Habilitation à diriger des recherches / Manuscript in english

defended on September 13, 2000  (slides)

The power of backward error analysis / Puissance de l'analyse inverse des erreurs
Valérie Frayssé, CERFACS

Main document (Postscript)

Introduction

  • Computability in finite precision
    • Numerical approximation
    Finite precision approximation
    Numerical and arithmetical coupling
    Exact versus finite precision computation
    Computation in the neighbourhood of a singularity
    Where does the backward error analysis à la Wilkinson stand in this picture
    Conclusion
  • Backward error analysis à la Wilkinson

    • Backward error analysis for finite precision computations

  • Models
  • A priori backward error analysis
  • A posteriori backward error analysis
  • Two examples of backward error analysis
    • Backward error analysis for the generalised eigenproblem
    Linear systems of the type $A^*Ax = b$
  • Singular problems
    • Backward error analysis for well-posed singular problems
    Condition number and distance to singularity
  • Backward error and pseudosolutions
  • The toolbox PRECISE

    • Backward error analysis for approximation methods in exact arithmetic

  • Homotopic perturbations
  • Pseudospectra
    • Pseudospectra of matrices and nonnormality
    Pseudospectra of operators
  • Convergence of iterative methods in Linear Algebra
    • Convergence of the Power method
    Embedded iterative solvers

    Conclusion

    Bibliography

    List of publications

    Curriculum vitae (in French)

    Attached documents

  • Computations in the neighbourhood of algebraic singularities

    • F. Chatelin,  V. Frayssé and T. Braconnier. Num. Funct. Anal. Opt., 16:287--302, 1995.
  • A note on the normwise perturbation theory for the regular generalized eigenproblem $Ax = \lambda B x$

    • V. Frayssé and V. Toumazou. J. Numer. Linear Algebra Appl., 5:1--10, 1998.
  • Structured backward error and condition number for linear systems of the type ${A^*A} x = b$

    • V. Frayssé, S. Gratton and V. Toumazou. BIT, 40:74--83, 2000.
  • A relaxation strategy for inexact matrix-vector products for Krylov methods

    • A. Bouras and V. Frayssé. CERFACS Technical Report TR/PA/00/15, 2000.
    Submitted to the Journal of Numerical Linear Algebra with Applications

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