Question

4. Show that the sequence of iterates r(k) defined by r(k) 1 +(rk)2 converges for an arbitrary choice of x(0) є R. Carry out the first l0 itera- tions in Matlab, and comment on the order of convergence.

Answer 1

Matlab code for sequence of iteration clear all close all %Function for sequence Displaying the function fprintf 'For the function \n') disp(£) initial guess xk-2; fprint f ("For initial guess %0.2f\n" , xk ) exact-0.0022360; %loop for iterations for i= 1:1000 xk-f (xk) %printing result for 10 iteration if i<-10 fprint f ( -\tAfter 2d iteration xk-%f\n',1,xk) end error(i)-abs (xk-x_exact); end plot of order of convergence loglog ( error ( 1 : end-1 ) , error ( 2 : end) , . r . ' } title(loglog plot for order of convergence) xlabel error(k) ylabel('error (k+1)) %fitting of error plot ab-polyfit (log (error (1:end-1)),log (error (2:end)),1) yy-polyval (ab, log(error (1:end-1))); hold on loglog (error (1:end-1), exp (yy)); legend 'Actual data', 'fitted data,location', northwest') fprintf(nFrom the plot it can be shown that the order of convergence is linear.In') plot(xx) 号응용号号%%号号%%号 End of Code %용융융용용융융욤욤응응용용융융 For the function For initial guess 2.00 After I iteration xk=0.400000 After 2 iteration xk=0.344828 After 3 iteration xk=0.308183 After 4 iteration xk 0.281451

After 5 iteration xk=0.260793 After 6 iteration xk 0.244185 After 7 iteration xk-0.230444 After 8 iteration xk-0.218824 After 9 iteration xk=0.208825 After 10 iteration xk-0.200099 From the plot it can be shown that the order of convergence is linear. loglog plot for order of convergence Actual data fitted data 101 2 10 102 10 error Published with MATLAB® R2018a

%%Matlab code for sequence of iteration
clear all
close all

%Function for sequence
f=@(x) x./(1+x.^2);
%Displaying the function
fprintf('For the function \n')
disp(f)
%initial guess

xk=2;
fprintf('For initial guess %0.2f\n',xk)
x_exact=0.0022360;
%loop for iterations
for i=1:1000
    xk=f(xk);
    %printing result for 10 iteration
    if i<=10
        fprintf('\tAfter %d iteration xk=%f\n',i,xk)
    end
    xx(i)=xk;
    error(i)=abs(xk-x_exact);
end

%Plot of order of convergence

loglog(error(1:end-1),error(2:end),'r.')
title('loglog plot for order of convergence')
xlabel('error(k)')
ylabel('error(k+1)')

%fitting of error plot
ab=polyfit(log(error(1:end-1)),log(error(2:end)),1);
yy=polyval(ab,log(error(1:end-1)));
hold on
loglog(error(1:end-1),exp(yy));
legend('Actual data','fitted data','location','northwest')
fprintf('\nFrom the plot it can be shown that the order of convergence is linear.\n')

%plot(xx)
%%%%%%%%%%%%%% End of Code %%%%%%%%%%%%%%%%