Iterative1
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The document describes the iterative method for finding roots of equations. It discusses: 1) How to set up an iterative process to find a root by taking an initial approximation x0 and repeatedly applying the function pi(x) to get closer approximations x1, x2, etc. until the change between successive values is very small. 2) An example of using the method to find the root of the quadratic equation 2x^2 - 4x + 1 = 0. It shows calculating successive approximations that converge to the root. 3) How the iterative method can be implemented in C++ using a function that takes the initial value and returns the updated value at each iteration until the change is negligible.
Iterative1
- 1. P R E P A R E D B Y M D . M U J A H I D I S L A M M D . R A F I Q U L I S L A M K H A Z A F A H M I D A A K T E R
- 2. Let the given equation be f(x) = 0 and the value of x to be determined. By using the Iteration method you can find the roots of the equation. To find the root of the equation first we have to write equation like below x = pi(x) Let x=x0 be an initial approximation of the required root α then the first approximation x1 is given by x1 = pi(x0). Similarly for second, third and so on. Approximation x2 = pi(x1) x3 = pi(x2) x4 = pi(x3) xn = pi(xn-1)
- 3. First we take a initial value. Then determine function value by using initial value. Again we find the function value using function previous value until current value & previous value different is 0.0000001 . When we reached our expected position then we surely will get root .
- 4. Consider the quadratic equation 2x2-4x+1=0 =>x=(1/2)x2+(1/4) =>xn+1=(1/2)xn 2+(1/4) How to Determine the root of this equation by iterative method? Solution are given bellow…………..
- 5. Let , x0=1 .Now from the equation we can get several value using this method . Those value are given bellow by using a table . So x1=(1/2)*1*1+(1/4) 0.50+0.25 =0.75 Now determine current value by using previous values again and again .The process are given bellow.
- 6. n Xn 1 1.0 2 0.75 . . . . . . . . 13 0.292894 14 0.292893 15 0.292893
- 7. How using this method in our PC? Answer : We can using this method to determine root by using different Language such as Java, C, C++, Pascal etc. We use a specific Language that is C++ .Those are presented bellow.
- 8. #include<iostream> #include<cmath> using namespace std; double rec(const int,const int,const int,double); int main() { int a,b,c; cin>>a>>b>>c; double x1,x2,x3; cin>>x1; for(int i=1;;i++){ x2=rec(a,b,c,x1); if(fabs(x1-x2)<=0.0000001) {cout<<x1;break;} else{ x1=x2; } } } double rec(const int a,const int b,const int c,double x){ return (-((a*x*x+c)/b)); }
- 9. We define variable type & define function type . Input initial values & passing it through the function . Through the function manipulate it & return values again & again .Then we check condition again & again .
- 10. Any question? Please give your valuable Suggestion to improve our knowledge………