Iterative optical vector-matrix processors (survey of selected achievable operations)

An iterative optical vector-matrix multiplier with a microprocessor-controlled feedback loop capable of performing a wealth of diverse operations was described. A survey and description of many of its operations demonstrates the versatility and flexibility of this class of optical processor and its use in diverse applications. General operations described include: linear difference and differential equations, linear algebraic equations, matrix equations, matrix inversion, nonlinear matrix equations, deconvolution and eigenvalue and eigenvector computations. Engineering applications being addressed for these different operations and for the IOP are: adaptive phased-array radar, time-dependent system modeling, deconvolution and optimal control

An Exponential Matrix Method for Numerical Solutions of Hantavirus Infection Model

In this paper, a new matrix method based on exponential polynomials and collocation points is proposed to obtain approximate solutions of Hantavirus infection model corresponding to a class of systems of nonlinear ordinary differential equations. The method converts the model problem into a system of nonlinear algebraic equations by means of the matrix operations and the collocation points. The reliability and efficiency of the proposed scheme is demonstrated by the numerical applications and all numerical computations have been made by using a computer program written in Maple

On the fast assemblage of finite element matrices with application to nonlinear heat transfer problems

The finite element method is a well-established method for the numerical solution of partial differential equations (PDEs), both linear and nonlinear. However, the repeated reassemblage of finite element matrices for nonlinear PDEs is frequently pointed out as one of the bottlenecks in the computations. The second bottleneck being the large and numerous linear systems to be solved arising from the use of Newton's method to solve nonlinear systems of equations. In this paper, we will address the first issue. We will see how under mild assumptions the assemblage procedure may be rewritten using a completely loop-free algorithm. Our approach leads to a small matrix-matrix multiplication for which we may count on highly optimized algorithms.Comment: 18 pages, 13 figures, 10 tables. Submitted manuscrip

The Eight Epochs of Math as Regards Past and Future Matrix Computations

This survey paper gives a personal assessment of epoch-making advances in matrix computations, from antiquity and with an eye toward tomorrow. It traces the development of number systems and elementary algebra and the uses of Gaussian elimination methods from around 2000 BC on to current real-time neural network computations to solve time-varying matrix equations. The paper includes relevant advances from China from the third century AD on and from India and Persia in the ninth and later centuries. Then it discusses the conceptual genesis of vectors and matrices in Central Europe and in Japan in the fourteenth through seventeenth centuries AD, followed by the 150 year cul-de-sac of polynomial root finder research for matrix eigenvalues, as well as the superbly useful matrix iterative methods and Francis’ matrix eigenvalue algorithm from the last century. Finally, we explain the recent use of initial value problem solvers and high-order 1-step ahead discretization formulas to master time-varying linear and nonlinear matrix equations via Zhang neural networks. This paper ends with a short outlook upon new hardware schemes with multilevel processors that go beyond the 0–1 base 2 framework which all of our past and current electronic computers have been using

An Exponential Matrix Method for Numerical Solutions of Hantavirus Infection Model

Abstract In this paper, a new matrix method based on exponential polynomials and collocation points is proposed to obtain approximate solutions of Hantavirus infection model corresponding to a class of systems of nonlinear ordinary differential equations. The method converts the model problem into a system of nonlinear algebraic equations by means of the matrix operations and the collocation points. The reliability and efficiency of the proposed scheme is demonstrated by the numerical applications and all numerical computations have been made by using a computer program written in Maple