Speaker: Frank Uhlig, Auburn University

Title: Francis’s QR Algorithm for the Eigen Problem

Abstract:

(I) Francis’ Algorithm, its inner workings
a) Reduction to Hessenberg form
b) Francis’ algorithm with implicit steps
b1) Francis shifts
b2) The shift(s) induced unitary similarity on Hn;n creates a Hessenberg matrix with bulge
b3) Chasing the bulge to regain Hessenberg form
c) Convergence
(II) Why Francis’ Algorithm ?
a) Classical QR algorithm of John Francis and Vera Kublanovskaja (1961/62), or m = 1
b) Implicit QR algorithm of Francis, m = 2 and the Implicit Q Theorem
c) Wilkinson shift strategy, (actually due to Francis)
d) Multishift Francis algorithm of Karen Braman, Ralph Byers, and Roy Mathias (2002), m > 2
(III) Who is Francis ?
a) His vita
b) His surroundings and influences of the time
(IV) One Extension (of many); to generalized Orthogonality and Polynomial Root Finding
a) The DQR Algorithm for generalized tridiagonal matrices (Uhlig 1997)
b) A crude comparisons of Francis and DQR on Tridiagonals :
c) Polynomial roots via Francis and Euclid plus DQR : (Uhlig 1999)

Bio: Frank Uhlig is Professor Emeritus of Mathematics, Auburn University.  His mathematical interest lie in Matrix Theory with all of its applications from Numerical Analysis to Mechanics and Education.  Mr. Uhlig is now working with ICIAM, the International Council of Industrial and Applied Math.