Related papers: Accurate and Efficient Expression Evaluation and L…
A method for evaluating matrix polynomials have recently been developed that require one fewer matrix product ($1M$) than the Paterson--Stockmeyer (PS) method. Since the computational cost for large-scale matrices is asymptotically…
Linear algebraic expressions are the essence of many computationally intensive problems, including scientific simulations and machine learning applications. However, translating high-level formulations of these expressions to efficient…
Word-level verification of arithmetic circuits with large operands typically relies on arbitrary-precision arithmetic, which can lead to significant computational overhead as word sizes grow. In this paper, we present a hybrid algebraic…
We consider the following problem: Given a rational matrix $A \in \setQ^{m \times n}$ and a rational polyhedron $Q \subseteq\setR^{m+p}$, decide if for all vectors $b \in \setR^m$, for which there exists an integral $z \in \setZ^p$ such…
We study output-sensitive algorithms and complexity for multiobjective combinatorial optimization problems. In this computational complexity framework, an algorithm for a general enumeration problem is regarded efficient if it is…
In this paper, we explain a procedure based on a classical result of Sturm that can be used to determine rigorously whether a given trigonometric polynomial is nonnegative in a certain interval or not. Many examples are given. This…
As a partial answer to a question of Rao, a deterministic and customizable efficient algorithm is presented to test whether an arbitrary graphical degree sequence has a bipartite realization. The algorithm can be configured to run in…
In many classification tasks there is a requirement of monotonicity. Concretely, if all else remains constant, increasing (resp. decreasing) the value of one or more features must not decrease (resp. increase) the value of the prediction.…
Dijkstra observed that verifying correctness of a program is difficult and conjectured that derivation of a program hand-in-hand with its proof of correctness was the answer. We illustrate this goal-oriented approach by applying it to the…
It is well known that the behavior of dense linear algebra algorithms is greatly influenced by factors like target architecture, underlying libraries and even problem size; because of this, the accurate prediction of their performance is a…
We are interested in the fast computation of the exact value of integrals of polynomial functions over convex polyhedra. We present speed ups and extensions of the algorithms presented in previous work. We present the new software…
We design and analyze new protocols to verify the correctness of various computations on matrices over the ring F[x] of univariate polynomials over a field F. For the sake of efficiency, and because many of the properties we verify are…
Explicit time advancement for continuous finite elements requires the inversion of a global mass matrix. For spectral element simulations on quadrilaterals and hexahedra, there is an accurate approximate mass matrix which is diagonal,…
Automated mathematical reasoning is a challenging problem that requires an agent to learn algebraic patterns that contain long-range dependencies. Two particular tasks that test this type of reasoning are (1) mathematical equation…
The exact complexity of geometric cuts and bisections is the longstanding open problem including even the dimension one. In this paper, we resolve this problem for dimension one (the real line) by designing an exact polynomial time…
We study the top Lyapunov exponents of random products of positive $2 \times 2$ matrices and obtain an efficient algorithm for its computation. As in the earlier work of Pollicott, the algorithm is based on the Fredholm theory of…
The Automatic Amortized Resource Analysis (AARA) derives program-execution cost bounds using types. To do so, AARA often makes use of cost-free types, which are critical for the composition of types and cost bounds. However, inferring…
Initiated by Mulmuley, Vazirani, and Vazirani (1987), many algebraic algorithms have been developed for matching and related problems. In this paper, we review basic facts and discuss possible improvements with the aid of fast computation…
We give polynomial-time algorithms for the exact computation of lowest-energy (ground) states, worst margin violators, log partition functions, and marginal edge probabilities in certain binary undirected graphical models. Our approach…
An efficient algorithm for computing eigenvectors of a matrix of integers by exact computation is proposed. The components of calculated eigenvectors are expressed as polynomials in the eigenvalue to which the eigenvector is associated, as…