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 → ∞.

Let A k ; k 0, be a sequence of m m nonnegative matrices and let A(z) = P 1 k=0 A k z k be such that A(1) is an irreducible stochastic matrix. The unique power-bounded solution of the nonlinear matrix equation G = P 1 k=0 A k G k has been shown to play a key role in the analysis of Markov chains of M/G/1 type. Assuming that the matrix A(z) is rational, we show that the solution of this matrix e...

Let A be a real continuous nxn matrix on an interval T3 and let the n-vector x be a solution of the differential equation x = Ax on r. If [oc,g]er, g is called a conjugate point of a if the equation has a nontrivial solution vector x = (x1,...,xn) such that x1(a) = ... = xk(a) = x k+1(P) = ... = x (0) = 0 for some ke[l,n-l]. It is shown that the absence on (t..,t ) of a point conjugate to t, wi...

Let A k ; k 0, be a sequence of mm nonnegative matrices and let A(z) = P 1 k=0 A k z k be such that A(1) is an irreducible stochastic matrix. The unique power-bounded solution of the nonlinear matrix equation G = P 1 k=0 A k G k has been shown to play a key role in the analysis of Markov chains of M/G/1 type. Assuming that the matrix A(z) is rational, we show that the solution of this matrix eq...

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.

A new algorithmic setting is proposed for the discrete-time finite-horizon linear quadratic (LQ) optimal control problem with constrained or unconstrained final state, no matter whether the problem is cheap, singular, or regular. The proposed solution, based on matrix pseudoinversion, is completed and made practically implementable by a nesting procedure for welding optimal subarcs that enables...

The matrix equation X + AXA = Q arises in Green’s function calculations in nano research, where A is a real square matrix and Q is a real symmetric matrix dependent on a parameter and is usually indefinite. In practice one is mainly interested in those values of the parameter for which the matrix equation has no stabilizing solutions. The solution of interest in this case is a special weakly st...

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 ...

in this paper, we derive the necessary and sufficient conditions for the quaternion matrix equation xa=b to have the least-square bisymmetric solution and give the expression of such solution when the solvability conditions are met. futhermore, we consider the maximal and minimal inertias of the least-square bisymmetric solution to this equation. as applications, we derive sufficient and necess...