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Parasitic roots of BDF and Tendler's formulas

The parasitic roots $\lambda_i$ of a multistep method are the roots, which are not 1. In below table the root 1 is indexed with 0. The roots are for the matrix polynomial

$$ \rho(\lambda) = A_m \lambda^m + A_{m-1} \lambda^{m-1} + \cdots + A_1 \lambda + A_0 $$

for the multistep method

$$ \sum\limits_{i=0}^{m} A_i Y_{k+i} = h \sum\limits_{i=0}^{m} B_i Y'_{k+i}. $$

Tendler's formulas can be found in Stiffly Stable Integration Process Using Composite Methods.

1. The BDF

BDF1

root real imaginary absolute value
0 1.00000000 0.00000000 1.00000000

BDF2

root real imaginary absolute value
0 1.00000000 0.00000000 1.00000000
1 0.33333333 0.00000000 0.33333333

BDF3

root real imaginary absolute value
0 1.00000000 0.00000000 1.00000000
1 0.31818182 0.28386355 0.42640143
2 0.31818182 -0.28386355 0.42640143

BDF4

root real imaginary absolute value
0 1.00000000 0.00000000 1.00000000
1 0.26926080 0.49200027 0.56086152
2 0.26926080 -0.49200027 0.56086152
3 0.38147841 -0.00000000 0.38147841

BDF5

root real imaginary absolute value
0 1.00000000 -0.00000000 1.00000000
1 0.21004369 0.67686976 0.70871082
2 0.21004369 -0.67686976 0.70871082
3 0.38484682 0.16212131 0.41760076
4 0.38484682 -0.16212131 0.41760076

BDF6

root real imaginary absolute value
0 1.00000000 0.00000000 1.00000000
1 0.14527451 -0.85107039 0.86338027
2 0.14527451 0.85107039 0.86338027
3 0.37615366 -0.28847439 0.47403486
4 0.37615366 0.28847439 0.47403486
5 0.40612327 0.00000000 0.40612327

2. Tendler's cyclic formulas

Tendler3

root real imaginary absolute value
0 1.00000000 -0.00000000 1.00000000
1 0.55371901 0.00000000 0.55371901
2 0.00000000 0.00000000 0.00000000

Tendler4

root real imaginary absolute value
0 1.00000000 0.00000000 1.00000000
1 0.35406989 0.00000000 0.35406989
2 -0.18613598 0.00000000 0.18613598
3 -0.02584359 0.00000000 0.02584359

Tendler5

root real imaginary absolute value
0 1.00000000 0.00000000 1.00000000
1 -0.42931855 0.00000000 0.42931855
2 0.31485709 -0.00000000 0.31485709
3 -0.09291397 -0.00000000 0.09291397
4 0.00602978 -0.00000000 0.00602978

Tendler6

root real imaginary absolute value
0 1.00000000 0.00000000 1.00000000
1 0.52827598 -0.00000000 0.52827598
2 -0.33496615 -0.38537896 0.51060676
3 -0.33496615 0.38537896 0.51060676
4 -0.00423359 0.01617638 0.01672120
5 -0.00423359 -0.01617638 0.01672120

Tendler7

root real imaginary absolute value
0 1.00000000 -0.00000000 1.00000000
1 -0.23749387 -0.62295902 0.66669430
2 -0.23749387 0.62295902 0.66669430
3 0.57052707 0.00000000 0.57052707
4 -0.04931498 0.01790043 0.05246325
5 -0.04931498 -0.01790043 0.05246325
6 0.02535626 0.00000000 0.02535626