Question

2. [6pt] We attempt to find all solutions to f(x) = 0, where f(x) = e" – 3x – 1. (a) Sketch y = f(x) for -1 < x <3. How many solutions & does f(x) = 0 have? (b) Write code to implement the bisection method. Using the initial interval (1,3), write down the sequence of approximations X1, 22, 23, 24, 25 produced from your code. (c) What is the theoretical maximum value of 25 – £|? How large must we take n to ensure that an - El < 10-10? (d) We now look at the fixed point problem x = g(x) with g(x) = ln(3.c +1). Show that this is equivalent to finding the roots of f.. (e) Implement the fixed point iteration method for x = g(x) given above. Using the initial point Xo = 1, write down the iterates X1, X2, X3, 3X4, X5. (f) Plot the two sequences (In) produced above as functions of n, with n = 0,1,...,100. Is one method faster than the other?

NEED HELP ESPECIALLY ON C,D,E,F

Answer 1

clear all close all function for which root have to find fun=@(x) exp(x)-3. *x-1; x0=-1;x1=3; maxit=1000; [root1, iterl]=bisection method (fun, x0, x1, maxit, 10^-5); fprintf('For error limit 10^-5 number of iteration =&d and root=&f \n',iteri, root1) [root2, iter2] bisection_method (fun, x0, x1, maxit,10^-10); fprintf('For error limit 10^-10 number of iteration =%d and root=$f \n', iter2, root 2) g=@(x) log (3*x+1); [root, iter ]=fixed_method(g,1,maxit,10^-10); fprintf('For error limit 10^-10 with initial guess i, number of iteration = $d and root=$f\n',iter, root) [root, iter]=fixed_method (9,2,maxit,10^-10); fprintf('For error limit 10^-10 with initial guess 2, number of iteration =%d and root=$f\n', iter, root) Matlab function for Bisection Method function (root, iter]=bisection method(fun, x0, x1, maxit, tol) if fun (x0) <=0 t=x0; x0=xl; xl=t; end fprintf('\nRoot using Bisection method for f(x)=') disp(fun) f (x1) should be positive f(x0) should be negative k=10; count=0; while k>tol count=count+1; xx (count)=(x0+x1)/2; mm=double(fun (XX (count))); if mm>=0 x0=xx (count); else x1=xx count); end err(count)=abs(fun (x1)); k=abs(fun(xl)); if count>=maxit break end if count<=5 fprintf('\tAfter $d iteration root using Bisection method is 8f\n',count,xx(count))

end end root-xx (end); iter=count; end $Matlab function for Fixed point Method function [root, iter]=fixed_method(fun, x0,maxit, tol) k=10; count=0; fprintf('\nRoot using fixed point method for f(x)=') disp(fun) while k>tol count=count+1; xx=fun (x0); k=abs(xx-x0); x0=xx; rr (count)=xx; &k=abs(fun (xx)); if count <=5 fprintf('\tAfter $d iteration root using fixed point method is '\n',count,rr(count)) end if count>=maxit break end end Some countertop AS root=rr(count); iter=count; end Root using Bisection method for f(x)= @(x)exp (x)-3. *X-1 After 1 iteration root using Bisection method is 1.000000 After 2 iteration root using Bisection method is 0.000000 After 3 iteration root using Bisection method is 0.500000 After 4 iteration root using Bisection method is 0.250000 After 5 iteration root using Bisection method is 0.125000 For error limit 10^-5 number of iteration =20 and root=0.000004 Root using Bisection method for f(x)= @(x)exp(x)-3. *x-1 After 1 iteration root using Bisection method is 1.000000 After 2 iteration root using Bisection method is 0.000000 After 3 iteration root using Bisection method is 0.500000 After 4 iteration root using Bisection method is 0.250000 After 5 iteration root using Bisection method is 0.125000 For error limit 10^-10 number of iteration =37 and root=0.000000 Root using fixed point method for f(x) = (x)log (3*x+1)

After 1 iteration root using fixed point method is 1.386294 ion root using fixed point method is 1.640720 tion root using fixed point method is 1.778701 After 4 iteration root using fixed point method is 1.846264 After 5 iteration root using fixed point method is 1.877752 For error limit 10^-10 with initial guess i, number of iteration = 30 and root=1.903814 Root using fixed point method for f(x) = @(x)log (3*x+1) After 1 iteration root using fixed point method is 1.945910 After 2 iteration root using fixed point method is 1.922456 After 3 iteration root using fixed point method is 1.912112 After 4 iteration root using fixed point method is 1.907516 After 5 iteration root using fixed point method is 1.905467 For error limit 10-10 with initial guess 2, number of iteration =26 and root=1.903814 Published with MATLAB® R2018a

clear all
close all
%function for which root have to find
fun=@(x) exp(x)-3.*x-1;
x0=-1;x1=3;
maxit=1000;
[root1,iter1]=bisection_method(fun,x0,x1,maxit,10^-5);
fprintf('For error limit 10^-5 number of iteration =%d and root=%f\n',iter1,root1)
[root2,iter2]=bisection_method(fun,x0,x1,maxit,10^-10);
fprintf('For error limit 10^-10 number of iteration =%d and root=%f\n',iter2,root2)

g=@(x) log(3*x+1);
[root,iter]=fixed_method(g,1,maxit,10^-10);
fprintf('For error limit 10^-10 with initial guess 1, number of iteration =%d and root=%f\n',iter,root)

[root,iter]=fixed_method(g,2,maxit,10^-10);
fprintf('For error limit 10^-10 with initial guess 2, number of iteration =%d and root=%f\n',iter,root)

%Matlab function for Bisection Method
function [root,iter]=bisection_method(fun,x0,x1,maxit,tol)
if fun(x0)<=0
    t=x0;
    x0=x1;
    x1=t;
end
fprintf('\nRoot using Bisection method for f(x)=')
disp(fun)
%f(x1) should be positive
%f(x0) should be negative
k=10; count=0;
while k>tol
    count=count+1;
    xx(count)=(x0+x1)/2;
    mm=double(fun(xx(count)));
    if mm>=0
        x0=xx(count);
    else
        x1=xx(count);
    end
    err(count)=abs(fun(x1));
    k=abs(fun(x1));
    if count>=maxit
        break
      
    end
    if count<=5
        fprintf('\tAfter %d iteration root using Bisection method is %f\n',count,xx(count))
    end
end

root=xx(end); iter=count;
end

%Matlab function for Fixed point Method
function [root,iter]=fixed_method(fun,x0,maxit,tol)
    k=10; count=0;
    fprintf('\nRoot using fixed point method for f(x)=')
    disp(fun)
    while k>tol
        count=count+1;
        xx=fun(x0);
        k=abs(xx-x0);
        x0=xx;
        rr(count)=xx;
        %k=abs(fun(xx));
        if count <=5
            fprintf('\tAfter %d iteration root using fixed point method is %f\n',count,rr(count))
        end
        if count>=maxit
                break
        end
    end

    root=rr(count);
    iter=count;
end

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