Blocked Algorithms for Robust Solution of Triangular Linear Systems

  • Carl Christian Kjelgaard MikkelsenEmail author
  • Lars Karlsson
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10777)


We consider the problem of computing a scaling \(\alpha \) such that the solution \({\varvec{x}}\) of the scaled linear system \({\varvec{Tx}} = \alpha {\varvec{b}}\) can be computed without exceeding an overflow threshold \(\varOmega \). Here \({\varvec{T}}\) is a non-singular upper triangular matrix and \({\varvec{b}}\) is a single vector, and \(\varOmega \) is less than the largest representable number. This problem is central to the computation of eigenvectors from Schur forms. We show how to protect individual arithmetic operations against overflow and we present a robust scalar algorithm for the complete problem. Our algorithm is very similar to xLATRS in LAPACK. We explain why it is impractical to parallelize these algorithms. We then derive a robust blocked algorithm which can be executed in parallel using a task-based run-time system such as StarPU. The parallel overhead is increased marginally compared with regular blocked backward substitution.


Triangular linear systems Overflow Blocked algorithms Robust algorithms 



This work is part of a project that has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 671633.


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Carl Christian Kjelgaard Mikkelsen
    • 1
    Email author
  • Lars Karlsson
    • 1
  1. 1.Department of Computing ScienceUmeå UniversityUmeåSweden

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