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Parasitic roots of various cyclic linear multistep methods

Original post is here eklausmeier.goip.de/blog/2025/12-06-parasitic-roots-of-various-cyclic-linear-multistep-methods.


1. Tischer's formulas

All cyclic linear multistep methods were designed to only have root at 1, and all other parasitic roots to be zero.

See Tischer, Peter E. and Sacks-Davis, Ron: “A New Class of Cyclic Multistep Formulae for Stiff Systems”.

2. Donelson & Hansen formulas

See Donelson III, John and Hansen, Eldon: “Cyclic Composite Multistep Predictor-Corrector Methods”, SIAM Journal on Numerical Analysis, Vol 8, 1971, pp.137—157.

DH1

root real imaginary absolute value
0 1.00000000 0.00000000 1.00000000
1 0.00000000 0.00000000 0.00000000
2 0.00000000 0.00000000 0.00000000

DH2

root real imaginary absolute value
0 -0.31605416 0.94874115 1.00000000
1 -0.31605416 -0.94874115 1.00000000
2 1.00000000 0.00000000 1.00000000

DH3

root real imaginary absolute value
0 1.00000000 0.00000000 1.00000000
1 0.04526646 0.00000000 0.04526646
2 0.00000000 0.00000000 0.00000000

DH4

root real imaginary absolute value
0 1.00000000 0.00000000 1.00000000
1 -0.09090909 0.00000000 0.09090909
2 0.00000000 -0.00000000 0.00000000

DH5

root real imaginary absolute value
0 1.00000000 0.00000000 1.00000000
1 -0.01125522 0.93329189 0.93335976
2 -0.01125522 -0.93329189 0.93335976
3 0.00000000 0.00000000 0.00000000

DH6

root real imaginary absolute value
0 1.00000000 0.00000000 1.00000000
1 -0.37892727 -0.00000000 0.37892727
2 0.13825231 0.00000000 0.13825231
3 0.00000000 0.00000000 0.00000000

3. Mihelcic's formulas

See Matija Mihelčić (Mihelcic) and K. Wingerath: “A(α)-stable Cyclic Composite Multistep Methods of Orders 6 and 7 for Numerical Integration of Stiff Ordinary Differential Equations”, ZAMM, Band 61, 1981, pp.261—264

Mihelcic4

root real imaginary absolute value
0 1.00000000 0.00000000 1.00000000
1 0.50107560 0.00000000 0.50107560
2 -0.42506560 0.00000000 0.42506560

Mihelcic5

root real imaginary absolute value
0 1.00000000 0.00000000 1.00000000
1 -0.38334289 0.00000000 0.38334289
2 -0.21218358 0.00000000 0.21218358

Mihelcic6

root real imaginary absolute value
0 1.00000000 -0.00000000 1.00000000
1 -0.90745413 -0.00000000 0.90745413
2 0.41922349 0.00000000 0.41922349
3 0.06530404 -0.00000000 0.06530404

Mihelcic7

root real imaginary absolute value
0 1.00000000 0.00000000 1.00000000
1 0.16549909 -0.00000000 0.16549909
2 0.09946611 0.00000000 0.09946611
3 0.02567147 -0.00000000 0.02567147
4 -0.00642008 0.00000000 0.00642008