Related papers: An Optimizing Symbolic Algebra Approach for Genera…
Traditional optimization methods rely on the use of single-precision floating point arithmetic, which can be costly in terms of memory size and computing power. However, mixed precision optimization techniques leverage the use of both…
Matrix preconditioning is a critical technique to accelerate the solution of linear systems, where performance heavily depends on the selection of preconditioning parameters. Traditional parameter selection approaches often define fixed…
Quantum simulation of the interactions of fermions and bosons -- the fundamental particles of nature -- is essential for modeling complex quantum systems in material science, chemistry and high-energy physics and has been proposed as a…
We formulate the loop-free, binary superoptimization task as a stochastic search problem. The competing constraints of transformation correctness and performance improvement are encoded as terms in a cost function, and a Markov Chain Monte…
For multiparametric mixed-integer convex programming problems such as those encountered in hybrid model predictive control, we propose an algorithm for generating a feasible partition of a subset of the parameter space. The result is a…
Computer algebra systems play an important role in science as they facilitate the development of new theoretical models. The resulting symbolic equations are often implemented in a compiled programming language in order to provide fast and…
The approximate computation of all gravitational forces between $N$ interacting particles via the fast multipole method (FMM) can be made as accurate as direct summation, but requires less than $\mathcal{O}(N)$ operations. FMM groups…
Algebraic multigrid (AMG) methods derive their optimal efficiency from the interplay between a relaxation process and a corresponding coarse grid correction. In many standard formulations, relaxation and coarse-graining are analyzed and…
In this paper, we present a generalized version of the matrix chain algorithm to generate efficient code for linear algebra problems, a task for which human experts often invest days or even weeks of works. The standard matrix chain problem…
As mathematical computing becomes more democratized in high-level languages, high-performance symbolic-numeric systems are necessary for domain scientists and engineers to get the best performance out of their machine without deep knowledge…
This paper presents a generalised symbolic algorithm for solving systems of linear algebraic equations with multi-diagonal coefficient matrices. The algorithm is given in a pseudocode. A theorem which gives the condition for correctness of…
Quantum algorithms for computational linear algebra promise up to exponential speedups for applications such as simulation and regression, making them prime candidates for hardware realization. But these algorithms execute in a model that…
Dynamic programming (DP) is an algorithmic design paradigm for the efficient, exact solution of otherwise intractable, combinatorial problems. However, DP algorithm design is often presented in an ad-hoc manner. It is sometimes difficult to…
Large language models (LLMs) have demonstrated remarkable capabilities in code generation and structured reasoning; however, their performance often degrades on complex tasks that require consistent multi-step planning. Recent work has…
We present a novel certified and complete algorithm to compute arrangements of real planar algebraic curves. It provides a geometric-topological analysis of the decomposition of the plane induced by a finite number of algebraic curves in…
We present a fast, adaptive multiresolution algorithm for applying integral operators with a wide class of radially symmetric kernels in dimensions one, two and three. This algorithm is made efficient by the use of separated representations…
We present an algorithm to parallelize the inverse fast multipole method (IFMM), which is an approximate direct solver for dense linear systems. The parallel scheme is based on a greedy coloring algorithm, where two nodes in the hierarchy…
Code-generating Large Language Models (LLMs) have become essential tools in modern software development, enhancing productivity and accelerating development. This paper aims to investigate the fine-tuning of code-generating LLMs using…
Sympiler is a domain-specific code generator that optimizes sparse matrix computations by decoupling the symbolic analysis phase from the numerical manipulation stage in sparse codes. The computation patterns in sparse numerical methods are…
In this work we present a variant of the fast multipole method (FMM) for efficiently evaluating standard layer potentials on geometries with complex coordinates in two and three dimensions. The complex scaled boundary integral method for…