Sigmoid-like functions and root finding methods

An efficient method for finding an initial approximation to a real root of nonlinear equation f(x)=0f(x)=0 is proposed. The presented approach is based on numerical integration of the transform functions tanhm(f(x)),arctanm(f(x))(m>0) and sgn(f(x)) of sigmoidal type. Combining numerical integration method and rapidly convergent iterative methods, we construct a hybrid method of great efficiency. The introduced sigmoid-like functions are also convenient for the detection of a cluster of zeros or a multiple zero lying in a given interval. The presented numerical examples demonstrate the feasibility and efficiency of the proposed procedures.