Construction of Optimal Derivative-Free Techniques without Memory

Algorithm 4

nMax = 2; beta = SetAccuracy[1, 2000];

gamma = SetAccuracy[0, 2000];

For[j = 1, j <= NumberOfGuesses, j++,

 {x = SetAccuracy[initialValues[[j]], 2000],

   Do[

    fx = SetAccuracy[f[x], 2000];

    k = SetAccuracy[x + beta fx, 2000];

    fk = SetAccuracy[f[k], 2000];

    FD = SetAccuracy[(fk - fx)/(beta fx), 2000];

    y = SetAccuracy[x - fx/FD, 2000];

    fy = SetAccuracy[f[y], 2000];

    z = SetAccuracy[y - (fy/FD)*(1 + ((2 + beta FD)/(1

        + beta FD)) (fy/fx)), 2000];

    fz = SetAccuracy[f[z], 2000];

    FDxz = SetAccuracy[(fz - fx)/(z - x), 2000];

    FDxy = SetAccuracy[(fx - fy)/(x - y), 2000];

    FDxk = SetAccuracy[(fk - fx)/(k - x), 2000];

    x = SetAccuracy[z - fz/(FDxz + ((FDxk - FDxy)/(k - y)

       - (FDxk - FDxz)/(k - z)

       - (FDxy - FDxz)/(y - z))*(x - z)

   + gamma (z - x)*(z - k)*(z - y)), 2000];

    , {n, nMax}];

 Print[Column[{

   N[x, 128],

   N[f[x]]}

   , Background -> {{LightGreen, LightGray}},

   Frame -> True]];

 }

  ];