information:classification, information:filtering, structural:scholar_edition, stylistic:tone
ITP method

Given κ 1 ∈ (0, ∞), κ 2 ∈ [ 1, 1 + ϕ) {\displaystyle \kappa _{1}\in (0,\infty),\kappa _{2}\in \left[1,1+\phi \right)}, n 1 / 2 ≡ ⌈ (b 0 − a 0) / 2 ϵ ⌉ {\displaystyle n_{1/2}\equiv \lceil (b_{0}-a_{0})/2\epsilon \rceil } and n 0 ∈ [ 0, ∞) {\displaystyle n_{0}\in [0,\infty)} where ϕ {\displaystyle \phi } is the golden ration 1 2 (1 + 5) {\displaystyle {\tfrac {1}{2}}(1+{\sqrt {5}})}, in each iteration j = 0, 1, 2... {\displaystyle j=0,1,2...} the ITP method calculates the point x ITP {\displaystyle x_{\text{ITP}}} following three steps:

The value of the function f (x ITP) {\displaystyle f(x_{\text{ITP}})} on this point is queried, and the interval is then reduced to bracket the root by keeping the sub-interval with function values of opposite sign on each end. This three step procedure guarantees that the minmax properties of the bisection method are enjoyed by the estimate as well as the superlinear convergence of the secant method. And, is observed to outperform both bisection and interpolation based methods under smooth and non-smooth functions.