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

Introduction to sequential smoothing and prediction

The book deals extensively with polynomial filters, namely, fixed-memory, expanding-memory and fading-memory filters. A general approach to tracking and smoothing of data, applicable in many fields

Algorithms Seminar, 2001-2002

These seminar notes constitute the proceedings of a seminar devoted to the analysis of algorithms and related topics. The subjects covered include combinatorics, symbolic computation, asymptotic analysis, number theory, as well as the analysis of algorithms, data structures, and network protocols

Efficient Numerical Solution of Large Scale Algebraic Matrix Equations in PDE Control and Model Order Reduction

Matrix Lyapunov and Riccati equations are an important tool in mathematical systems theory. They are the key ingredients in balancing based model order reduction techniques and linear quadratic regulator problems. For small and moderately sized problems these equations are solved by techniques with at least cubic complexity which prohibits their usage in large scale applications. Around the year 2000 solvers for large scale problems have been introduced. The basic idea there is to compute a low rank decomposition of the quadratic and dense solution matrix and in turn reduce the memory and computational complexity of the algorithms. In this thesis efficiency enhancing techniques for the low rank alternating directions implicit iteration based solution of large scale matrix equations are introduced and discussed. Also the applicability in the context of real world systems is demonstrated. The thesis is structured in seven central chapters. After the introduction chapter 2 introduces the basic concepts and notations needed as fundamental tools for the remainder of the thesis. The next chapter then introduces a collection of test examples spanning from easily scalable academic test systems to badly conditioned technical applications which are used to demonstrate the features of the solvers. Chapter four and five describe the basic solvers and the modifications taken to make them applicable to an even larger class of problems. The following two chapters treat the application of the solvers in the context of model order reduction and linear quadratic optimal control of PDEs. The final chapter then presents the extensive numerical testing undertaken with the solvers proposed in the prior chapters. Some conclusions and an appendix complete the thesis