ROOTFINDING

 Fixed Point Method

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ALGORITHM CODE:

 FixedPointIteration[x0_,max_]:=Module[{},p0=N[x0];k=0;

   Print["0  p=",PaddedForm[p0,{15,15}]];

   While[k<max,Module[{},p1=g[p0];k=k+1;

      Print[" p"k, "=", PaddedForm[p1,{15,15}]];

      p0=p1;];];

   p=p0;

   Print[" "];

   Print["The function is g[x]=", g[x]];

   Print["  p=",PaddedForm[p,{15,15}]];

   Print["g[p]=",PaddedForm[g[p],{15,15}]];];

EXAMPLE: x^3+x-1=0

 g[x_]=(1-x)^1/3

 FixedPointIteration[0.5,12]

 0  p=  0.500000000000000

  p =  0.793700525984100

 2  p =  0.590880113275177

 3  p =  0.742363932168006

 4  p =  0.636310203481661

 5  p =  0.713800814144207

 6  p =  0.659006145622400

 7  p =  0.698632605730219

 8  p =  0.670448496228072

 9  p =  0.690729120589141

 10  p =  0.676258924926827

 11  p =  0.686645536864490

 12  p =  0.679222339897004

 13  p =  0.684544005469716

 The function is g[x]= (1-x)^1/3

   p=  0.684544005469716

 g[p]=  0.680737373803562

 Plot[{x^3+x-1,(1-x)^1/3, x},{x,0,1}]