Aitken process
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Aitken's delta-squared process is a series acceleration method used to accelerate the convergence of a sequence. It was introduced in 1926 by Alexander Aitkin. The method generates a sequence that converges to the root of a function more quickly than the original sequence. The summary provides an example applying Aitken's process to find the root of x^2+2x-2=0, starting with x1=1 and calculating successive terms x2, x3, etc. until the difference between terms is less than a small value. Pseudocode is also provided to implement the Aitken's process algorithm.
Aitken process
- 1. Prepared by Md. Mujahid Islam Md.Rafiqul Islam Khaza Fahmida Akter
- 2. In Numerical analysis, Aitkin's delta-squared process is a series acceleration method, used for accelerating the rate of convergence of a sequence. It is named after Alexander Aitkin , who introduced this method in 1926.Its early form was known to Seki Kowa (end of 17th century) and was found for rectification of the circle, i.e. the calculation of π. It is most useful for accelerating the convergence of a sequence that is converging linearly.
- 3. • Start with a suitable x1 • Take x2=g(x1) • Take x3=g(x2) • Take x4=x3-(x3-x2)2/(x3-2x2+x1) • Take x5=g(x4) • Take x6=g(x5) • Take x7=x6-(x6-x5)2/(x6-2x5+x4) Continue this process to the required accuracy.
- 4. Consider the following equation determine root according to the aitken’s process X2+2x-2=0
- 5. g(xn)=Xn+1=1-xn 2 Let x1=1.0 Then x2=g(1.0)=0.5 x3=0.875 x4=0.714285 x5=0.7445976 x6=0.7225635 x7=0.7320507 Continue this process until fulfill this condition .
- 7. • #include<cstdio> • #include <cmath> • #include<iostream> • using namespace std; • #define ESP 0.0000001 • #define F(x) 1-(0.5*x*x) • int main() • { • double x1,x2,x3,x4,y; • cin>>x1; • for(int i=1;;i=i+3){ • x2=F(x1); • x3=F(x2); • if(fabs(x3-x2)<ESP){cout<<x3;break;} • else{ • y=x3-2*x2+x1; • x1=x3-(((x3-x2)*(x3-x2))/y); • } • } • }
- 8. Short Description of Aitken’s process Algorithm Example Graphically representation Code
- 10. THANKS TO ALL