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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