General-purpose linear solvers [CRUNCH

Abstract

Between 1999 and 2011, “By the Numbers” appeared as an occasional column in this magazine. The purpose was to share some algorithms and numerical methods—ones that might be useful to fellow I&Mers, but many of which are not typically covered in a beginning course on numerical analysis. Justin Dyer co-authored about two-thirds of those 23 installments with me. Not accomplished, however, was the presentation of a few broadly useful algorithms and methods that are commonplace, in one sense or another, in introductory courses covering elementary numerical methods or signalprocessing concepts. “CRUNCH” is a follow-on column that continues the number-crunching theme of its predecessor. At present, its lifetime is imagined to be quite finite—perhaps a year or so worth of issues. Its sole purpose is that of laying out some of those useful topics—e.g., a general-purpose linear solver, a fast algorithm for discrete orthogonal transforms, windowing, fast convolution and correlation, elementary classical spectrum analysis, etc.—that were bypassed during the days of “By the Numbers.” Solution of Systems of Linear Equations Where do linear systems arise? Everywhere! We make our mathematical models of problems—physical or otherwise— linear, if at all reasonable—at least at the outset. In linearity there are no real surprises. And there is a wealth of mathematical tools available to be used, just as long as linearity holds. We could make a list of situations in which linear algebraic systems may be found: in “network” problems, in boundaryvalue problems involving differential or difference equations, in data fitting, in eigenvalue problems, etc. The list is long. To be general, we could write the linear system of CRUNCH April 2015 Stephen A. Dyer General-Purpose Linear Solvers Between 1999 and 2011, “By the Numbers” appeared as an occasional column in this magazine. The purpose was to share some algorithms and numerical methods—ones that might be useful to fellow I&Mers, but many of which are not typically covered in a beginning course on numerical analysis. Justin Dyer co-authored about two-thirds of those 23 installments with me. Not accomplished, however, was the presentation of a few broadly useful algorithms and methods that are commonplace, in one sense or another, in introductory courses covering elementary numerical methods or signal-processing concepts. “CRUNCH” is a follow-on column that continues the numbercrunching theme of its predecessor. At present, its lifetime is imagined to be quite finite—perhaps a year or so worth of issues. Its sole purpose is that of laying out some of those useful topics—e.g., a general-purpose linear solver, a fast algorithm for discrete orthogonal transforms, windowing, fast convolution and correlation, elementary classical spectrum analysis, etc.—that were bypassed during the days of “By the Numbers.”