In numerical analysis, the ITP method (Interpolate Truncate and Project method) is the first root-finding algorithm that achieves the superlinear convergence of the secant method while retaining the optimal worst-case performance of the bisection method. It is also the first method with guaranteed average performance strictly better than the bisection method under any continuous distribution. In practice it performs better than traditional interpolation and hybrid based strategies (Brent's Method, Ridders, Illinois), since it not only converges super-linearly over well behaved functions but also guarantees fast performance under ill-behaved functions where interpolations fail. The ITP method follows the same structure of standard bracketing strategies that keeps track of upper and lower bounds for the location of the root; but it also keeps track of the region where worst-case performance is kept upper-bounded. As a bracketing strategy, in each iteration the ITP queries the value of the function on one point and discards the part of the interval between two points where the function value shares the same sign. The queried point is calculated with three steps: it interpolates finding the regula falsi estimate, then it perturbs/truncates the estimate (similar to Regula falsi § Improvements in regula falsi) and then projects the perturbed estimate onto an interval in the neighbourhood of the bisection midpoint. The neighbourhood around the bisection point is calculated in each iteration in order to guarantee minmax optimality (Theorem 2.1 of ). The method depends on three hyper-parameters κ 1 ∈ ( 0 , ∞ ) , κ 2 ∈ [ 1 , 1 + ϕ ) {\displaystyle \kappa _{1}\in (0,\infty ),\kappa _{2}\in \left[1,1+\phi \right)} and n 0 ∈ [ 0 , ∞ ) {\displaystyle n_{0}\in [0,\infty )} where ϕ {\displaystyle \phi } is the golden ratio 1 2 ( 1 + 5 ) {\displaystyle {\tfrac {1}{2}}(1+{\sqrt {5}})} : the first two control the size of the truncation and the third is a slack variable that controls the size of the interval for the projection step.
Root finding problem Given a continuous function f {\displaystyle f} defined from [ a , b ] {\displaystyle [a,b]} to R {\displaystyle \mathbb {R} } such that f ( a ) f ( b ) ≤ 0 {\displaystyle f(a)f(b)\leq 0} , where at the cost of one query one can access the values of f ( x ) {\displaystyle f(x)} on any given x {\displaystyle x} . And, given a pre-specified target precision ϵ > 0 {\displaystyle \epsilon >0} , a root-finding algorithm is designed to solve the following problem with the least amount of queries as possible: Problem Definition: Find x ^ {\displaystyle {\hat {x}}} such that | x ^ − x ∗ | ≤ ϵ {\displaystyle |{\hat {x}}-x^{*}|\leq \epsilon } , where x ∗ {\displaystyle x^{*}} satisfies f ( x ∗ ) = 0 {\displaystyle f(x^{*})=0} . This problem is very common in numerical analysis, computer science and engineering; and, root-finding algorithms are the standard approach to solve it. Often, the root-finding procedure is called by more complex parent algorithms within a larger context, and, for this reason solving root problems efficiently is of extreme importance since an inefficient approach might come at a high computational cost when the larger context is taken into account. This is what the ITP method attempts to do by simultaneously exploiting interpolation guarantees as well as minmax optimal guarantees of the bisection method that terminates in at most n 1 / 2 ≡ ⌈ log 2 ( ( b 0 − a 0 ) / 2 ϵ ) ⌉ {\displaystyle n_{1/2}\equiv \lceil \log _{2}((b_{0}-a_{0})/2\epsilon )\rceil } iterations when initiated on an interval [ a 0 , b 0 ] {\displaystyle [a_{0},b_{0}]} .
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