lajos/2012autumn

Travels

  • October 15-20: Wolfram Technology Conference, Champaign, Illinois
  • October 27- November 2: Vacation leave to Hungary

Paper to be written and submitted

Absolute monotonicity of rational functions, with David.

– Counterexample to Griend-Kraaijevanger conjecture: IRK, s=3, p=2

– Exact values for the radius of absolute monotonicity of the stability function in the following classes:

  • IRK, s=2, p=3 (fully human-readable proof)
  • IRK, s=3, p=6 (fully human-readable proof)
  • IRK, s=3, p=5 (partly Mathematica proof)
  • IRK, s=4, p=7 (partly Mathematica proof, to be finished)
  • SDIRK, s=3, p=5: such method does not exist
  • SDIRK, s=3, p=4 (human-readable proof)
  • SDIRK, s=3, p=3 (human-readable proof)
  • SDIRK, s=3, p=2 (fully human-readable proof)
  • SDIRK, s=4, p=2 (fully human-readable proof)

– Further results (that might sometimes be useful):

  • Bounds on the coefficients of a polynomial having all derivatives non-negative
  • Describe the proof of the uniqueness of the numerator of the stability function for \(p=2\) and \(3\le s \le 8\), including the “Linear combination lemma”.

To be investigated

  • Following Igor’s work (based on a paper of Hundsdorfer-Mozartova-Spijker), what can we say about the non-negativity of the sequences arising in connection with multistep methods and generated by linear recursions?

  • Investigate David’s observations on the monotonicity properties and on the patterns in connection with the numbers \(R_{s, p, k}\), i.e., the optimal threshold factors for explicit general linear methods with \(s\) stages, \(k\) steps, and order at least \(p\).

  • Work with David and Matteo on finding the maximal disks/ellipses contained in the spectrum in connection with predicting upper bounds on the effective time steps.