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Presentation on Numerical Method (Trapezoidal Method)

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- 1. Trapezoidal Method
- 2. Acknowledgement Md. Jashim Uddin Assistant Professor Dept. Of Natural Sciences Dept. Of Computer Science and Engineering Daffodil International University
- 3. 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
- 4. Team : Root Finder Group 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
- 5. What is Trapezoidal Method ? In numerical analysis, the trapezoidal rule or method is a technique for approximating the definite integral. 푥푛 푥0 f(x) dx It also known as Trapezium rule. 1
- 6. General Formula of Integration In general Integration formula when n=1 its Trapezoidal rule. I=h[n푦0+ 푛2 2 Δ푦0+ 2푛3−3푛2 12 Δ2푦0+ 푛4−4푛3+4푛2 24 Δ3푦0 + ⋯ ] After putting n=1, Trapezoidal Rule = ℎ 2 [푦0 + 푦푛 + 2(푦1 + 푦2 + 푦3 + ⋯ . 푦푛−1)] 2
- 7. How it works ? Trapezoid is an one kind of rectangle which has 4 sides and minimum two sides are parallel Area A= 푏1+푏2 2 ℎ 3
- 8. 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, 푏 f(x) dx ≈ 푎 (b−a) 2 [f(a) +f(b)] 4
- 9. The trapezoidal rule approximation improves With More strips , from This figure we can clearly See it 5
- 10. 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
- 11. Advantages There are many alternatives to the trapezoidal rule, but this method deserves attention because of • Its ease of use • Powerful convergence properties • Straightforward analysis 7
- 12. 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
- 13. Example: 푥1 푥2 푥3 =2 =3 =4 =1 =5 5 1 + 푥2 푑푥 1 h = 5−1 4 =1 Trapezoidal Rule = 1 2 [ 푓(1) + 푓(5) + 2(푓(2) + 푓(3) + 푓(4)] = 1 2 [ (1 + 12) + (1 + 52) + 2((1 + 22) + (1 + 32) + (1 + 42)] = 1 2 × 92 = 46 9
- 14. Problem & Algorithm Problem: Here we have to find integration for the (1+푥2)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
- 15. 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: "); 11
- 16. 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; } 12
- 17. Live Preview Live Preview of Trapezoidal Method 5 1 + 푥2 푑푥 1 Lower limit =1 Upper limit =5 Interval h=4 13
- 18. 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
- 19. References http://en.wikipedia.org/wiki/Trapezoidal_rule http://blogs.siam.org/the-mathematics-and-history- of-the-trapezoidal-rule/ And various relevant websites 15
- 20. Thank You

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