Date post: | 16-Jul-2015 |
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Engineering |
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Acknowledgement
Md. Jashim Uddin
Assistant Professor
Dept. Of Natural Sciences
Dept. Of Computer Science and
Engineering
Daffodil International University
Content What is Trapezoidal Method
General Formula of Integration
How it works
History of Trapezoidal Method
Advantages
Application of Trapezoidal Rule
Example
Problem & Algorithm
C code for Trapezoidal Rule
Live Preview
Conclusion
References
Team : Root FinderGroup Member :
• Syed Ahmed Zaki ID:131-15-2169
• Fatema Khatun ID:131-15-2372
• Sumi Basak ID:131-15-2364
• Priangka Kirtania ID:131-15-2385
• Afruza Zinnurain ID:131-15-2345
What is Trapezoidal Method ?
In numerical analysis, the trapezoidal rule or method is a
technique for approximating the definite integral.
𝑥0𝑥𝑛
f(x) dx
It also known as Trapezium rule.
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General Formula of Integration
In general Integration formula when n=1 its
Trapezoidal rule.
I=h[n𝑦0+ 𝑛2
2∆𝑦0+
2𝑛3−3𝑛2
12∆2𝑦0+
𝑛4−4𝑛3+4𝑛2
24∆3𝑦0 +⋯ ]
After putting n=1,
Trapezoidal Rule = ℎ
2[𝑦0 + 𝑦𝑛 + 2(𝑦1 + 𝑦2 + 𝑦3 +⋯ . 𝑦𝑛−1)]
2
How it works ?
Area A=𝑏1+𝑏2
2ℎ
Trapezoid is an one kind of rectangle which has 4 sides and minimum two sides are parallel
3
The trapezoidal rule works
by approximating the region
under the graph of the
function as a trapezoid and
calculating its area in limit.
It follows that,
𝑎𝑏
f(x) dx ≈ (b−a)2
[f(a) +f(b)]
4
The trapezoidal rule
approximation improves
With More strips , from
This figure we can clearly
See it
5
History Of Trapezoidal Method
• Trapezoidal Rule,” by Nick Trefethen and André Weideman. It deals with a fundamental and classical issue in numerical analysis—approximating an integral.
• By focusing on up-to-date covergence of recent results
Trefethen
6
There are many alternatives to the trapezoidal rule,
but this method deserves attention because of
• Its ease of use
• Powerful convergence properties
• Straightforward analysis
Advantages
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Application of Trapezoidal Rule
• The trapezoidal rule is one of the family members of
numerical-integration formula.
• The trapezoidal rule has faster convergence.
• Moreover, the trapezoidal rule tends to become
extremely accurate than periodic functions
8
Example:
𝑥1 𝑥2 𝑥3=1 =5
1
5
1 + 𝑥2 𝑑𝑥
h = 5−1
4=1
Trapezoidal Rule = 1
2[ 𝑓(1) + 𝑓(5) + 2(𝑓(2) + 𝑓(3) + 𝑓(4)]
=2 =3 =4
= 1
2[ (1 + 12) + (1 + 52) + 2((1 + 22) + (1 + 32) + (1 + 42)]
= 1
2× 92
= 469
Problem & Algorithm
Problem: Here we have to find integration for the (1+𝑥2)dx with lower limit =1 to upper limit = 5
Algorithm:
Step 1: input a,b,number of interval n
Step 2: h=(b-a)/n
Step 3: sum=f(a)+f(b)
Step 4: If n=1,2,3,……i
Then , sum=sum+2*y(a+i*h)
Step 5: Display output=sum *h/2
10
C Code for Trapezoidal Method
#include<stdio.h>
float y(float x)
{
return (1+x*x);
}
int main()
{
float a,b,h,sum;
int i,n;
printf("Enter a=x0(lower limit), b=xn(upper limit), number of
subintervals: ");
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scanf("%f %f %d",&a,&b,&n);
h=(b-a)/n;
sum=y(a)+y(b);
for(i=1;i<n;i++)
{
sum=sum+2*y(a+i*h);
}
printf("\n Value of integral is %f \n",(h/2)*sum);
return 0;
}
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Live Preview
Live Preview of Trapezoidal Method
1
5
1 + 𝑥2 𝑑𝑥
Lower limit =1
Upper limit =5
Interval h=4
13
Conclusion
Trapezoidal Method can be applied accurately for
non periodic function, also in terms of periodic
integrals.
when periodic functions are integrated over their
periods, trapezoidal looks for extremely accurate.
14
Periodic Integral Function
http://en.wikipedia.org/wiki/Trapezoidal_rule
http://blogs.siam.org/the-mathematics-and-
history-of-the-trapezoidal-rule/
And various relevant websites
References
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