Zeroes and Extrema of Functions via Random Measures

TL;DR

The paper introduces a differentiation-free framework (PPZ) to locate all zeroes and extrema of real or complex functions within a bounded window by modeling the zero set as a realization of a Poisson-type random measure. It defines a tractable intensity function that concentrates on zeros and presents a thinning-based algorithm to sample the zero set, with an adaptive variant for refining tolerance. The approach yields a global alternative to iterative optimization, demonstrated on simple real functions, complex polynomials, the Riemann zeta function, and high-dimensional multivariate cases. It also discusses extensions to gradient zeros, vector-valued targets, and potential Cox-process generalizations for richer dependence structures, underscoring practical applicability and avenues for Bayesian automation.

Abstract

We present methods that provide all zeroes and extrema of a function that do not require differentiation. Using point process theory, we are able to describe the locations of zeroes or maxima, their number, as well as their distribution over a given window of observation. The algorithms in order to accomplish the theoretical development are also provided, and they are exemplified using many illustrative examples, for real and complex functions.

Figures (6)

• Figure 1: The APPPZ algorithm over $\mathcal{W}=[-15,15]$ for the function $f(x)=cos(x)$ (black color). We run the algorithm for 1, 5, and 10 iterations, with the latter displayed (left plot). The derivative $f^{^{\prime}}(x)=-sin(x)$ is displayed in green and the intensity function is displayed in blue color. The zeros are displayed as red points. Note that the zeroes of $cos(x)$ are the values where the extrema for the function $-sin(x)$ are attained. The zeroes are presented in Table \ref{['CosineTable']}. The right plot presents the results of the PPPZ on the zeroes of the derivative (i.e., the points of extrema of $cos(x)$).
• Figure 2: The APPPZ algorithm over $\mathcal{W}=[-15,15]$ for the function $f(x)=\sin (x/20)+\cos ^{2}(x)$ (left plot in black color, algorithm was run for 10 iterations). The right plot presents the results of the PPPZ on the zeroes of the derivative (i.e., the points of extrema of $\sin (x/20)+\cos ^{2}(x)$). The derivative is displayed in green and the intensity function is displayed in blue color. The zeros are displayed as red points.
• Figure 3: The APPPZ algorithm over $\mathcal{W}=[0,5]$ for the function $f(x)=35(x-3)^{5}(x-2)^{10}$, in black color. We run the algorithm for 10 iterations in two cases with the left plot showing the true zeroes at $x=2,3$, and the right plot illustrating what can happen when the function is very close to zero for a range of its values; for the given tolerance $Tol= 1e-10$, the adaptive algorithm ends up proposing three zeroes. The derivative is displayed in green and the intensity function is displayed in blue color. The zeros are displayed as red points.
• Figure 4: (left) Roots of a complex polynomial $p_{10}(s)=\prod\limits_{k=1}^{10}(s-{ \if@compatibility \mathchar"0118 {} \mathchar"0118 } _{k}),$ with randomly generated real and imaginary parts of the roots ${ \if@compatibility \mathchar"0118 {} \mathchar"0118 } _{k}$, independently, as uniform random variables over $Unif(-1,1)$. The true roots denoted by x's (black) and the recovered roots as o's (red). Note that all roots are recovered well except for one (with real part near $0.9)$. (right) The APPPZ algorithm over $\mathcal{W}=\{{ \if@compatibility \mathchar"011B {} \mathchar"011B } +it:0<{ \if@compatibility \mathchar"011B {} \mathchar"011B } <1.3,13<t<43\}\subset \mathbb{C}$ for the ${ \if@compatibility \mathchar"0111 {} \mathchar"0111 }$ function. The zeros of the ${ \if@compatibility \mathchar"0110 {} \mathchar"0110 }$ function are displayed as red points and the zeroes of the ${ \if@compatibility \mathchar"0111 {} \mathchar"0111 }$ function on the strip ${ \if@compatibility \mathchar"011B {} \mathchar"011B }=1$, as green points. The red horizontal lines correspond to the well known zeroes of the strip ${ \if@compatibility \mathchar"011B {} \mathchar"011B }=0.5$ with imaginary part $t=14.134725$, $21.022040$, $25.010858$, $30.424876$, $32.935062$, $37.586178$, and $40.918719$.
• Figure 5: (left) The function $f(x,y)=(\sin (x/20)+\cos ^{2}(x))(\sin(y/20)+\cos ^{2}(y)),$$(x,y)\in \mathcal{W}=[-10,10]^{2}\subset \Re ^{2}.$ (right) The APPPZ results from 5 iterations ($K=10$, $Q=0.5$, $N=1000,$$31.32483$hrs). All the points of extrema are displayed as red circles.

Theorems & Definitions (1)

• Remark 1: Adaptive PPPZ algorithm