(*  Computing the "exact value"  *)

p=N[3,3000];  f[x_] := Cos[x]-x;
For[i=1, i<=25, i++, 
  { p=p-f[p]/f'[p], }
  ]
exactvalue=p;


(*  Newton's method  *)

p=N[3,3000];  f[x_] := Cos[x]-x;
For[i=1, i<=13, i++, 
  { p=p-f[p]/f'[p], 
    y = Abs[p - exactvalue], 
    Print[i], Print[Log[y]/Log[10.]]
    }
  ]


(*  Secant method  *)

p0=N[0,3000];  p1 = N[2,3000];  f[x_] := Cos[x]-x;
For[i=1, i<=16, i++, 
  {p=p1-f[p1]*(p1-p0)/(f[p1]-f[p0]), 
    y = Abs[p - exactvalue], 
    Print[i], Print[Log[y]/Log[10.]], 
    p0 = p1, p1 = p
    }
  ]


(*  Fixed point iteration  *)

p=N[1,3000];  g[x_] := Cos[x];
For[i=1, i<=100, i++, 
  {p=g[p], y = Abs[p - exactvalue], 
    If[Mod[i,1]\[Equal]0, 
      {Print[i], Print[Log[y]/Log[10.]]}
      ],
    }
  ]