ROOTFINDING

 Secant  Method

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ALGORITHM CODE:

 SecantMethod[x0_,x1_,max_]:=Module[{},k=1;p0=N[x0];p1=N[x1];

  Print["p0=",PaddedForm[p0,{16,16}]", f[p0]=",NumberForm[f[p0],16]];

  Print["p1=", PaddedForm[p1,{16,16}],", f[p1]=", NumberForm[f[p1],16]];

  p2=p1;

  p1=p0;

  While[k<max, p0=p1;p1=p2; p2=p1-(f[p1](p1-p0))/(f[p1]-f[p0]);

   k=k+1;

   Print["p "k, "=", PaddedForm[p2,{16,16}],", f[","p"k,"]=",NumberForm[f[p2],16]];];

  Print["p =",NumberForm[p2,16]];

  Print[" Δp=±"Abs[p2-p1]];

  Print["f[p]=",NumberForm[f[p2],16]];]

 EXAMPLE:

 f[x_]=x^3+x-1

 -1+x+x³

 SecantMethod[0,1,9]

 p0= , f[p0]=  0.0000000000000000 -1.

 p1=  1.0000000000000000 , f[p1]= 1.

 2  p =  0.5000000000000000 , f[ 2 p ]= -0.375

 3  p =  0.6363636363636363 , f[ 3 p ]= -0.1059353869271225

 4  p =  0.6900523560209423 , f[ 4 p ]= 0.01863614179998446

 5  p =  0.6820204196481856 , f[ 5 p ]= -0.000736518493373528

 6  p =  0.6823257814098928 , f[ 6 p ]= -4.847148849740357*10^-6

 7  p =  0.6823278043590257 , f[ 7 p ]= 1.272670357987948*10^-9

 8  p =  0.6823278038280184 , f[ 8 p ]= -2.275957200481571*10^-15

 9  p =  0.6823278038280193 , f[ 9 p ]= 1.665334536937735*10^-16

  p= 0.6823278038280193

 9.99201*10^-16  Δp=±

 f[p]= 1.665334536937735*10^-16

 Plot[-1+x+x³,{x,0,1}]