Multipoint Iterative Methods for Finding All the Simple Zeros in an Interval

Algorithm 3

digits = 2000;

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

  {k = 0; X = SetAccuracy[setInitial[[i]], digits];

  While[Abs[f[X]] > 10^∧−1000 && k <= 20,

   {k = k + 1; fX = SetAccuracy[f[X], digits];

    f1X = SetAccuracy[f′[X], digits];

    Y = SetAccuracy[X  −  fX/f1X, digits];

    fY = SetAccuracy[f[Y], digits];

    Z = SetAccuracy[Y  −  (fX/(fX  −  2 fY)) (fY/f1X), digits];

    fZ = SetAccuracy[f[Z], digits]; t = fY/fX; s = fZ/fX; u = fZ/fY;

    Weight = SetAccuracy[1 + 2 t + 4 s + 5 t^∧2 + 12 t^∧3 + u/(1 + u^∧2),

     digits];

    X = SetAccuracy[Z  −  Weight  (fZ/f1X), digits];};];

  Print[Column[{

    “The number of full iterations is:”  k,

    “The zero is:”  N[X, 64],

    “The residual norm of the approximate zero is:”  N[Abs[f[X]],

     5]}, Background -> {{LightRed, LightBrown, LightBlue}},

   Frame -> All]];}]; // AbsoluteTiming