Solution of one variable equations (Q2739960)

This paper deals with the approximate solution of the equation \(f(x)=0, f\in C^{(r)}[a,b]\), \(f(a)f(b)<0\). The authors propose a method of inverse interpolation for the solution of the given equation and compare it with the method of sections and with the Newton method. It is proved that the approximation error in the method of inverse interpolation is less than the approximation error in the Newton and in the sections methods in the case when \(f(x)\in C^2[a,b]\) is a strictly monotone strictly convex function. The case of parametric representation of \(f\) and of complex roots of the equation \(f(z)=0\), \(z\in C\) is studied. The approximation method based on the interpolation formulas with base exponential functions is presented.