Analyzing Approximate Roots of Monotone Functions

Concetti Chiave

The author explores the computation of approximate roots for monotone functions, highlighting conditions and methods for efficient root-finding.

Sintesi

The content delves into the computation of approximate roots for monotone functions. It discusses conditions under which polynomiality results extend to multi-dimensional functions and presents algorithms for efficient root-finding. The analysis covers various scenarios, including diagonal and ex-diagonal monotonicity conditions, providing insights into the complexity of finding roots in higher dimensions.

Statistiche

Given a function f : [a, b] → R, if f(a) < 0 and f(b) > 0 and f is continuous, the Intermediate Value Theorem implies that f has a root in [a, b].
The number of required evaluations using the bisection method is polynomial in the number of accuracy digits.
For a two-dimensional function with a single monotonicity condition, the number of required evaluations is polynomial in accuracy.
For d-dimensional functions satisfying all d2 − d "ex-diagonal" monotonicity conditions, the number of evaluations is polynomial in accuracy.

Citazioni

"In general, finding an approximate root might require an exponential number of evaluations even for a two-dimensional function."
"If f satisfies only d2 − d − 2 ex-diagonal conditions, then the number of required evaluations may be exponential in accuracy."

Approfondimenti chiave tratti da

Computing approximate roots of monotone functions

by Alexandros H... alle arxiv.org 03-01-2024

https://arxiv.org/pdf/2310.07333.pdf

Domande più approfondite

Is there an algorithm that requires a number of evaluations that is polynomial in the number of digits

In the context provided, the algorithm presented in Theorem 1.3 demonstrates that for a two-dimensional positive-switching function with certain monotonicity conditions, an approximate root can be found using O(log2(1/ε)) evaluations. This result shows that there is indeed an algorithm that requires a number of evaluations polynomial in the number of digits when specific conditions are met.

Does every Lipschitz-continuous positive-switching function need Ω((L/ε)d−1) evaluations

Not every Lipschitz-continuous positive-switching function needs Ω((L/ε)d−1) evaluations. The lower bound mentioned in Proposition 1.5 applies to functions satisfying d2 − 2 monotonicity conditions, including all diagonal and ex-diagonal conditions except for two ex-diagonal ones. For such functions, the number of required evaluations may be exponential in (L/ε).

How does Tarski's fixed-point theorem relate to finding approximate roots

Tarski's fixed-point theorem is closely related to finding approximate roots through its connection with Brouwer's fixed-point theorem and Miranda's root theorem. Tarski's theorem states that every order-preserving function from a complete lattice to itself has a fixed point. By discretizing the value space into a lattice and applying Tarski's theorem conceptually, Fearnley et al., Chen and Li have shown algorithms for finding fixed points efficiently based on this principle.