Transformation methods for finding multiple roots of nonlinear equations (Q711289)

The problem of finding a root of multiplicity \((m\geq 1)\) of a nonlinear equation on an interval \((a,b)\) is considered. The function \(f(x)\) is transformed to a hyper tangent function combined with a simple difference formula whose value changes from \((-1)\) to 1 as \((x)\) passes through the root of function. Then the so-called numerical integration method is applied to the transformed equation. A Steffensen-type iterative method which does not require any derivatives of \(f(x)\) nor is quite effected by an initial approximation is proposed. It is shown that the convergence order of the proposed method becomes cubic by simultaneous approximation to the root and its multiplicity.