In this paper, firstly, we discuss the following matrix completion problem in spectral norm: ?(A B B* X)?2 < 1 subject to (A X) ? 0. The feasible condition for above is established, case, general positive semidefinite solution and its minimum rank are presented. Secondly, applying result of problem, also study approximation problem: ?A BXB*?2 A BXB* 0, where Cm?m?, Cm?n, X Cn?n?.

We define texture mapping as an optimization problem for which the goal is to preserve the maximum amount of information in the mapped texture. We derive a solution that is optimal in the least-squares sense and that corresponds to the pseudo-inverse of the texturemapping transformation. In practice, a first-order approximation of the least-squares solution is used as an initial estimate for th...

We consider periodic solution for coupled systems of mass-spring. Three practical cases of these systems are explained and introduced. An analytical technique called Parameter Expansion Method PEM was applied to calculate approximations to the achieved nonlinear differential oscillation equations. Comparing with exact solutions, the first approximation to the frequency of oscillation produces t...

Last time, we proved that a sketch dimension m % 1 δ2W (AK(xLS) ∩ Sn−1) is sufficient to ensure this property. A related but slightly different notion of approximation is that of solution approximation, in which we measure the quality in terms of some norm between x̂ and xLS. Defining the (semi)-norm ‖u‖A : = √ uTATAu/n, let us say that x̂ is a δ-accurate solution approximation if ‖xLS − x̂‖A ≤ δ....