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A parallel algorithm for solving a series of matrix equations with a constant tridiagonal matrix and different right-hand sides is proposed and studied. The process of solving the problem is represented in two steps. The first preliminary…
Symbolic integration is an important module of a typical Computer Algebra System. As for now, Mathematica, Matlab, Maple and Sage are all mainstream CAS. They share the same framework for symbolic integration at some points. In this book…
This document describes an algorithm to scale a complex vector by the reciprocal of a complex value. The algorithm computes the reciprocal of the complex value and then scales the vector by the reciprocal. Some scaling may be necessary due…
We present an iterative method to diagonalise large matrices. The basic idea is the same as the conjugated gradient (CG) method, i.e, minimizing the Rayleigh quotient via its gradient and avoiding reintroduce errors to the directions of…
In this paper, we propose an orbital iteration based parallel approach for electronic structure calculations. This approach is based on our understanding of the single-particle equations of independent particles that move in an effective…
A new robust algorithm for the numerical computation of biarcs, i.e. $G^1$ curves composed of two arcs of circle, is presented. Many algorithms exist but are based on geometric constructions, which must consider many geometrical…
Metasurfaces are an emerging technology that may supplant many of the conventional optics found in imaging devices, displays, and precision scientific instruments. Here, we develop a method for designing optical systems composed of multiple…
The vision of super computer at every desk can be realized by powerful and highly parallel CPUs or GPUs or APUs. Graphics processors once specialized for the graphics applications only, are now used for the highly computational intensive…
We present novel, deterministic, efficient algorithms to compute the symmetries of a planar algebraic curve, implicitly defined, and to check whether or not two given implicit planar algebraic curves are similar, i.e. equal up to a…
Quantum algorithms have been developed for efficiently solving linear algebra tasks. However, they generally require deep circuits and hence universal fault-tolerant quantum computers. In this work, we propose variational algorithms for…
The geometric mean of two matrices is considered and analyzed from a computational viewpoint. Some useful theoretical properties are derived and an analysis of the conditioning is performed. Several numerical algorithms based on different…
We propose an efficient algorithm for computing a common eigenvector of a finite set of square matrices. As an immediate consequence we obtain an algorithm for determining whether the matrices admit a simultaneous triangulation, and, if so,…
A matrix-compression algorithm is derived from a novel isogenic block decomposition for square matrices. The resulting compression and inflation operations possess strong functorial and spectral-permanence properties. The basic observation…
This article presents a new approach to the real-time solution of inverse problems on embedded systems. The class of problems addressed corresponds to ordinary differential equations (ODEs) with generalized linear constraints, whereby the…
The study of some parametric integrals is presented with a combined approach of analytical development, the usage of a Computed Algebra System (CAS) and of the Online Encyclopedia of Integer Sequences. The methodology for the solution…
We propose a parallel version of the cross interpolation algorithm and apply it to calculate high-dimensional integrals motivated by Ising model in quantum physics. In contrast to mainstream approaches, such as Monte Carlo and quasi Monte…
We propose a simple technique that, if combined with algorithms for computing functions of triangular matrices, can make them more efficient. Basically, such a technique consists in a specific scaling similarity transformation that reduces…
Bayesian probabilistic numerical methods are a set of tools providing posterior distributions on the output of numerical methods. The use of these methods is usually motivated by the fact that they can represent our uncertainty due to…
Some iterative techniques are defined to solve reversible inverse problems and a common formulation is explained. Numerical improvements are suggested and tests validate the methods.
We propose a message passing neural network architecture designed to be equivariant to column and row permutations of a matrix. We illustrate its advantages over traditional architectures like multi-layer perceptrons (MLPs), convolutional…