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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.