A weakly stable algorithm for general Toeplitz systems

We show that a fast algorithm for the QR factorization of a Toeplitz or Hankel matrix A is weakly stable in the sense that R^T.R is close to A^T.A. Thus, when the algorithm is used to solve the semi-normal equations R^T.Rx = A^Tb, we obtain a weakly stable method for the solution of a nonsingular Toeplitz or Hankel linear system Ax = b. The algorithm also applies to the solution of the full-rank Toeplitz or Hankel least squares problem.Comment: 17 pages. An old Technical Report with postscript added. For further details, see http://wwwmaths.anu.edu.au/~brent/pub/pub143.htm

Transformation Techniques for Toeplitz and Toeplitz-plus-Hankel Matrices Part II. Algorithms

In the first part of the paper transformations mapping Toeplitz and Toeplitz-plus-Hankel matrices into generalized Cauchy matrices were studied. In this second part fast algorithms for LU-factorization and inversion of generalized Cauchy matrices are discussed. It is shown that the combination of transformation pivoting techniques leads to algorithms for indefinite Toeplitz and Toeplitz-plus-Hankel matrices that are more stable than the classical ones Special attention is paid to the symmetric and hermitian cases

The mu-calculus and Model Checking

International audienceThis chapter presents a part of the theory of the mu-calculus that is relevant to the, broadly understood, model-checking problem. The mu-calculus is one of the most important logics in model-checking. It is a logic with an exceptional balance between expressiveness and algorithmic properties. The chapter describes in length the game characterization of the semantics of the mu-calculus. It discusses the theory of the mu-calculus starting with the tree model property, and bisimulation invariance. Then it develops the notion of modal automaton: an automaton-based model behind the mu-calculus. It gives a quite detailed explanation of the satisfiability algorithm, followed by the results on alternation hierarchy, proof systems, and interpolation. Finally, the chapter discusses the relations of the mu-calculus to monadic second-order logic as well as to some program and temporal logics. It also presents two extensions of the mu-calculus that allow us to address issues such as inverse modalities