We show how to incorporate exact line searches into Newton’s method for solving the quadratic matrix equation AX2 + BX + C = 0, where A, B and C are square matrices. The line searches are relatively inexpensive and improve the global convergence properties of Newton’s method in theory and in practice. We also derive a condition number for the problem and show how to compute the backward error o...

A nonlinear dynamic output feedback control method for a 5-degree-offreedom AMB is presented in this paper. This method is based on the system structure of AMB and a quadratic-like Lyapunov function. It is shown that backstepping and completing square techniques enable the construction of the voltage input by using measured output only. The control law guarantees the asymptotic stability of the...

A gradient recovery method for the Crouzeix–Raviart element is proposed and analyzed. The proposed method is based on local discrete least square fittings. It is proven to preserve quadratic polynomials and be a bounded linear operator. Numerical examples indicate that it can produce a superconvergent gradient approximation for both elliptic equations and Stokes equations. In addition, it provi...

We will assume that the twisted L-series are primitive (i.e. the corresponding twisted modular forms are newforms). There is no loss of generality in making this assumption: if this is not the case, then f would be a quadratic twist of a newform of smaller level, which we can choose instead. The question of efficiently computing the family of central values L(f,D, 1), for fundamental discrimina...

The problems studied are the separable variational inequalities with linearly coupling constraints. Some existing decomposition methods are very problem specific, and the computation load is quite costly. Combining the ideas of proximal point algorithm PPA and augmented Lagrangian method ALM , we propose an asymmetric proximal decomposition method AsPDM to solve a wide variety separable problem...

This monograph covers some recent advances in a range of acceleration techniques frequently used convex optimization. We first use quadratic optimization problems to introduce two key families methods, namely momentum and nested schemes. They coincide the case form Chebyshev method. discuss methods detail, starting with seminal work Nesterov structure convergence proofs using few master templat...

As a generalization of the famous four square theorem of Lagrange, Ramanujan and Willerding found all positive definite integral quaternary quadratic forms that represent all positive integers. In this paper, we find all positive definite integral quinary quadratic forms that represent all positive definite integral binary quadratic forms. We also discuss recent results on positive definite int...