Modified Newton Formula of Tangent Parabolas on the Number Axis

Abstract

The paper proposes a modified Newton's formula - tangent parabolas on the real axis. The analytical formula contains a square root (radical) and is applicable for the multiplicity of the root no more than two. It is shown that for a single root the formula with a radical has the third order of the rate of convergence of the residual to zero, while Newton's formula converges with the second order of the rate. For a root multiplicity of two, the order of speed for a formula with a radical is two. The formula with the radical is replaced by a series of eleven terms, that is, it is extended to the numerical axis for any multiplicity of the root. For a root of multiplicity one, an iterative formula of eleven terms is proposed. For the multiplicity of the root two or more, an iterative algorithm with the parameter 0<q<1 is proposed. At the first stage, before the start of the iterative cycle, the parameter q is calculated only once, which depends only on the multiplicity of the root. At the second stage, the iterative formula with the parameter q with the second order of the discrepancy rate works in the cycle. The examples show that the new formula has a smaller number of iterations than in the Newton formula and in the modified Newton formula for the multiplicity of the root one. The accelerated algorithm with the second order of the residual rate is efficient over a wider area than the modified Newton formula. The iterative algorithm is applicable to a function that is twice continuously differentiable on the interval [a, b] and takes opposite signs at its ends.