Related papers: Computing square roots in quaternion algebras

The classical quadratic formula and some of its lesser known variants for solving the quadratic equation are reviewed. Then, a new formula for the roots of a quadratic polynomial is presented.

In this paper, we present some applications of quaternions and octonions. We present the real matrix representations for complex octonions and some of their properties which can be used in computations where these elements are involved.…

This paper establishes new upper bounds for the right eigenvalues of monic matrix polynomials over the quaternion division algebra. The noncommutative nature of quaternion multiplication presents fundamental challenges in eigenvalue…

It is well known that Clifford (geometric) algebra offers a geometric interpretation for square roots of -1 in the form of blades that square to minus 1. This extends to a geometric interpretation of quaternions as the side face bivectors…

In this paper we associate an invariant to a biquaternion algebra $B$ over a field $K$ with a subfield $F$ such that $K/F$ is a quadratic separable extension and $\operatorname{char}(F)=2$. We show that this invariant is trivial exactly…

Let A be a modular abelian variety over \Q of arbitrary even dimension. We establish criteria to prevent a given quaternion algebra over a totally real number field to be the endomorphism algebra of A over \bar\Q. We accomplish this by…

We give a geometric classification of 4-dimensional superalgebras over an algebraic closed field.

Quadratically optimized polynomials are described which are useful in multi-bosonic algorithms for Monte Carlo simulations of quantum field theories with fermions. Algorithms for the computation of the coefficients and roots of these…

We consider Clifford algebras over the field of real or complex numbers as a quotient algebra without fixed basis. We present classification of Clifford algebra elements based on the notion of quaternion type. This classification allows us…

In this article we propose a consistent scheme of doing computations directly in four dimensions using conventional quantum field theory methods. This work is continuation of a stochastic quantization program reported earlier.

A formal description of quaternions by means of exterior calculus is presented. Considering a three-dimensional space-time characterized by three time-like coordinates, we have been able to consistently recover a suitable formulation of…

Using the complex Klein-Gordon field as a model, we quantize the quaternionic scalar field in the real Hilbert space. The lagrangian formulation has accordingly been obtained, as well as the hamiltonian formulation, and the energy and…

We present algorithms to compute generalized root systems of Nichols algebras of diagonal type and of contragredient Lie superalgebras. As a consequence, we obtain an algorithm to compute the Lyndon words in the Kharchenko PBW basis…

In this paper we develop a new method which is a generalization of the Obreshkoff -Ehrlich method for the cases of algebraic, trigonometric and exponential polynomials. This method has a cubic rate of convergence. It is efficient from the…

For quadratic spaces which represent 1 there is a characterization of hermitian compositions in the language of algebras-with-involutions using the even Clifford algebra. We extend this notion to define a generalized composition based on…

This paper presents a high-order accurate numerical quadrature algorithm for evaluating integrals over curved surfaces and regions defined implicitly via a level set of a given function restricted to a hyperrectangle. The domain is divided…

Quaternions, introduced by Hamilton in 1843 as a generalization of complex numbers, have found, in more recent years, a wealth of applications in a number of different areas which motivated the design of efficient methods for numerically…

An algorithm is presented for the efficient and accurate computation of the coefficients of the characteristic polynomial of a general square matrix. The algorithm is especially suited for the evaluation of canonical traces in determinant…

Let $k$ be a finite field, and $L$ be a $q$-linearized polynomial defined over $k$ of $q$-degree $r$ ($L=\sum^r_{i=0}a_iZ^{q^i}$, with $a_i\in k$). This paper provides an algorithm to compute a characteristic polynomial of $L$ over a large…

We review known factorization results in quaternion matrices. Specifically, we derive the Jordan canonical form, polar decomposition, singular value decomposition, the QR factorization. We prove there is a Schur factorization for commuting…