Fixed Point Interation
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The document outlines a practical report on implementing the fixed point iteration method for numerical analysis, including the creation of an m-file to compute errors and convergence rates. It demonstrates the theoretical foundation, numerical implementation, and convergence analysis for finding roots of the equation f(x)=0, leading to a real root of approximately x=1.3141. The report concludes that the method is linearly convergent with a convergence rate of 0.3099.
![Theory
Fixed point iteration method is based on the principle of finding
a sequence {xn}, each element of which successively
approximates a real root of equation in [a,b]. We re-write f(x)=0
as x=Φ(x).Thus, a root α of the given equation satisfies α= Φ(x)
Therefore, the point remains fixed under the mapping Φ and so
a root of the equation is a fixed point of Φ(x) which is called the
iteration function.
Iterations are generated by the formula xn+1 = Φ(x)
The sequence {xn} of iterations may not converge to a limit. If it
converges, then it converges to α and the
number of iterations required depend upon the desired degree
of accuracy of the root α.](https://image.slidesharecdn.com/20463985-230523145655-ded92f93/85/Fixed-Point-Interation-3-320.jpg)
![Script File
x0=1; %initial approx
MaxIter=20;
tolX=1e-8;
%iteration Method
x=x0;
xold=x0;
minx=x0;
maxx=x0;
for i =1:MaxIter
x=sqrt(2-log(x)); %function
if x<minx
minx=x;
end
if x>maxx
maxx=x;
end
err(i)=abs(x-xold);
xold=x;
if(err(i)<tolX)
break
end
%display error in iterations
disp(['Error in iteration
',num2str(i),' is =
',num2str(err(i),'%e')])
end
plot(x,x,'*')
hold on
X=linspace(minx,maxx,51);
Y=sqrt(2-log(X));
plot(X,Y)](https://image.slidesharecdn.com/20463985-230523145655-ded92f93/85/Fixed-Point-Interation-5-320.jpg)
Fixed Point Interation
- 1. PRACTICAL Name- Saloni Singhal M.Sc. (Statistics) II-Sem. Roll No: 2046398 Course- MATH-409 L Numerical Analysis Lab Submitted To: Dr. S.C. Pandey
- 2. OBJECTIVE 1. Create an M-file to implement Fixed Point Iteration method. 1. Calculate the error and convergence of this method.
- 3. Theory Fixed point iteration method is based on the principle of finding a sequence {xn}, each element of which successively approximates a real root of equation in [a,b]. We re-write f(x)=0 as x=Φ(x).Thus, a root α of the given equation satisfies α= Φ(x) Therefore, the point remains fixed under the mapping Φ and so a root of the equation is a fixed point of Φ(x) which is called the iteration function. Iterations are generated by the formula xn+1 = Φ(x) The sequence {xn} of iterations may not converge to a limit. If it converges, then it converges to α and the number of iterations required depend upon the desired degree of accuracy of the root α.
- 4. Convergence The presentation of f(x)=0 as x= Φ(x) is not unique, therefore, the convergence of {xn } depends upon the nature of Φ(x). Using Lagrange Mean Value Theorem: We conclude the result:
- 5. Script File x0=1; %initial approx MaxIter=20; tolX=1e-8; %iteration Method x=x0; xold=x0; minx=x0; maxx=x0; for i =1:MaxIter x=sqrt(2-log(x)); %function if x<minx minx=x; end if x>maxx maxx=x; end err(i)=abs(x-xold); xold=x; if(err(i)<tolX) break end %display error in iterations disp(['Error in iteration ',num2str(i),' is = ',num2str(err(i),'%e')]) end plot(x,x,'*') hold on X=linspace(minx,maxx,51); Y=sqrt(2-log(X)); plot(X,Y)
- 6. Output
- 7. Plot of the iterated values and the final fixed point This shows the final iterated value =1.3141 satisfying x=Φ(x)
- 8. Log Plot of error in i and i+1 iterations Codes:
- 9. Conclusion • For the given equation f(x)=x2+log(x)-2=0 • the initial approximated root is x0=1 and f’(1)=0.35<1 so convergence is assured. • The rate of convergence of err is 0.3099 • Since the rate of convergence is 1(calculated from log plot), so the method is linearly convergent. • The real root is x =1.3141 • And as f(1.3141) is in accepted region as tolx =1e-8