arXiv subjects

David K. Zhang

Publications and source records attributed to David K. Zhang.

3 recordsLinked to original sources

arXiv2025

Automatic Verification of Floating-Point Accumulation Networks

Floating-point accumulation networks (FPANs) are key building blocks used in many floating-point algorithms, including compensated summation and double-double arithmetic. FPANs are notoriously difficult to analyze, and algorithms using FPANs are often published without rigorous correctness proofs. In fact, on at least one occasion, a published error bound for a widely used FPAN was later found to be incorrect. In this paper, we present an automatic procedure that produces computer-verified proofs of several FPAN correctness properties, including error bounds that are tight to the nearest bit. Our approach is underpinned by a novel floating-point abstraction that models the sign, exponent, and number of leading and trailing zeros and ones in the mantissa of each number flowing through an FPAN. We also present a new FPAN for double-double addition that is faster and more accurate than the previous best known algorithm.

math.NA↗

arXiv2019

Discovering New Runge-Kutta Methods Using Unstructured Numerical Search

Runge-Kutta methods are a popular class of numerical methods for solving ordinary differential equations. Every Runge-Kutta method is characterized by two basic parameters: its order, which measures the accuracy of the solution it produces, and its number of stages, which measures the amount of computational work it requires. The primary goal in constructing Runge-Kutta methods is to maximize order using a minimum number of stages. However, high-order Runge-Kutta methods are difficult to construct because their parameters must satisfy an exponentially large system of polynomial equations. This paper presents the first known 10th-order Runge-Kutta method with only 16 stages, breaking a 40-year standing record for the number of stages required to achieve 10th-order accuracy. It also discusses the tools and techniques that enabled the discovery of this method using a straightforward numerical search.

math.NA↗

arXiv2015

The biexciton puzzle

The Stochastic Variational Method (SVM) is used to show that the effective mass model correctly estimates the binding energies of excitons and trions, but fails to predict the experimental binding energy of the biexciton. Using high-accuracy variational calculations, it is demonstrated that the biexciton binding energy in transition metal dichalcogenides is smaller than the trion binding energy, contradicting experimental findings. It is also shown that an excited state of the biexciton is in very good agreement with experimental data. This excited state corresponds to an hole attached to a negative trion and may be a possible resolution of the discrepancy between theory and experiment.

cond-mat.mes-hall↗