(*  Initializing the variable "exactvalue" 
      used below for computing the error of each method  *)

exactvalue=N[7^(1/5), 3000];



(*  Fixed point iteration given by (a)  *)

p=N[1,3000]; g[x_] := x-(x^5-7)/(x^2);
For[i=1, iČ14, i++, 
  {p=g[p], y = Abs[p - exactvalue], 
    Print[i], Print[1.0*p], Print[Log[y]/Log[10.]], Print[ ], 
    }
  ]

g'[x]

1.0*g'[exactvalue]



(*  Fixed point iteration given by (b)  *)

p=N[1,3000];  g[x_] := x-(x^5-7)/(5*x^4);
For[i=1, iČ14, i++, 
  {p=g[p], y = Abs[p - exactvalue], 
    Print[i], Print[1.0*p], Print[Log[y]/Log[10.]], Print[ ], 
    }
  ]

g'[x]

1.0*g'[exactvalue]



(*  Fixed point iteration given by (c)  *)

p=N[1,3000];  g[x_] := x-(x^5-7)/12;
For[i=1, iČ20, i++, 
  {p=g[p], y = Abs[p - exactvalue], 
    Print[i], Print[1.0*p], Print[Log[y]/Log[10.]], Print[ ], 
    }
  ]

g'[x]

1.0*g'[exactvalue]



(*  Fixed point iteration given by (d)  *)

p=N[1,3000];  g[x_] := x-(x^5-7)/25;
For[i=1, iČ20, i++, 
  {p=g[p], y = Abs[p - exactvalue], 
    Print[i], Print[1.0*p], Print[Log[y]/Log[10.]], Print[ ], 
    }
  ]

g'[x]

1.0*g'[exactvalue]