The paper proposes a new method, called the Fast Quadratic Transform (FQT), to solve the general indefinite two-variable quadratic equation in integers. The paper presents the new approach, discusses its properties, and provides a comparative evaluation with the classical technique. The FQT is demonstrated to be markedly superior for all cases in which it applies, including examples for more th...

Let E be an elliptic curve over Q without complex multiplication, and for each prime p of good reduction, let nE(p) = |E(Fp)|. For any integer b, we are studying in this paper elliptic pseudoprimes to the base b. More precisely, let QE,b(x) be the number of primes p 6 x such that bE ≡ b (modnE(p)), and π E,b (x) be the number of compositive nE(p) such that b nE(p) ≡ b (modnE(p)) (also called el...

We describe the solution of a bound constrained convex quadratic problem with limited memory resources. The problem arises from physical simulations occurring within video games. The motivating problem is outlined, along with a simple interior point approach for its solution. Various linear algebra issues arising in the implementation are explored, including preconditioning, ordering and a numb...

In this paper we answer certain questions posed by V.I. Arnold, namely, we study periods of continued fractions for solutions of quadratic equations in the form x + px = q with integer p and q, p + q ≤ R. Our results concern the average sum of period elements and Gauss–Kuzmin statistics as R → ∞.

In this paper, we consider minimizing the ratio of two indefinite quadratic functions subject to two quadratic constraints. Using the extension of Charnes– Cooper transformation, we transform the problem to a homogenized quadratic problem. Then, we show that, under certain assumptions, it can be solved to global optimality using semidefinite optimization relaxation.

This report covers the application of a piecewise-linear approximation to a quadratic program, QP. In this case the program is one that solves 3D conformal radiation treatment problem and returns optimal beam weights. Four different methods of approximating the quadratic problem were tested and compared to each other and the original problem. The methods are δ method, λ method, σ method, and γ ...

Split quaternions are noncommutative and contain nontrivial zero divisors. Generally speaking, it is difficult to solve equations in such an algebra. In this paper, by using the roots of any split two real nonlinear systems, we derive explicit formulas for computing x2+bx+c=0 quaternion