/2), an
improved O(h6
) estimate may be formed using:
D=
4
3
D(h2)−
1
3
D(h1)
D=
16
15
D(h2)−
1
15
D(h1)

Numerical integration, also known as quadrature, is intrinsically a much more accurate
procedure than numerical differentiation. Quadrature approximates the definite
Integral. Numerical integration is a widely encountered problem in economics.
All rules of quadrature are derived from polynomial interpolation of the
integrand. Therefore, they work best if f (x) can be approximated by a polynomial.
Methods of numerical integration can be divided into two groups:
1.Newton–Cotes formulas (The Trapezoidal Rule and Simpson’s Rule)
2.Gaussian quadrature.
Newton–Cotes formulas are characterized by
equally spaced abscissas, and include well-known methods such as the trapezoidal
rule and Simpson’s rule. They are most useful if f (x) has already been computed at
equal intervals, or can be computed at low cost. Since Newton–Cotes formulas are
based on local interpolation, they require only a piecewise fit to a polynomial
INTRODUCTION TO NUMERICAL INTEGRATION :

The trapezoidal rule uses trapezoids
to approximate area, which usually is
a much more accurate
approximation, even with just a few
subintervals.
Notice that the height of the
trapezoids will actually be Δx
(horizontal) and bases will be the parallel
f(xi) (vertical) on each side of the
subinterval.
( )trapezoid
1 2
A =
2
b b
h
+
×
THE TRAPEZOIDAL RULE

What is Trapezoidal Method ?
1

Observe that the coefficients are 1, 2, 2, 2, 2, . . . 2, 1
TRAPEZOIDAL RULE

The area of the first trapezoid would be
0 1( ) ( )
2
f x f x b a
n
+ −   
× ÷    
0 1 11 2( ) ( ) ( ) ( )( ) ( )
2 2 2
n nf x f x f x f xf x f xb a
n
−+ ++−   
+ +×× × + ÷ 
   
Area =
and then total area would be
Letting the number of trapezoids n approach infinity, you
improve the approximation to the exact answer.
( ) ( ) ( )0 1 1 2 1lim ( ) ( ) ( ) ( ) ( ) ( )
2
n n
n
b a
f x f x f x f x f x f x
n
−
→∞
−
+ + + +×× × + +  

There are many alternatives to the trapezoidal rule,
but this method deserves attention because of :
•Its ease of use
•The trapezoidal rule has faster convergence.
•Moreover, the trapezoidal rule tends to become extremely accurate than
periodic functions
•Powerful convergence properties
•Straightforward analysis
Advantages
7

Thomas Simpson (1710-1761) used second-degree polynomials
(a section of a quadratic) to approximate the graph for each subinterval.
Before we get into Simpson’s Rule, we need to list a theorem for evaluating
integrals of polynomials of degree 2 or less.
THE SIMPSON’S RULE

THE SIMPSON’S RULE
Notice the coefficients are 1, 4, 2, 4, . . . 2, 4, 1

EXAMPLE OF SIMPSON’S RULE :
Given, f=(cos x); n=12; low value=-1; high value=2;
Integrate the function over the values of x using Simpson’s 1/3 rule :
clear all;
close all;
clc;
h= -0.1;
n=12;
sum=0;
f=inline(‘ cos (x)’);
sum=sum + f(a)
for i=2 : 2 :n ;
x(i)=a + i*n;
sum=sum+4*f(x(i))
end
for i =3 : 2 : (n-1) ;
x( i )=a + i *n;
sum=sum+2*f(x(i))
end
sum=sum + f(a +n * h)
p=(sum*h)/3

ERRORS :
These two expressions tell how much of an error you can expect to
get using either the Trapezoidal Rule or Simpson’s Rule.

 http://en.wikipedia.org
 www.google.com
 http://slideshare.com
 And various relevant websites
References
15

MATLAB : Numerical Differention and Integration

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