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A new symbolic algorithm to compute sums of squares multipliers (certificates) to witness the membership of non-negative univariate polynomials in a saturated univariate quadratic module is presented. Certificates are first computed in…
The FLAME methodology makes it possible to derive provably correct algorithms from a formal description of a linear algebra problem. So far, the methodology has been successfully used to automate the derivation of direct algorithms such as…
Symbolic Mathematical tasks such as integration often require multiple well-defined steps and understanding of sub-tasks to reach a solution. To understand Transformers' abilities in such tasks in a fine-grained manner, we deviate from…
Vectorization via Single Instruction, Multiple Data (SIMD) architectures is a cornerstone of high-performance computing. To fully exploit hardware potential, developers often resort to explicit vectorization using intrinsics, as…
These course notes are about computing modular forms and some of their arithmetic properties. Their aim is to explain and prove the modular symbols algorithm in as elementary and as explicit terms as possible, and to enable the devoted…
A novel algorithm for computing the action of a matrix exponential over a vector is proposed. The algorithm is based on a multilevel Monte Carlo method, and the vector solution is computed probabilistically generating suitable random paths…
Two essential problems in Computer Algebra, namely polynomial factorization and polynomial greatest common divisor computation, can be efficiently solved thanks to multiple polynomial evaluations in two variables using modular arithmetic.…
Complex vector light modes with a spatial variant polarization distribution have become topical of late, enabling the development of novel applications in numerous research fields. Key to this is the remarkable similarities they hold with…
Machine-part cell formation is used in cellular manufacturing in order to process a large variety, quality, lower work in process levels, reducing manufacturing lead-time and customer response time while retaining flexibility for new…
With the rapid development of 3D printing, the demand for personalized and customized production on the manufacturing line is steadily increasing. Efficient merging of printing workpieces can significantly enhance the processing efficiency…
Some explicit algorithms for higher order symplectic integration of a large class of Hamilton's equations have recently been discussed by Mushtaq \emph{et. al}. Here we present a Python program for automatic numerical implementation of…
In this paper a vectorized algorithm for simultaneously computing up to eight singular value decompositions (SVDs, each of the form $A=U\Sigma V^{\ast}$) of real or complex matrices of order two is proposed. The algorithm extends to a batch…
Specialized computational units that perform small matrix multiplications as primitive operations are typically present in modern AI accelerators. However, these Matrix Multiplication Units (MMUs) are often underutilized for many…
In a series of papers it has been shown that for many linear algebra operations it is possible to generate families of algorithms by following a systematic procedure. Although powerful, such a methodology involves complex algebraic…
Symmetry plays a central role in accelerating symbolic computation involving polynomials. This chapter surveys recent developments and foundational methods that leverage the inherent symmetries of polynomial systems to reduce complexity,…
When writing high-performance code for numerical computation in a scripting language like MATLAB, it is crucial to have the operations in a large for-loop vectorized. If not, the code becomes too slow to use, even for a moderately large…
Vectorized glyphs are widely used in poster design, network animation, art display, and various other fields due to their scalability and flexibility. In typography, they are often seen as special sequences composed of ordered strokes. This…
Efficient Matlab codes in 2D and 3D have been proposed recently to assemble finite element matrices. In this paper we present simple, compact and efficient vectorized algorithms, which are variants of these codes, in arbitrary dimension,…
We propose to store several integers modulo a small prime into a single machine word. Modular addition is performed by addition and possibly subtraction of a word containing several times the modulo. Modular Multiplication is not directly…
This work presents the Polynomiogram framework, an integrated computational platform for exploring, visualizing, and generating art from polynomial root systems. The main innovation is a flexible sampling scheme in which two independent…