Now showing items 17-20 of 20

    • A root-finding algorithm for cubics 

      Northshield, Sam (Proceedings of the American Mathematical Society, 2013)
      Newton's method applied to a quadratic polynomial converges rapidly to a root for almost all starting points and almost all coefficients. This can be understood in terms of an associative binary operation arising from 2 x ...
    • A short proof and generalization of Lagrange's theorem on continued fractions 

      Northshield, Sam (American Mathematical Monthly, 2011)
      We present a short new proof that the continued fraction of a quadratic irrational eventually repeats. The proof easily generalizes; we construct a large class of functions which, when iterated, must eventually repeat when ...
    • Square Roots of 2x2 Matrices 

      Northshield, Sam (2010)
      This paper is designed to pique the interest of undergraduate students who are familiar with the concepts of linear algebra. We investigate five methods of computing square roots of two-by-two matrices. Each method gives ...
    • Sums across Pascal's triangle modulo 2 

      Northshield, Sam (Congressus Numerantium, 2010)
      We consider sums of the binomial coefficients C(i + j, i) modulo 2 over lines ai + bj = n. Many interesting sequences (old and new) arise this way.