Preconditioning technique for symmetric M-matrices

Abstract Linear systems with M-matrices occur in a wide variety of areas including numerical partial differential equations, input-output production and growth models in economics, linear complementarity problems in operations research and Markov chains in stochastic analysis.In this paper, we propose a new preconditioner for solving a system with symmetric positive definite M-matrix by the preconditioned conjugate gradient (PCG) method. We show that our preconditioner increases the convergence rate of the PCG method and reduces the operation cost. Numerical results are given.     

Approximate solution of systems of Volterra integral equations with error analysis

This paper describes a procedure for solving the system of linear Volterra integral equations by means of the Sinc collocation method. A convergence and an error analysis are given; it is shown that the Sinc solution produces an error of order O(exp(?c N ^1/2)), where c>0 is a constant. This approximation reduces the system of integral equations to an explicit system of algebraic equations. The analytical results are illustrated with numerical examples that exhibit the exponential convergence rate.     

Fuzzy relation equations (I): the general and specialized solving algorithms

 In this article, we develop a new method and an algorithm to solve a system of fuzzy relation equations. We first introduce a solution-base-matrix and then give a tractable mathematical logic representation of all minimal solutions. Next, we design a new universal algorithm to get them. Two simplification rules are found to simplify the solution-base-matrix. We show that a polynomial time algorithm to find all minimal solutions for a general system of fuzzy relation equations simply does not exist with expectation of P=N P. Hence, the problem of solving fuzzy relation equations is an N P-hard problem in terms of computational complexity. Our universal algorithm is still useful when one does not solve a large number of equations. In many real applications the problem of solving fuzzy relation equations can be simplified in polynomial time problems. In this article, we will discuss several cases of practical applications which have such polynomial algorithms.     

More on the controllability of linear time-invariant systems

This study shows that a controllable system [xdot] = Ax + Bu, where x is an n-vector, can be driven from an arbitrary initial condition x(0) = x₀ to an arbitrary final condition x(t_f = x_f by a polynomial control function of degree M = 2r + 1, where r = n ? rank (B), through a polynomial trajectory of degree M. A simple algorithm for finding u by solving a set of linear equations, the solution of which yields the polynomial coefficients, is given.     

Approximate algorithms for partitioning problems

We consider the problem of optimally assigning the modules of a parallel/pipelined program over the processors of a multiple processor system under certain restrictions on the interconnection structure of the program as well as the multiple computer system. We show that for a variety of such problems, it is possible to find if a partition of the modular program exists in which the load on any processor is whithin a certain bound. This method when combined with a binary search over a fixed range, provides an optimal solution to the partitioning problem.The specific problems we consider are partitioning of (1) a chain structured parallel program over a chain-like computer system, (2) multiple chain-like programs over a host-satellite system, and (3) a tree structured parallel program over a host-satellite system.For a problem withN modules andM processors, the complexity of our algorithm is no worse thanO(Mlog(N)log(W _T/)), whereW _T is the cost of assigning all modules to one processors, and the desired accuracy. This algorithm provides an improvement over the recently developed best known algorithm that runs inO(MNlog(N)) time.This Research was supported by a grant from the Division of Research Extension and Advisory Services, University of Engineering and Technology Lahore, Pakistan. Further support was provided by NASA Contracts NAS1-17070 and NAS1-18107 while the author was resident at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, Virginia, USA.     

Characteristic lines for two- and three-dimensional flows

An algorithm based on the method of characteristics for one-dimensional flow is generalized to apply to multidimensional flow. InN dimensions the method has 2N+2 ordinary differential equations defined on a unique set of 2N+1 characteristic lines, modulo the appropriate rotation group.     

Dependence of performance indices of a call center on the distribution of calls’ sojourn time in the orbit

The operation of a call center is described in the form of a retrial queuing system. The dependence of its performance indices on the Markovian and two-phase Erlang distributions of calls’ sojourn time in the orbit is considered. An analytical model of an M / M / c / / / E ₂ retrial queuing system is developed. An asymptotic analysis of some characteristics of M / M / c / 0 / 2/ / M and M / M / c / 0 / 2 / / E ₂ systems is performed. An application is developed for solving M / M / c / 0 / N / / N / / E ₂ and M / M / c / 0 / N / / M systems using sparse matrices. Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 170–183, March–April 2009.     

A two-stage approach in solving the state probabilities of the multi-queue M/G/1 model

The M/G/1 model is the fundamental basis of the queueing system in many network systems. Usually, the study of the M/G/1 is limited by the assumption of single queue and infinite capacity. In practice, however, these postulations may not be valid, particularly when dealing with many real-world problems. In this paper, a two-stage state-space approach is devoted to solving the state probabilities for the multi-queue finite-capacity M/G/1 model, i.e. q-M/G/1/K_i with K_i buffers in the ith queue. The state probabilities at departure instants are determined by solving a set of state transition equations. Afterward, an embedded Markov chain analysis is applied to derive the state probabilities with another set of state balance equations at arbitrary time instants. The closed forms of the state probabilities are also presented with theorems for reference. Applications of Little's theorem further present the corresponding results for queue lengths and average waiting times. Simulation experiments have demonstrated the correctness of the proposed approaches.     

Optimal Speed-Up Parallel Image Template Matching Algorithms on Processor Arrays with a Reconfigurable Bus System,

The image template matching problem is one of the fundamental problems of and has many practical applications in image processing, pattern recognition, and computer vision. It is a useful operation for filtering, edge detection, image registration, and object detection [13]. In this paper, we first design twoO[(M²/p²)log logM] andO[(M²/p²)+(M/p)log logp] time parallel image template matching algorithms on a 3-D processor array with a reconfigurable bus system usingp²N²processors with each processor containingO(1) andO(M/p) restricted memory for 1 ≤p≤M≤N, respectively, for anN×Ndigital image and anM×Mtemplate. By increasing the number of processors, these two proposed algorithms can be run inO(M²/p²) time for speeding up the time complexity usingp²M^1/cN²andp^2+1/cN²processors, respectively, wherecis a constant andc≥1. Furthermore, anO(1) time can be also obtained from these two proposed algorithms by usingM²+1^/cN²processors. These results improve the best known bounds and achieve both optimal and optimal speed-up in their time and processor complexities.     

Extrapolation for solving a system of weakly singular nonlinear Volterra integral equations of the second kind

This article discusses an extrapolation method for solving a system of weakly singular nonlinear Volterra integral equations of the second kind. Based on a generalization of the discrete Gronwall inequality and Navot's quadrature rule, the modified trapeziform quadrature algorithm is presented. The iterative algorithm for solving the discrete system possesses a high accuracy order O(h ^2+α). After the asymptotic expansion of errors is proved, we can obtain an approximation with a higher accuracy order using extrapolation. An a posteriori error estimation is provided. Some numerical results are presented to illustrate the efficiency of our methods.     

An algorithm for solving the matrix equation X = FXF T + S

A now algorithm for solving the matrix equation X = FXF ^T + S, which is important in the control system design, is presented in this paper. The algorithm is based on the QR algorithm for finding the eigenvalues of a matrix and works efficiently for large dimensional problems. A simple example is given to illustrate the algorithm. The method is also applicable to other types of equations such as the Lyapunov equation A ^T X + XA + B = 0.     

Systolic arrays for multidimensional discrete transforms

An active area of research in supercomputing is concerned with mapping certain finite sums, such as discrete Fourier transforms, onto arrays of processors. This paper presents systolic mapping techniques that exploit the parallelism inherent in discrete Fourier transforms. It is established that, for anM-dimensional signal, parallel executions of such transforms are closely related to mappings of an (M + 1)-dimensional finite vector space into itself. Three examples of such parallel schemes are then described for the discrete Fourier transform of a two-dimensional finite extent sequence of sizeN ₁ ×N ₂. The first is a linear array ofN ₁ +N ₂ – 1 processors and takesO(N ₁ N ₂) steps. The second is anN ₁ ×N ₂ rectangular array of processors and takesO(N ₁ +N ₂) steps, and the third is a hexagonal array which usesN ₁ N ₂ + (N ₂ – 1)(N ₁ +N ₂ – 1) processors andO(N ₁ +N ₂) steps. All three mappings are optimal in that they achieve asymptotically the highest speedup possible over the sequential execution of the same transform, and can easily be generalized to higher dimensions.     

The convergence conditions of the psd iterative method for linear systems

In this paper, the PSD iterative method was proposed by Evans and Missirlis [4], for solving a large nonsingular system of linear equations Ax=b A general necessary condition for con-vergence of the PSD iterative method is obtained. The convergence theorems of the PSD iterative method are established under the condition that the coefficient matrix A is an H-matrix, our theorems improve and extend some known results.     

Vector choice in the texture spectrum approach

Recently the texture spectrum approach has been proposed as a statistical method for texture analysis, and applied to remotely sensed images. In the present study, this method is generalized to a space of M vectors and N grey level intervals for the elements of texture units, instead of M = 8 and N = 3 in the earlier studies of the texture spectrum. In this way, the texture unit set can be defined in a neighbourhood of 3 pixels by 3 pixels, 5 pixels by 5 pixels, 7 pixels by 7 pixels or other forms and sizes, and the co-occurrence matrix approach is unified to the texture spectrum method with the extreme case of M = 1. Several combinations of M and N have been evaluated to classify an imagery composed of six natural textures. The results show maximum discrimination for M = 5, followed by M = 4. In this way we minimize the calculation lime needed to maximize the accuracy of the classification.     

A geometric process model for M/PH(M/PH)/1/K queue with new service machine procurement lead time

In this article, we consider a geometric process model for M/PH(M/PH)/1/K queue with new service machine procurement lead time. A maintenance policy (N???1,?N) based on the number of failures of the service machine is introduced into the system. Assuming that a failed service machine after repair will not be ‘as good as new’, and the spare service machine for replacement is only available by an order. More specifically, we suppose that the procurement lead time for delivering the spare service machine follows a phase-type (PH) distribution. Under such assumptions, we apply the matrix-analytic method to develop the steady state probabilities of the system, and then we obtain some system performance measures. Finally, employing an important Lemma, the explicit expression of the long-run average cost rate for the service machine is derived, and the direct search method is also implemented to determine the optimal value of N for minimising the average cost rate.     

An efficient approximation for the Generalized Assignment Problem

We present a simple family of algorithms for solving the Generalized Assignment Problem (GAP). Our technique is based on a novel combinatorial translation of any algorithm for the knapsack problem into an approximation algorithm for GAP. If the approximation ratio of the knapsack algorithm is α and its running time is O(f(N)), our algorithm guarantees a (1+α)-approximation ratio, and it runs in O(M⋅f(N)+M⋅N), where N is the number of items and M is the number of bins. Not only does our technique comprise a general interesting framework for the GAP problem; it also matches the best combinatorial approximation for this problem, with a much simpler algorithm and a better running time.     

The use of Richardson extrapolation for the numerical solution of low Mach number flow in confined regions

We use artificial compressibility together with Richardson extrapolation in the Mach numberM as a method for solving the time dependent Navier-Stokes equation for very low Mach number flow and for incompressible flow. The question of what boundary conditions one should use for low Mach number flow, especially at inflow and outflow boundaries, is investigated theoretically, and boundary layer suppressing boundary conditions are derived. For the case of linearization around a constant flow we show that the low Mach number solution will converge with the rateO(M²) to the true incompressible solution, provided that we choose the boundary conditions correctly. The results of numerical calculations for the time dependent, nonlinear equations and for flow situations with time dependent inflow velocity profiles are presented. The convergence rateM ² to incompressible solution is numerically confirmed. It is also shown that using Richardson extrapolation toM ²= 0 in order to derive a solution with very small divergence can with good result be carried through withM ² as large as 0.1 and 0.05. As the time step in numerical methods must be chosen approximately such thatt · (i/(M x)–v/x ²) is in the stability region of the time stepping method, and asM ²=0.05 is sufficiently small to yield good results, the restriction on the time step due to the Mach number is not serious. Therefore the equations can be integrated very fast by explicit time stepping methods. This method for solving very low Mach number flow and incompressible flow is well suited to parallel processing.     

Optimal regulators for linear systems with no control cost

This article is concerned with the theory of optimal feedback regulator for the linear system =Ax +Bu with the cost functional given byJ(u) = 1/2(Mx(T),x(T)). Due to absence of the usual positive definite quadratic cost for controls, this is a nonstandard problem.Two sets of results are presented: one for bounded and one for unbounded controls. For bounded controls, the control law is given by solving a system of coupled nonlinear differential equations of the Riccati type; and for unbounded controls, the optimal control law is determined by solving a parameterized family of matrix Riccati differential equations.     

On exact computational analysis of distributions of numbers in systems for M/G/1/N + 1 and GI/M/1/N + 1 queues using roots

In this paper, we obtain the distribution of number in system and other measures of efficiency for the M/G/1/N + 1 queuing system in terms of the roots of the associated characteristic equation (CE). Results for the GI/M/1/N + 1 queuing system have also been obtained from those of M/G/1/N + 1. Numerical results in the form of tables and graphs have been presented for a variety of service-(interarrival-) time distributions, e.g. Erlang (E_k), generalized Erlang (GE_k) and hyperexponential (HE_k).     

Newton Iterative Parallel Finite Element Algorithm for the Steady Navier-Stokes Equations

A combination method of the Newton iteration and parallel finite element algorithm is applied for solving the steady Navier-Stokes equations under the strong uniqueness condition. This algorithm is motivated by applying the Newton iterations of m times for a nonlinear problem on a coarse grid in domain Ω and computing a linear problem on a fine grid in some subdomains Ω _j ⊂Ω with j=1,…,M in a parallel environment. Then, the error estimation of the Newton iterative parallel finite element solution to the solution of the steady Navier-Stokes equations is analyzed for the large m and small H and h≪H. Finally, some numerical tests are made to demonstrate the the effectiveness of this algorithm.