GENERATION OF ITERATION FORMULAS

If an equation has k occurrences of x, then by algebraic isolation we can usually compute k iterators of the form x = F(x).   E.g., from $x^{1.001}-x \ln x =0$ we obtain 3 iterators:

x^{1.001} - x \ln x = 0 & \Longrightarrow & x = \left ( x \ln x... ... x^{1.001} } {\ln x} \\ & \Longrightarrow & x = e^{x^{1.001}/x}

In general, it is possible to obtain infinitely many iterators from one equation.

The algorithm for zero-finding is as follows: 

for x[0] in set_of_initial_values do
    for m = 1 to k do
        for i = 1 to maximum_iterations do
            x[i] := F[m](x[i-1]);
            if computation_failed or outside_domain(x[i])
                 then next m
            else if convergence_achieved
                 then return x[i] 
            else if i>3 and diverging
                 then break
            else if acceleration_possible
                 then x[i] := acceleration(x[i],x[i-1],...)
            end if
        end for i
    end for m
end for set_of_initial_values;
previous next