The Transformation of Numerical Analysis by the Computer: an Example from the Work of John von Neumann

Publisher Summary This chapter discusses the time when the mathematicians of the nineteenth and early twentieth centuries developed numerical methods for practical hand calculation to solve small systems of linear equations, invert small matrices, approximate integrals, and solve ordinary differential equations. These methods remained relatively unchanged until the introduction of the computer after the second world war. Keying on the computer's great power to carry out long sequences of calculations rapidly and without human intervention, John von Neumann and a few other visionary mathematicians saw the opportunity to extend numerical techniques to a much wider domain of mathematics and mathematical applications. The computer differed in several important respects from the desk calculator or punched-card equipment that had previously driven numerical analyses First, the computer was much faster—on the order of 10 3 to 10 5 times faster—than hand methods in its ability to carry out multiplications. The chapter describes two of von Neumann's forays into computer-oriented numerical analysis.