Path-Following Method to Determine the Field of Values of a Matrix with High Accuracy

Sebastien Loisel, Peter Maxwell

Research output: Contribution to journalArticlepeer-review

5 Citations (Scopus)
38 Downloads (Pure)


We describe a novel and efficient algorithm for calculating the field of values boundary, W(.), of an arbitrary complex square matrix: The boundary is described by a system of ordinary differential equations which are solved using Runge-Kutta (Dormand-Prince) numerical integration to obtain control points with derivatives, then finally Hermite interpolation is applied to produce a dense output. The algorithm computes W(.) both efficiently and with low error. Formal error bounds are proven for specific classes of matrix. Furthermore, we summarize the existing state of the art and make comparisons with the new algorithm. Finally, numerical experiments are performed to quantify the cost-error trade-off between the new algorithm and existing algorithms.

Original languageEnglish
Pages (from-to)1726-1749
Number of pages24
JournalSIAM Journal on Matrix Analysis and Applications
Issue number4
Publication statusPublished - 27 Nov 2018


  • Dormand- Prince
  • eigenvalue crossing
  • eigenvalue perturbation
  • field of values
  • Johnson's algorithm
  • numerical range
  • parametrized eigenvalue problem
  • Runge-Kutta

ASJC Scopus subject areas

  • Analysis


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