Helmut Wielandt's contributions to the numerical solution of complex eigenvalue problems

1. What It's All About. In 1943 and 1944 Helmut Wielandt wrote a series of ve papers, with the themècontributions to the mathematical treatment of complex eigenvalue problems' 20, 24, 21, 22, 23]. The papers deal with eigenvalue problems for algebraic operators, for linear Fredholm integral operators, and for linear ordinary diierential operators in the form of boundary value problems. In particular, they discuss the location of eigenvalues of matrices in the complex plane 20]; as well as the computation of eigenvalues and eigenfunctions via the power method 24, 21, 22], and inverse iteration 23]. Only a single paper, 24], was published in a journal; the others exist as technical reports. In this commentary we focus on the computational aspects of Wielandt's work on eigenvalue problems. 2. Impressions. Helmut Wielandt is well known for his work in matrix and group theory 6]. While reading his papers I realised with surprise that he also did pioneering work in computational numerical analysis. In fact, Wielandt was a computational numerical analyst of the same calibre as Jim Wilkinson, whose groundbreaking work on round-oo error analysis 26] and the algebraic eigenvalue problem 27] continues to innuence computational linear algebra to this day. Both, Wielandt and Wilkinson had a thorough understanding of the mathematical theory in addition to an extraordinary intuition regarding the eeects caused by nite precision arithmetic. Both used a good measure of common sense to solve problems: aiming for the simplest possible approach; avoiding excessive formalism ; and not hesitating to argue on the back of the envelope, so to speak, if that was enough to get the point across. For instance, both illustrated the numerical behaviour of their methods with plenty of well-chosen examples; and both provided estimates of the computational errors and the operation counts. Wielandt's criteria in 1944 for designing numerical methods agree with the principles that have been formalised over the last thirty years and are in use today. His goal was to design reliable numerical methods. By this he meant methods that are simple and insensitive to computational errors 21, Section I.3(A)]. The last feature corresponds to what we now call numerical stability. Because computations were still done by hand at times, Wielandt valued simple and repetitive algorithms, as do today's programmers of vector and parallel machines. He was also aware of problems associated with nite precision arithmetic, such as overrow and catastrophic cancellation, which continue to …