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The goal of Nonnegative Matrix Factorization (NMF) is to represent a large nonnegative matrix in an approximate way as a product of two significantly smaller nonnegative matrices. This paper shows in detail how an NMF algorithm based on Newton iteration can be derived using the general Karush-Kuhn-Tucker (KKT) conditions for first-order optimality. This algorithm is suited for parallel execution on systems with shared memory and also with message passing. Both versions were implemented and tested, delivering satisfactory speedup results.

By introducing the block estimate technique and directly using the Newton iteration method, the author constructs Cantor families of time periodic solutions to a class of nonlinear wave equations with periodic boundary conditions. The Lyapunov-Schmidt decomposition used by J. Bourgain, W. Craig and C. E. Wayne is avoided. Thus this work simplifies their framework for KAM theory for PDEs.

Low orbit has features of strong invisibility and penetration, but needs more shutdown energy comparable to high orbit under the same range, which strongly requires studying the problem of delivery capacity optimization for multi-stage launch vehicles. Based on remnant apparent velocity and constraints models, multi-constraint closed-loop guidance with constraints of trajectory maximum height and azimuth was proposed, which adopted elliptical orbit theory and Newton iteration algorithm to optimize trajectory and thrust direction, reached to take full advantage of multi-stage launch vehicle propellant, and guided low orbit vehicle to enter maximum range trajectory. Theory deduction and numerical example demonstrate that the proposed guidance method could extend range and achieve precise control for orbit maximum height and azimuth.