Some efficient methods for enclosing simple zeros of nonlinear equations (Q1196877)

Three methods for solving univariate equations are presented. The methods are derivative free and render nested sequences of intervals converging to a zero of the equation. The common feature of the methods are so- called double length secant steps, which aim to improve enclosing intervals that have been obtained by preceding bracketing phases. The interval widths of the first algorithm converge \(Q\)-quadratically to zero. Two function evaluations each iteration are needed finally (that is, from a certain stage of the computation). The second algorithm uses quadratic interpolation polynomials, converges with \(Q\)-order 4, and needs three function evaluations finally each iteration. The third algorithm arises from the second one by replacing a secant step with a simple bisection. In this case, convergence is of \(R\)-order \((3+\sqrt {13})/2\), and three function evaluations each step are needed.