Numerical Integration -Mathematical Modeling and Simulation-Lecture Slides, Slides of Mathematical Modeling and Simulation

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Modeling and Simulation

Lecture: Numerical Integration

Numerical Integration

• When we have integral from point a to point b, such an

integral is known a definite integral and it is a solution of a

simple ordinary differential equation.

• Numerical integration can used when the function is a

difficult to find or when its data form is given.

• Common numerical integration techniques are:
  • Trapezoidal rule,
  • Simpson’s method,
  • Romberg Integration, and
  • Gaussian quadratures.

You can fit a straight line and / or a polynomial between two data points.

Single trapezoid

You can just fit a straight line

as approximation to actual

function and say what is area

under this straight line.

This area will have large error

as function and straight line

are not same.

Single trapezoid

Many subdivisions

can be done

between x=a and

x=b and then a

straight line fit gives

better comparison to

the actual curve.

Many trapezoid

Many trapezoids

• The same area for two sub-intervals is given by following:

where, f(x0) and f(x2) are same as fa and fb respectively.

• For four sub-divisions within same boundaries, the area is:
• When there are eight sub-divisions in the same range [a, b], the area is

 f(x ) f(x ) f(x ) 

(b a) T (^) o

( ) 1 2

1 2 2 2

 f(x ) f(x ) f(x ) f(x ) f(x ) 

(b a) T (^) o

( ) 1 2 3 4

2 2 2 2 2 4

    

 

f(x ) f(x ) f(x ) f(x )]

[ f(x ) f(x ) f(x ) f(x ) f(x )

(b a) T (^) o

( )

5 6 7 8

1 2 3 4

3

Integral : Area under the curve

• When many sufficiently

small strips are added

we can generate the area

under the curve from x =

a to x = b with very

small error.

f(x)

   a b x

b

a

gb ga f(x)dx

Strips = 18

Here with 18 Strips of different heights and same Width we get the

area under the curve.

Integral for N trapezoids

• The integral in general using N compound trapezoids within

the range, is given by

• where,
• and
• The error in the computed integral value is given by

 ( ) 2 ( ) 2 ( ) 2 ( ) ( )

2

f x 0 f x 1 f x 2 f xN 1 f xN

x Integral     

   

x a i x i N i

   ,  0 , 1 , 2 , 3 ,,

2

  f

b a x

error

N

b a x

(  )  

• For two subdivisions, we need to have:
• Thus

f  f ( 2. 5 )  0. 16 o

f 1  f ( 3. 5 ) 0. 0816

Integral 

( 4. 5 ) 0. 2

f  f 

 f(x ) f(x ) f(x ) 

(b a) T o

( ) 1 2

1 2

2 2

 



 

Example 1 on integration

• If we divide the interval into four equal parts, then

fo  f ( 2. 5 ) 0. 16

 x ( 4. 5  2. 5 )/ 4  0. 5

f 1  f ( 3. 0 ) 0. 111

f 2  f ( 3. 5 ) 0. 0816

f 3  f ( 4. 0 ) 0. 0625

f 1  f ( 4. 5 ) 0.

[ 0. 16 2. ( 0. 111 0. 0816 0. 0625 ) 0. 0494 ]

Integral 

 f(x ) f(x ) f(x ) f(x ) f(x ) 

(b a)

T o

( ) 1 2 3 4

2

Example 1 on integration

% Program Name: Integral_1.m

% A MATLAB script file to compute Integral of f(x)

% over an interval [a, b] by trapezoidal rule.

% It calls an external function trapez.m using

% inputs of a, b and N (number of sub-divisions).

N = 8; a = 2.5; b = 4.5;

exact_value = 0.177778;

estimated_value = trapez(a, b, N)

error = abs(100(exact_value -*

estimated_value)/exact_value)

Example 1 on integration

• It uses a function trapez.m as a subroutine:

function z = trapez(a, b, N)

% function for integration by trapezoidal rule: trapez.m

% to be used with integral_1.m code.

delta_x = (b - a)/N;

x(1) = a;

if(x(1) == 0), error('A is zero, change it')

end

sum = 0;

f(1) = 1.0/(x(1)^2);

for i=2: N+

x(i) = x(i-1) + delta_x;

f(i) = 1/(x(i)^2);

end

if(N > 1)

for i=2: N, sum = sum + f(i); end

end

z = double((delta_x/2)(f(1) + 2sum + f(N+1)));**

Example 1 on integration

Simpson Rule

• When we formed trapezoids, straight lines between points of a

curve were joined; much better approximation is obtained if we

join points by segments of parabolas.

• Suppose we have two intervals for which are known, the

interpolation formula of the second order polynomial can be

obtained.

• Integrating the polynomial in those two intervals of  x , we

obtain the integration formula:

 ( ) 4 ( ) ( )

0

f x f x f x

x

f x dx o

x

x

k



Simpson’s Rule

-. If we have 2I intervals and apply Simpson's rule repeatedly I

times , the integration formula is

• where, the step size is
• and

2 ( ) 4 ( ) ( )]

[ ( ) 4 ( ) 2 ( ) 4 ( ) 2 ( )

3

( )

2 1

1 2 3 4

N N N

o

b

a

f x f x f x

f x f x f x f x f x

x f x dx

   

    

 

 



 

N

b a x

(  )  

x a i x for i N i

   ,  0 , 1 , 2 , 3 ,,