4 Presentations

• 
  10:30 AM - 10:55 AM

  Loss of orthogonality in large scale matrix computations

  • Chris Paige, presenter, School of Computer Science, McGill University, Montreal, Canada

  Many large scale matrix algorithms are based on orthogonality, but for efficiency this orthogonality is often obtained via short term recurrences. This can lead to both loss of orthogonality and loss of linear independence of computed vectors, yet with well designed algorithms high accuracy can still be obtained. Here we discuss a nice theoretical indicator of loss of orthogonality and linear independence, and for such short term recurrence algorithms show how it can lead to a related higher dimensional orthogonality that can be used to analyze and prove the effectiveness of such algorithms. We illustrate advantages and shortcomings of such algorithms with Cornelius Lanczos’ symmetric matrix tridiagonalization process, which is the basis for many of our most useful large sparse matrix algorithms.
• 
  10:55 AM - 11:20 AM

  A tridiagonalization method for symmetric and quasi-definite saddle-point systems

  • Dominique Orban, presenter, GERAD - Polytechnique Montréal

  We propose an iterative method for symmetric saddle-point systems that splits the system into a least-squares and a least-norm problem. Our method typically requires fewer operator-vector products than MINRES, yet performs a comparable amount of work per iteration and has comparable storage requirements. We illustrate a generalization to elliptic norms.
• 
  11:20 AM - 11:45 AM

  Exploiting variable arithmetic in GMRES

  • David Titley-Peloquin, presenter, McGill University
  • Serge Gratton, CERFACS et ENSEEIHT, Toulouse
  • Ehouarn Simon, Université de Toulouse, INP, IRIT
  • Philippe L. Toint, University of Namur

  Variable floating-point arithmetic precision (beyond IEEE single/double) is becoming increasingly available to users. We show how this can be exploited in MGS-GMRES for inexact matrix-vector products and inexact reorthogonalization without affecting the algorithm's convergence. This is joint work with Serge Gratton and Ehouarn Simon (INPT-ENSEEIHT) and Philippe Toint (Namur).
• 
  11:45 AM - 12:10 PM

  Krylov.jl : A Julia basket of hand-picked Krylov methods

  • Alexis Montoison, presenter,
  • Dominique Orban, GERAD - Polytechnique Montréal

  Krylov.jl provides Julia implementations of certain of the most useful Krylov method for linear systems, least squares, and least norm problems, together with facilities for saddle-point systems. Those methods have been optimized to ensure performance in terms of time and memory. We illustrate those features on our implementation of DQGMRES, and memory improvements to MINRES. We also present future improvements, an implementation of a new method named BiLQ and multiple precision support.