ROOTFINDING

 Bisection Method

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ALGORITHM CODE:

 Bisection[a0_,b0_,m_]:=Module[{},a=N[a0];b=N[b0]; c=(a+b)/2; k=0;

  output={{k,a,c,b,f[c]}};

  While[k<m,

   If[Sign[f[b]]==Sign[f[c]],

    b=c,  a=c;];

   c=(a+b)/2;

   k=k+1;

   output=Append[output,{k,a,c,b,f[c]}];];

  Print[NumberForm[TableForm[output,

     TableHeadings->{None,{"k","a_k","c_k", "b_k","f[c_k]"}}],16]];

  Print["  c=",NumberForm[c,16]];

  Print[" Δc=±",(b-a)/2];

  Print["f[c]=", NumberForm[f[c],16]];]

 EXAMPLE:

 f[x_]=x^3+x-1

 -1+x+x³

 Bisection[0,1,10]

 {

 {k, a_k, c_k, b_k, f[c_k]},

 {0, 0., 0.5, 1., -0.375},

 {1, 0.5, 0.75, 1., 0.171875},

 {2, 0.5, 0.625, 0.75, -0.130859375},

 {3, 0.625, 0.6875, 0.75, 0.012451171875},

 {4, 0.625, 0.65625, 0.6875, -0.061126708984375},

 {5, 0.65625, 0.671875, 0.6875, -0.02482986450195312},

 {6, 0.671875, 0.6796875, 0.6875, -0.006313800811767578},

 {7, 0.6796875, 0.68359375, 0.6875, 0.003037393093109131},

 {8, 0.6796875, 0.681640625, 0.68359375, -0.001646004617214203},

 {9, 0.681640625, 0.6826171875, 0.68359375, 0.0006937412545084953},

 {10, 0.681640625, 0.68212890625, 0.6826171875, -0.0004766195779666305}

}

   c= 0.68212890625

  Δc=± 0.000488281

 f[c]= -0.0004766195779666305

Plot[{x^3 + x - 1, (1 - x)^(1/3), x}, {x, 0, 1}]