Take Two

muller[f_,x1_,x2_,x3_]:=
	While[f1*f2*f3≠0,
f1=f[x1];
f2=f[x2];
f3=f[x3];
Solve[{a (x1^2) +b x1 +c ==f1, a (x2^2) +b x2+c ==f2, a (x3^2) +b x3+c ==f3},{a,b,c}];
If[b≥0, {r1=((2 c)/(-b-(Sqrt[(b^2)-4 a c]))), r2=((-b-(Sqrt[(b^2)-4 a c]))/(2 a))},{r1=((-b+(Sqrt[(b^2)-4 a c]))/(2 a)), r2=((2 c)/(-b-(Sqrt[(b^2)-4 a c])))}];
If[Abs[x3-r1]≺ Abs[x3-r2], {x1=x2,x2=x3,x3=r1},{x1=x2,x2=x3,x3=r2}]];
{{x1,f1},{x2,f2},{x3,f3}}

f[x_]:=(E^x)-(1/(x^2))

Plot[{(E^x)-(1/(x^2)),-41.301(x^2)+70.091x-27.071}, {x,0,1}]

muller[f, 0.3,0.5,1]

Clear[muller,f,x1,x2,x3,f1,f2,f3,matrix,a,b,c,r1,r2,list]

muller[f_,x1_,x2_,x3_]:=
	While[f1*f2*f3≠0,
f1=f[x1];
f2=f[x2];
f3=f[x3];
matrix=RowReduce[{{(x1^2),x1,1,f1},{(x2^2),x2,1,f2},{(x3^2),x3,1,f3}}];
a=matrix[[1,4]];
b=matrix[[2,4]];
c=matrix[[3,4]];
list=List[x1,x2,x3]
If[b≥0, {r1=((2 c)/(-b-(Sqrt[(b^2)-4 a c]))), r2=((-b-(Sqrt[(b^2)-4 a c]))/(2 a))},{r1=((-b+(Sqrt[(b^2)-4 a c]))/(2 a)), r2=((2 c)/(-b-(Sqrt[(b^2)-4 a c])))}];
If[Abs[x3-r1]≺ Abs[x3-r2], {x3/.r1,x2=x3,x3=r1},{x1=x2,x2=x3,x3=r2}]];
{{x1,f1},{x2,f2},{x3,f3}}

Clear[muller,x1,x2,x3,f1,f2,f3,list,matrix,r1,r2,f]

muller[f_,list_]:=x1=N[list[[1]]];
x2=N[list[[2]]];
x3=N[list[[3]]];
f1=f[x1];
f2=f[x2];
f3=f[x3];
While[f1≠f2≠f3,
matrix=RowReduce[{{(x1^2),x1,1,f1},{(x2^2),x2,1,f2},{(x3^2),x3,1,f3}}];
a=matrix[[1,4]];
b=matrix[[2,4]];
c=matrix[[3,4]];
If[b≥0, {r1=((2 c)/(-b-(Sqrt[(b^2)-4 a c]))), r2=((-b-(Sqrt[(b^2)-4 a c]))/(2 a))},{r1=((-b+(Sqrt[(b^2)-4 a c]))/(2 a)), r2=((2 c)/(-b-(Sqrt[(b^2)-4 a c])))}];
list=Delete[list,1];
If[Abs[x3-r1]<Abs[x3-r2], AppendTo[list,r1],AppendTo[list,r2]];
x1=list[[1]];
x2=list[[2]];
x3=list[[3]];
f1=f[x1];
f2=f[x2];
f3=f[x3]];
{{x1,f1},{x2,f2},{x3,f3}}

f[x_]:=(E^x)-(1/(x^2))

list=List[0.3,0.5,1]