DMS Linear Algebra/Algebra Seminar

Speaker: Frank Uhlig

Title: The Eight Epochs of Math as regards past and future Matrix Computations

Abstract:  This survey paper gives a personal assessment of Epoch making advances in Matrix Computations, from antiquity and with an eye towards tomorrow.  It traces the development of number systems and elementary algebra and the uses of Gaussian Elimination methods from around 4000 BC on to current real-time Neural Network computations to solve time-varying matrix equations. The paper includes relevant advances from China from the 3rd century AD on and from India and Persia in the 9th and later centuries. Then it discusses the conceptual genesis of vectors and matrices in central Europe and in Japan in the 14th through 17th 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’s 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.