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- research-articleApril 2020
Differential Equation Invariance Axiomatization
Journal of the ACM (JACM), Volume 67, Issue 1Article No.: 6, Pages 1–66https://doi.org/10.1145/3380825This article proves the completeness of an axiomatization for differential equation invariants described by Noetherian functions. First, the differential equation axioms of differential dynamic logic are shown to be complete for reasoning about analytic ...
- research-articleJuly 2019
Index Reduction for Differential-algebraic Equations with Mixed Matrices
Journal of the ACM (JACM), Volume 66, Issue 5Article No.: 35, Pages 1–34https://doi.org/10.1145/3341499Differential-algebraic equations (DAEs) are widely used for the modeling of dynamical systems. The difficulty in numerically solving a DAE is measured by its differentiation index. For highly accurate simulation of dynamical systems, it is important to ...
- research-articleOctober 2017
Polynomial Time Corresponds to Solutions of Polynomial Ordinary Differential Equations of Polynomial Length
Journal of the ACM (JACM), Volume 64, Issue 6Article No.: 38, Pages 1–76https://doi.org/10.1145/3127496The outcomes of this article are twofold.
Implicit complexity. We provide an implicit characterization of polynomial time computation in terms of ordinary differential equations: we characterize the class P of languages computable in polynomial time in ...
- research-articleMay 2016
Almost Optimal Local Graph Clustering Using Evolving Sets
Journal of the ACM (JACM), Volume 63, Issue 2Article No.: 15, Pages 1–31https://doi.org/10.1145/2856030Spectral partitioning is a simple, nearly linear time algorithm to find sparse cuts, and the Cheeger inequalities provide a worst-case guarantee for the quality of the approximation found by the algorithm. A local graph partitioning algorithm finds a ...
- research-articleDecember 2014
Multiway Spectral Partitioning and Higher-Order Cheeger Inequalities
Journal of the ACM (JACM), Volume 61, Issue 6Article No.: 37, Pages 1–30https://doi.org/10.1145/2665063A basic fact in spectral graph theory is that the number of connected components in an undirected graph is equal to the multiplicity of the eigenvalue zero in the Laplacian matrix of the graph. In particular, the graph is disconnected if and only if ...
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- articleApril 1981
- articleJuly 1976
Semi-Implicit Runge-Kutta Procedures with Error Estimates for the Numerical Integration of Stiff Systems of Ordinary Differential Equations
A -stable, semi-implicit Runge-Kutta procedures requiring at most one Jacobian evaluation per time step are developed for the approximate numerical integration of stiff systems of ordinary differential equations. A simple procedure for estimating the ...
- articleApril 1976
On the Linear Convergence of a Covariance Factorization Algorithm
An algorithm for factoring a covariance function into its Hurwitz factors, which is based on the Cholesky factors of a certain matrix, was proposed by F.L. Bauer and others. This algorithm bears a close connection to the theory of orthogonal polynomials,...
- articleApril 1976
Convergence Estimates for the Distribution of Trailing Digits
This paper analyzes the distribution of trailing digits (tail end digits) of positive real floating-point numbers represented in arbitrary base β and randomly chosen from a logarithmic distribution. The analysis shows that the nth digit for n ≥ 2 is ...
- articleJuly 1973
Toward Abstract Numerical Analysis
This paper deals with a technique for proving that certain problems of numerical analysis are numerically unsolvable. So that only methods which are natural for dealing with analytic problems may be presented, notions from recursive function theory have ...
- articleJanuary 1973
A-Stable Composite Multistep Methods
Consider the set of multistep formulas ∑l-1jmn-k αijxmn+j - h ∑l-1jmn-kβijxmn+j = 0, i = 1, ···, l, where xmn+j = ymn+j for j= -k, ···, -1 and xn = ƒn = ƒ(xn , tn). These formulas are solved simultaneously for the xmn+j with j = 0, ···, l - 1 in terms ...