Related papers: Exponentiation Using Laplace Expansion
We discuss the parallelization of algorithms for solving polynomial systems symbolically by way of triangular decomposition. Algorithms for solving polynomial systems combine low-level routines for performing arithmetic operations on…
In the present article, we review a continual effort on generalization of the Trotter formula to higher-order exponential product formulas. The exponential product formula is a good and useful approximant, particularly because it conserves…
This note is about using computational effects for scalability. With this method, the specification gets more and more complex while its semantics gets more and more correct. We show, from two fundamental examples, that it is possible to…
In this note explicit algorithms for calculating the exponentials of important structured 4 x 4 matrices are provided. These lead to closed form formulae for these exponentials. The techniques rely on one particular Clifford Algebra…
In this master's thesis, we introduce expansion systems as a general framework to describe a large variety of approximation algorithms, such as Taylor approximation, decimal expansion and continued fraction. We consider some basic…
We study Laplace-type operators on hybrid manifolds, i.e. on configurations consisting of closed two-dimensional manifolds and one-dimensional segments. Such an operator can be constructed by using the Laplace-Beltrami operators on each…
The evaluation of a matrix exponential function is a classic problem of computational linear algebra. Many different methods have been employed for its numerical evaluation [Moler C and van Loan C 1978 SIAM Review 20 4], none of which…
We propose a new theoretical method for the calculation of the interaction energy between macromolecular systems at large distances. The method provides a linear scaling of the computing time with the system size and is considered as an…
The two regular super-exponentials to base exp(1/e) are constructed. An efficient algorithm for the evaluation of these super-exponentials and their inverse functions is suggested and compared to the already published results.
This paper introduces an optimal representation for a right-to-left parallel elliptic curve scalar point multiplication. The right-to-left approach is easier to parallelize than the conventional left-to-right approach. However, unlike the…
This paper is devoted to the construction of exponential integrators of first and second order for the time discretization of constrained parabolic systems. For this extend, we combine well-known exponential integrators for unconstrained…
Polymorphism in programming languages enables code reuse. Here, we show that polymorphism has broad applicability far beyond computations for technical computing: parallelism in distributed computing, presentation of visualizations of…
In recent years, several problems regarding the partition regularity of exponential configurations have been studied in the literature, in some cases using the properties of specific ultrafilters. In this paper, we start to lay down the…
We present a practical application of parallel symbolic computation in General Relativity: the calculation of curvature invariants for large dimension. We discuss the structure of the calculations, an implementation of the technique and…
The calculation of physical quantities by lattice QCD simulations requires in some important cases the determination of the inverse of a very large matrix. In this article we describe how stochastic estimator methods can be applied to this…
The Parareal algorithm is used to solve time-dependent problems considering multiple solvers that may work in parallel. The key feature is a initial rough approximation of the solution that is iteratively refined by the parallel solvers. We…
Statistical applications often involve the calculation of intractable multidimensional integrals. The Laplace formula is widely used to approximate such integrals. However, in high-dimensional or small sample size problems, the shape of the…
Suzuki-Trotter decompositions of exponential operators like $\exp(Ht)$ are required in almost every branch of numerical physics. Often the exponent under consideration has to be split into more than two operators, for instance as local…
We represent in this note the solutions of the electronic Schr\"odinger equation as traces of higher-dimensional functions. This allows to decouple the electron-electron interaction potential but comes at the price of a degenerate elliptic…
When building a cylindrical algebraic decomposition (CAD) savings can be made in the presence of an equational constraint (EC): an equation logically implied by a formula. The present paper is concerned with how to use multiple ECs,…