Highly Accurate Golden Section Search Algorithms and Fictitious Time Integration Method for Solving Nonlinear Eigenvalue Problems

This study sets up two new merit functions,which are minimized for the detection of real eigenvalue and complex eigenvalue to address nonlinear eigenvalue problems.For each eigen-parameter the vector variable is solved from a nonhomogeneous linear system obtained by reducing the number of eigen-equation one less,where one of the nonzero components of the eigenvector is normalized to the unit and moves the column containing that component to the right-hand side as a nonzero input vector.1D and 2D golden section search algorithms are employed to minimize the merit functions to locate real and complex eigenvalues.Simultaneously,the real and complex eigenvectors can be computed very accurately.A simpler approach to the nonlinear eigenvalue problems is proposed,which implements a normalization condition for the uniqueness of the eigenvector into the eigenequation directly.The real eigenvalues can be computed by the fictitious time integration method(FTIM),which saves computational costs compared to the one-dimensional golden section search algorithm(1D GSSA).The simpler method is also combined with the Newton iterationmethod,which is convergent very fast.All the proposed methods are easily programmed to compute the eigenvalue and eigenvector with high accuracy and efficiency.

Proportional Fairness-Based Energy-Efficient Power Allocation in Downlink MIMO-NOMA Systems with Statistical CSI

In this paper, proportional fairness(PF)-based energy-efficient power allocation is studied for multiple-input multiple-output(MIMO) non-orthogonal multiple access(NOMA) systems. In our schemes, statistical channel state information(CSI) is utilized for perfect CSI is impossible to achieve in practice. PF is used to balance the transmission efficiency and user fairness. Energy efficiency(EE) is formulated under basic data rate requirements and maximum transmitting power constraints. Due to the non-convex nature of EE, a two-step algorithm is proposed to obtain sub-optimal solution with a low complexity. Firstly, power allocation is determined by golden section search for fixed power. Secondly total transmitting power is determined by fractional programming method in the feasible regions. Compared to the performance of MIMO-NOMA without PF constraint, fairness is obtained at expense of decreasing of EE.

A SEARCH TECHNIQUE FOR CHOOSING OPTIMUM PARAMETERS IN NUMERICAL WEATHER PREDICTION

The steepest descent(or ascent)search is employed for finding optimum diffusion coefficients in T42L9G model,with a view to improving the model's computational stability or prediction accuracy.The method of the steepest descent search is first described,in which the golden section search is chosen as the fundamental one- dimensional search used in the multi-dimentional steepest descent search,and then the optimization of the dif- fusion coefficients is described.

ON THE OPTIMALITY IN GENERAL SENSE FOR ODD-BLOCK SEARCH

In his classical article[3], J. Kiefer introduced the Fibonacci search as a direct optimal method. The optimality was proved under the restriction: the total number of tests is given in advance and fixed. To avoid this restriction, some different concepts of optimality were proposed and some corresponding optimal methods were obtained in [1], [2], [5] and [6]. In particular, the even-block search was treated in [1]. This paper deals with the odd-block search. The main result is Theorem 1.15.