Related papers: Modular SIMD arithmetic in Mathemagix
In this work, we extend modular techniques for computing Gr\"obner bases involving rational coefficients to (two-sided) ideals in free algebras. We show that the infinite nature of Gr\"obner bases in this setting renders the classical…
Matrix functions are a central topic of linear algebra, and problems requiring their numerical approximation appear increasingly often in scientific computing. We review various limited-memory methods for the approximation of the action of…
Matrix operations such as matrix inversion, eigenvalue decomposition, singular value decomposition are ubiquitous in real-world applications. Unfortunately, many of these matrix operations so time and memory expensive that they are…
Matrix computations have become increasingly significant in many data-driven applications. However, Moores law for digital computers has been gradually approaching its limit in recent years. Moreover, digital computers encounter substantial…
Over the past thirty years or so the authors have been teaching various programming for mathematics courses at our respective Universities, as well as incorporating computer algebra and numerical computation into traditional mathematics…
This paper presents a low-latency hardware accelerator for modular polynomial multiplication for lattice-based post-quantum cryptography and homomorphic encryption applications. The proposed novel modular polynomial multiplier exploits the…
The concept of a universal algorithm is discussed. Examples of this kind of algorithms are presented. Software implementations of such algorithms in C++ type languages are discussed together with means that provide for computations with an…
This paper deals with simultaneously fast and in-place algorithms for formulae where the result has to be linearly accumulated: some of the output variables are also input variables, linked by a linear dependency. Fundamental examples…
We propose an algorithm for generating explicit solutions of multiparametric mixed-integer convex programs to within a given suboptimality tolerance. The algorithm is applicable to a very general class of optimization problems, but is most…
Simplification of fractional powers of positive rational numbers and of sums, products and powers of such numbers is taught in beginning algebra. Such numbers can often be expressed in many ways, as this article discusses in some detail.…
Matrix multiplication is the dominant computation during Machine Learning (ML) inference. To efficiently perform such multiplication operations, Compute-in-memory (CiM) paradigms have emerged as a highly energy efficient solution. However,…
Since simulating quantum computers requires exponentially more classical resources, efficient algorithms are extremely helpful. We analyze algorithms that create single qubit and specific controlled qubit matrix representations of gates.…
Matrix multiplication is integral to various scientific and engineering disciplines, including machine learning, image processing, and gaming. With the increasing data volumes in areas like machine learning, the demand for efficient…
In this paper, we introduce distributed matrix multiplication (DMM)-friendly algebraic function fields for polynomial codes and Matdot codes, and present several constructions for such function fields through extensions of the rational…
Today's PCs can directly manipulate numbers not longer than 64 bits because the size of the CPU registers and the data-path are limited. Consequently, arithmetic operations such as addition, can only be performed on numbers of that length.…
The algorithms in the current sequential numerical linear algebra libraries (e.g. LAPACK) do not parallelize well on multicore architectures. A new family of algorithms, the tile algorithms, has recently been introduced. Previous research…
A long-standing open question in Integer Programming is whether integer programs with constraint matrices with bounded subdeterminants are efficiently solvable. An important special case thereof are congruency-constrained integer programs…
Computations over the rational numbers often encounter the problem of intermediate coefficient growth. A solution to this is provided by modular methods, which apply the algorithm under consideration modulo a number of primes and then lift…
We present the library Moore, which implements Interval Arithmetic in modern C++. This library is based on a new feature in the C++ language called concepts, which reduces the problems caused by template meta programming, and leads to a new…
We consider integer programming problems with bounded general-integer variables belonging to the general class of network flow problems. For those, we computationally investigate the effect on mixed-integer linear programming (MIP) solvers…