A New Algorithm for Computing the Geronimus Transformation with Large Shifts Articles uri icon

publication date

  • May 2010

start page

  • 101

end page

  • 139

issue

  • 54

volume

  • 1

International Standard Serial Number (ISSN)

  • 1017-1398

Electronic International Standard Serial Number (EISSN)

  • 1572-9265

abstract

  • A monic Jacobi matrix is a tridiagonal matrix which contains the parameters of the three-term recurrence relation satisfied by the sequence of monic polynomials orthogonal with respect to a measure. The
    basic Geronimus transformation with shift alfa transforms the monic Jacobi matrix associated with a measure dmu into the monic Jacobi matrix associated with dmu/(x − alfa) + Cdelta(x − alfa), for some constant C.
    In this paper we examine the algorithms available to compute this
    transformation and we propose a more accurate algorithm, estimate its
    forward errors, and prove that it is forward stable. In particular, we
    show that for C = 0 the problem is very ill-conditioned, and we present a new algorithm that uses extended precision.