Room P3.10, Mathematics Building

Carlos Tomei, PUC, Rio de Janeiro
The subtle convergence of Wilkinson's method

Wilkinson's iteration is frequently used to compute eigenvalues of symmetric matrices. Decades of experience led to believing that the algorithm performed extremely fast, Indeed, recently Nicolau Saldanha (PUC, Rio de Janeiro), Ricardo Leite (UFES) and I proved that this is so, for matrices whose spectrum does not contain three eigenvalues forming an arithmetic progression. Things may go slightly slower otherwise. The argument uses techniques from the theory of completely integrable systems, a new class of inverse variables for tridiagonal matrices and the construction of some Lyapunov functions. The counter-examples arise from an unexpected property related to the iteration of a discontinuous function on the plane.