28th December 2025, 58 min read

Stability Regions for eTendler Formulas

In Searching for Tendler-like formulas we developed new cyclic linear multistep formulas for the numerical solution of ordinary differential equations. These cyclic formulas are similar to Tendler's formulas. Order by order they improve on Widlund wedge angle and Widlund distance.

In Die verwendeten zyklischen Formeln im Programm TENDLER we listed the Widlund wedge α and Widlund distance δ for the seven formulas of Joel Tendler, which he developed during his PhD thesis in 1973.

We compute these values by using the QZ algorithm. Also see Stability Regions for BDF and Tendler's Formulas.

Compare Design Notes on System for the Analysis of Order- and Stepsize Changes for Cyclic Composite Multistep Methods. Many stability regions have been plotted here Sand/Osterby (1979).

The graphs were generated by using the C program stabregion2.c. The number of records, command line flag -r, was chosen to be sufficiently small to not generate too many data points but at the same time show enough details of the stability region.

stabregion2 -f eTendler3 -oj -r400
stabregion2 -f eTendler4 -oj -r400
stabregion2 -f eTendler5 -oj -r500
stabregion2 -f eTendler6 -oj -r500
stabregion2 -f eTendler7 -oj -r500
stabregion2 -f eTendler8 -oj -r500
stabregion2 -f eTendler9 -oj -r500