Clustering Complex Zeros of Triangular Systems of Polynomials

This paper gives the first algorithm for finding a set of natural $\epsilon$-clusters of complex zeros of a triangular system of polynomials within a given polybox in $\mathbb{C}^n$, for any given $\epsilon>0$. Our algorithm is based on a recent near-optimal algorithm of Becker et al (2016) for clustering the complex roots of a univariate polynomial where the coefficients are represented by number oracles. Our algorithm is numeric, certified and based on subdivision. We implemented it and compared it with two well-known homotopy solvers on various triangular systems. Our solver always gives correct answers, is often faster than the homotopy solver that often gives correct answers, and sometimes faster than the one that gives sometimes correct results.Comment: Research report V6: description of the main algorithm update

Modal Decomposition of Feedback Delay Networks

Feedback delay networks (FDNs) belong to a general class of recursive filters which are widely used in sound synthesis and physical modeling applications. We present a numerical technique to compute the modal decomposition of the FDN transfer function. The proposed pole finding algorithm is based on the Ehrlich-Aberth iteration for matrix polynomials and has improved computational performance of up to three orders of magnitude compared to a scalar polynomial root finder. We demonstrate how explicit knowledge of the FDN's modal behavior facilitates analysis and improvements for artificial reverberation. The statistical distribution of mode frequency and residue magnitudes demonstrate that relatively few modes contribute a large portion of impulse response energy

Inflation and inflation uncertainty in the United Kingdom: Evidence from GARCH modelling

This paper examines the relationship between inflation-uncertainty and the impact of inflation targeting using British data over the period 1972-2002. Uncertainty is proxied using the estimated conditional volatility from symmetric, asymmetric, and component GARCH-M models of inflation. The results indicate a positive relationship between past inflation and current uncertainty. We control for the indirect effect of lower average inflation throughout the last decade of inflation targeting and find that the adoption of an explicit target eliminates inflation persistence and reduces long-run uncertainty. Monetary authorities of implicit targeting countries should consider the extra benefits associated with formal targets

Numerical algebraic geometry approach to polynomial optimization, The

2017 Summer.Includes bibliographical references.Numerical algebraic geometry (NAG) consists of a collection of numerical algorithms, based on homotopy continuation, to approximate the solution sets of systems of polynomial equations arising from applications in science and engineering. This research focused on finding global solutions to constrained polynomial optimization problems of moderate size using NAG methods. The benefit of employing a NAG approach to nonlinear optimization problems is that every critical point of the objective function is obtained with probability-one. The NAG approach to global optimization aims to reduce computational complexity during path tracking by exploiting structure that arises from the corresponding polynomial systems. This thesis will consider applications to systems biology and life sciences where polynomials solve problems in model compatibility, model selection, and parameter estimation. Furthermore, these techniques produce mathematical models of large data sets on non-euclidean manifolds such as a disjoint union of Grassmannians. These methods will also play a role in analyzing the performance of existing local methods for solving polynomial optimization problems

Parallel schemes for global interative zero-finding.

by Luk Wai Shing.Thesis (M.Phil.)--Chinese University of Hong Kong, 1993.Includes bibliographical references (leaves 44-45).ABSTRACT --- p.iACKNOWLEDGMENTS --- p.iiChapter CHAPTER 1. --- INTRODUCTION --- p.1Chapter CHAPTER 2. --- DRAWBACKS OF CLASSICAL THEORY --- p.4Chapter 2.1 --- Review of Sequential Iterative Methods --- p.4Chapter 2.2 --- Visualization Techniques --- p.8Chapter 2.3 --- Review of Deflation --- p.10Chapter CHAPTER 3. --- THE IMPROVEMENT OF THE ABERTH METHOD --- p.11Chapter 3.1 --- The Durand-Kerner method and the Aberth method --- p.11Chapter 3.2 --- The generalized Aberth method --- p.13Chapter 3.3 --- The modified Aberth Method for multiple-zero --- p.13Chapter 3.4 --- Choosing the initial approximations --- p.15Chapter 3.5 --- Multiplicity estimation --- p.16Chapter CHAPTER 4. --- THE HIGHER-ORDER ITERATIVE METHODS --- p.18Chapter 4.1 --- Introduction --- p.18Chapter 4.2 --- Convergence analysis --- p.20Chapter 4.3 --- Numerical Results --- p.28Chapter CHAPTER 5. --- PARALLEL DEFLATION --- p.32Chapter 5.1 --- The Algorithm --- p.32Chapter 5.2 --- The Problem of Zero Component --- p.34Chapter 5.3 --- The Problem of Round-off Error --- p.35Chapter CHAPTER 6. --- HOMOTOPY ALGORITHM --- p.36Chapter 6.1 --- Introduction --- p.36Chapter 6.2 --- Choosing Q(z) --- p.37Chapter 6.3 --- The arclength continuation method --- p.38Chapter 6.4 --- The bifurcation problem --- p.40Chapter 6.5 --- The suggested improvement --- p.41Chapter CHAPTER 7. --- CONCLUSION --- p.42REFERENCES --- p.44APPENDIX A. PROGRAM LISTING --- p.A-lAPPENDIX B. COLOR PLATES --- p.B-