A comparison of two Fortran dialects for expressing parallel solutions for a problem in linear algebra

Recently, AMT has issued an extended version of Fortran Plus [1] which allows software to be developed without the developer needing to take explicit accout of the grid size of the target processor. Fortran-Plus and its extension, Fortran Plus Enhanced [2], have been developed for use on the AMT DAP 510 array processor. This machine has 1024 processors arranged in a square grid with nearest neighbour and wraparound connections. It is interesting to enquire whether the performance of code generated by the Fortran-Plus Enhanced compiler is, for a particular application, superior to that generated by the Fortran-Plus compiler from a program which recognises and is tailored to fit the characteristic features of the DAP 510. In this paper the performances of two implementations of an algorithm for the eigensolution of real tridiagonal symmetric matrices are compared. The algorithm is characterised by its heavy use of matrix operations, all of which can be efficiently implemented on an array processor. Some of the constituent operations commonly occur in other applications while others are specific to the problem being addressed.     

A new method for computing a solution of the Cauchy problem with polynomial data for the system of crystal optics

A new analytical method for solving an initial value problem (IVP) for the system of crystal optics with polynomial data and a polynomial inhomogeneous term is suggested. The found solution of the IVP is a polynomial. Theoretical and computational analysis of polynomial solutions and their comparison with non-polynomial solutions corresponding to smooth data are given. The applicability of polynomial solutions to physical processes is discussed. An implementation of this method has been made by symbolic computations in Maple 10.     

A linear algebra method to decompose forms whose length is lower than the number of variables into weighted sum of squares

In this paper, an algorithm based on linear algebra tools is proposed to compute a weighted sum of squares decomposition of a given form whose length is lower than the number of variables. Such an objective is pursued by using linear algebra techniques to perform tasks that are usually carried out through computational algebraic geometry tools. Several examples are reported to show that the use of linear algebra rather than algebraic geometry leads to a reduction of the execution times, without affecting the effectiveness of the algorithm. Applications of the given procedure to system analysis and to control design problems are reported as well as a detailed complexity analysis.     

A new type of solution method for the generalized linear complementarity problem over a polyhedral cone

This paper addresses the generalized linear complementarity problem （GLCP） over a polyhedral cone. To solve the problem, we first equivalently convert the problem into an affine variational inequalities problem over a closed polyhedral cone, and then propose a new type of method to solve the GLCP based on the error bound estimation. The global and R-linear convergence rate is established. The numerical experiments show the efficiency of the method.     

A method to evaluate the a switching unit spatial extensibility of and network

Switching units and networks have been analyzed as extensible fabrics, mostly in terms of their scheduling algorithms. The traditional literature on switching extensibility has provided complexity theory only relating to the total numbers of inputs （or outputs） and exchange lines. This paper analyzes switching extensibility in terms of not only the scheduling algorithm and also the fabric itself. It is found that determining extensibility from soft complexity related to the number of inputs （or outputs） of the scheduling algorithm and the fabric extensibility in previous studies without quantization is a flawed conception. A method is thus proposed to express the spatial extensibility of a switching unit or network in terms of the connections of a switching resource and capacity. The method calculates parameter ES （the efficiency of switching） of an m x n switching unit and obtains two functions of the switching unit to describe spatial extensibility along with the number of unilateral inputs or outputs. It is found that the range of ES is （0, 1] and three types of switching unit and two types of crosspoint networks have ES = 1. ES is calculated for banyan, Clos, parallel packet, fully interconnected and recirculation switching networks. The ES value for the banyan switching network is larger than that for other networks, and switching networks are classified into three types that have absolute/linear/denied spatial extensibility according to the limES value. It is demonstrated that a switching network has the largest ES value when it contains only the five types of switching unit for which ES ---- 1. Finally, a group-switching-first self-routing banyan switching network with lower blocking probability and time delay is deduced, and the ES method is contrasted with two other methods of evaluating spatial extensibility in terms of their mathematical expressions and intuitive graphics, for the five types of switching network listed above.     

A method to evaluate the spatial extensibility of a switching unit and network

Switching units and networks have been analyzed as extensible fabrics,mostly in terms of their scheduling algorithms.The traditional literature on switching extensibility has provided complexity theory only relating to the total numbers of inputs(or outputs)and exchange lines.This paper analyzes switching extensibility in terms of not only the scheduling algorithm and also the fabric itself.It is found that determining extensibility from soft complexity related to the number of inputs(or outputs)of the scheduling algorithm and the fabric extensibility in previous studies without quantization is a flawed conception.A method is thus proposed to express the spatial extensibility of a switching unit or network in terms of the connections of a switching resource and capacity.The method calculates parameter ES(the efciency of switching)of an m×n switching unit and obtains two functions of the switching unit to describe spatial extensibility along with the number of unilateral inputs or outputs.It is found that the range of ES is(0,1]and three types of switching unit and two types of crosspoint networks have ES=1.ES is calculated for banyan,Clos,parallel packet,fully interconnected and recirculation switching networks.The ES value for the banyan switching network is larger than that for other networks,and switching networks are classified into three types that have absolute/linear/denied spatial extensibility according to the limES value.It is demonstrated that a switching network has the largest ES value when it contains only the five types of switching unit for which ES=1.Finally,a group-switching-first self-routing banyan switching network with lower blocking probability and time delay is deduced,and the ES method is contrasted with two other methods of evaluating spatial extensibility in terms of their mathematical expressions and intuitive graphics,for the five types of switching network listed above.     

A new method to determine the periodic orbit of a nonlinear dynamic system and its period

Periodic motion is an important steady-state motion in the real world. In this paper, a new generalized shooting method for determining the periodic orbit of a nonlinear dynamic system and its period is presented by rebuilding the traditional shooting method. First, by changing the time scale, the period of the periodic orbit of a nonlinear system is drawn into the governing equation of this system explicitly. Then, the period is used as a parameter in the iteration procedure of the shooting method. The periodic orbit of the system and the period can be determined rapidly and precisely. The requirement of this method for the initial iteration conditions is not rigorous. This method can be used to analyze the forced nonlinear system and the parameter exciting system. As an example, the results of the Rössler equation for an eight-dimensional, nonlinear, flexible, rotor-bearing system are compared with those obtained by the Runge-Kutta integration algorithm. The validity of this method is verified by the numerical results obtained in the two examples.     

A minimum-order boundary element method to extract the 3-D inductance and resistance of the interconnects in VLSI

The high frequency resistance and inductance of the 3-D complex interconnect structures can be calculated by solving an eddy current electromagnetic problem. In this paper, a model for charactering such a 3-D eddy current problem is proposed, in which the electromagnetic fields in both the conducting and non-conducting regions are described in terms of the magnetic vector potential, and a set of the indirect boundary integral equations (IBIE) is obtained. The IBIEs can be solved by boundary element method, so this method avoids discretizing the domain of the conductors. As an indirect boundary element method, it is of minimum order. It does not restrict the direction of the current in conductors, and hence it can consider the mutual impedance between two perpendicular conductors. The numerical results can well meet the analytical solution of a 2-D problem. The mutual impedance of two perpendicular conductors is also shown under the different gaps between conductors and different frequencies.