Question

Consider the root of function f(x) = x ³ − 2x − 5. The function can be rearranged in the form x = g(x) in the following three ways: (a) x = g(x) = x³ − x − 5 (b) x = g(x) = (x ³ − 5)/2 (c) x = g(x) = thirdroot(2x + 5) For each form, apply fixed-point method with an initial guess x0 = 0.5 to approximate the root. Use the error tolerance = 10^-5 to terminate the iteration. If the iteration does not converge within 15 steps, stop and provide an explanation for the divergence. Keep at least 6 significant digits in your calculation.

Answer 1

a)

clc;
clear all;

format long
% Given f(x)=(sin(x)/2)+1/2-x, we rewrire (sin(x)/2)+1/2=x
g=@(x)x^3-2*x-5; %function

x1=0.5;%initial value
tol=1e-5; % tolerence (Stoping)
erorr=0.1;
n=1;
max_iter=15;
disp('_____________________________________________________________________________________')

disp('n x(n)) g(x(n)) error')
disp('_____________________________________________________________________________________')
while(erorr>tol&n<max_iter)

  
x1(n+1)=g(x1(n));
  
%x=x1;
erorr(n)=abs(x1(n)-x1(n+1));

fprintf('%d\t%15.8f \t %15.8f \t %15.8f \n',n ,x1(n),g(x1(n)),erorr(n))
n=n+1;
end
disp('Root of function is')
y=x1(end)
disp('Number of iteration')
n-1

%%%%%%%%%%%%%%%%%%%%%

_____________________________________________________________________________________
n x(n)) g(x(n)) error
_____________________________________________________________________________________
1   0.50000000    -5.87500000    6.37500000
2   -5.87500000    -196.02929688    190.15429688
3   -196.02929688    -7532525.85236584    7532329.82306897
4   -7532525.85236584    -427387575445953770000.00000000    427387575445946240000.00000000
5   -427387575445953770000.00000000    -78066674213739856000000000000000000000000000000000000000000000.00000000    78066674213739856000000000000000000000000000000000000000000000.00000000
6   -78066674213739856000000000000000000000000000000000000000000000.00000000    -475769978281058810000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000.00000000    475769978281058810000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000.00000000
7   -475769978281058810000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000.00000000    -Inf    Inf
8   -Inf    NaN    NaN
Root of function is

y =

NaN

Number of iteration

ans =

8

>>

b)

clc;
clear all;

format long
% Given f(x)=(sin(x)/2)+1/2-x, we rewrire (sin(x)/2)+1/2=x
g=@(x)(x^3-5)/2; %function

x1=0.5;%initial value
tol=1e-5; % tolerence (Stoping)
erorr=0.1;
n=1;
max_iter=15;
disp('_____________________________________________________________________________________')

disp('n x(n)) g(x(n)) error')
disp('_____________________________________________________________________________________')
while(erorr>tol&n<max_iter)

  
x1(n+1)=g(x1(n));
  
%x=x1;
erorr(n)=abs(x1(n)-x1(n+1));

fprintf('%d\t%15.8f \t %15.8f \t %15.8f \n',n ,x1(n),g(x1(n)),erorr(n))
n=n+1;
end
disp('Root of function is')
y=x1(end)
disp('Number of iteration')
n-1

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

_____________________________________________________________________________________
n x(n)) g(x(n)) error
_____________________________________________________________________________________
1   0.50000000    -2.43750000    2.93750000
2   -2.43750000    -9.74108887    7.30358887
3   -9.74108887    -464.66017665    454.91908778
4   -464.66017665    -50162178.07116126    50161713.41098461
5   -50162178.07116126    -63110142529143789000000.00000000    63110142529143738000000.00000000
6   -63110142529143789000000.00000000    -125680380630448420000000000000000000000000000000000000000000000000000.00000000    125680380630448420000000000000000000000000000000000000000000000000000.00000000
7   -125680380630448420000000000000000000000000000000000000000000000000000.00000000    -992595875594217240000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000.00000000    992595875594217240000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000.00000000
8   -992595875594217240000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000.00000000    -Inf    Inf
9   -Inf    -Inf    NaN
Root of function is

y =

-Inf

Number of iteration

ans =

9

>>

c)

clc;
clear all;

format long
% Given f(x)=(sin(x)/2)+1/2-x, we rewrire (sin(x)/2)+1/2=x
g=@(x)(2*x+5)^(1/3); %function

x1=0.5;%initial value
tol=1e-5; % tolerence (Stoping)
erorr=0.1;
n=1;
max_iter=15;
disp('_____________________________________________________________________________________')

disp('n x(n)) g(x(n)) error')
disp('_____________________________________________________________________________________')
while(erorr>tol&n<max_iter)

  
x1(n+1)=g(x1(n));
  
%x=x1;
erorr(n)=abs(x1(n)-x1(n+1));

fprintf('%d\t%15.8f \t %15.8f \t %15.8f \n',n ,x1(n),g(x1(n)),erorr(n))
n=n+1;
end
disp('Root of function is')
y=x1(end)
disp('Number of iteration')
n-1

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

_____________________________________________________________________________________
n x(n)) g(x(n)) error
_____________________________________________________________________________________
1   0.50000000    1.81712059    1.31712059
2   1.81712059    2.05151513    0.23439454
3   2.05151513    2.08799119    0.03647606
4   2.08799119    2.09355411    0.00556292
5   2.09355411    2.09439991    0.00084580
6   2.09439991    2.09452845    0.00012854
7   2.09452845    2.09454798    0.00001953
8   2.09454798    2.09455095    0.00000297
Root of function is

y =

2.094550949678837

Number of iteration

ans =

8

>>