Algebraically stable general linear methods and the G-matrix

Laura L Hewitt, Adrian T Hill

Research output: Contribution to journalArticlepeer-review

17 Citations (SciVal)


The standard algebraic stability condition for general linear methods (GLMs) is considered in a modified form, and connected to a branch of Control Theory concerned with the discrete algebraic Riccati equation (DARE). The DARE theory shows that, for an algebraically stable method, there is a minimal G-matrix, G *, satisfying an equation, rather than an inequality. This result, and another alternative reformulation of algebraic stability, are applied to construct new GLMs with 2 steps and 2 stages, one of which has order p=4 and stage order q=3. The construction process is simplified by method-equivalence, and Butcher’s simplified order conditions for the case p≤q+1.
Original languageEnglish
Pages (from-to)93-111
Number of pages19
JournalBIT Numerical Mathematics
Issue number1
Early online date30 Jan 2009
Publication statusPublished - Mar 2009


  • Discrete algebraic Riccati equation
  • General linear methods
  • Algebraic stability


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