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Stability Mountains for Hansen's Formulas
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Here we analyze the three methods from Eldon Hansen (1969). These are three cyclic linear multistep methods of order $p=5$ and $p=7$. The first one only needs 3 starting values for order 5. The second one only need 4 starting values and reaches order 7.
Both methods break the first Dahlquist barrier. They can do that because the Dahlquist barrier only holds for "simple" multistep methods, not for cyclic composite methods.
Theorem (First Dahlquist barrier): A linear multistep method with $k$ starting value can only be stable and have consistency order $p$ if
Proof: The shortest proof can be found in Butcher (2006), Thm 1.3.
Bibliography.
- Butcher, J.C.: “General linear methods, Acta Numerica, Vol. 15, May 2006, pp.157–256
- Hansen, E.: “Cyclic Composite Multistep Predictor-Corrector Methods” ACM '69: Proceedings of the 1969 24th national conference, pp.135–139
- Donelson, J., Hansen, E.: “Cyclic Composite Multistep Predictor-Corrector Methods, SIAM Journal on Numerical Analysis, Vol. 8, No. 1, March 1971, pp.137–157
- Stability Regions for Donelson and Hansen Formulas
- Stability Mountains for Donelson & Hansen 1-6
1. The formulas
Order 5.
Order 7.
The two cycles were not designed to be $A[\alpha]$-stable, and therefore are not $A[\alpha]$-stable.
The order 5 method.
Hansen5, p=5, k=3, l=3
0.0000 0.0000 0.0000
-57.0000 3249.0000 0.0000
24.0000 -1368.0000 0.0000
33.0000 -4041.0000 -57.0000
0.0000 2160.0000 24.0000
0.0000 0.0000 33.0000
-1.0000 0.0000 0.0000
24.0000 -1050.0000 0.0000
57.0000 -4041.0000 -1.0000
10.0000 1368.0000 24.0000
0.0000 753.0000 57.0000
0.0000 0.0000 10.0000
rho_0 0.000000000 0.000000000 0.000000000
rho_1 0.000000000 0.000000000 0.000000000
rho_2 0.000000000 0.000000000 0.000000000
rho_3 0.000000000 0.000000000 0.000000000
rho_4 0.000000000 0.000000000 0.000000000
rho_5 0.000000000 0.000000000 0.000000000
rho_6 -0.005555556 -0.013912037 -0.005555556 <-----
Roots at zero and infinity.
parasitic roots of Hansen5
nr real imag abs 3-th root
0 1.00000000 0.00000000 1.00000000 1.00000000
1 0.00000000 0.00000000 0.00000000 0.00002030
2 0.00000000 0.00000000 0.00000000 0.00000000
radius at infinity of Hansen5
nr real imag abs 3-th root
0 -70.18221266 0.00000000 70.18221266 4.12485816
1 0.20287329 0.00000000 0.20287329 0.58759076
2 -0.00097936 0.00000000 0.00097936 0.09930721
The order 7 method.
Hansen7, p=7, k=4, l=4
0.0000 0.0000 0.0000 0.0000
-1360.0000 184960.0000 0.0000 0.0000
-1350.0000 183600.0000 0.0000 0.0000
2160.0000 -992790.0000 -1360.0000 414984070.0000
550.0000 404720.0000 -1350.00001265001840.0000
0.0000 219510.0000 2160.0000-1380669030.0000
0.0000 0.0000 550.0000-446827600.0000
0.0000 0.0000 0.0000 147510720.0000
-9.0000 0.0000 0.0000 0.0000
456.0000 -51009.0000 0.0000 0.0000
2376.0000 -374904.0000 -9.0000 0.0000
1656.0000 149256.0000 456.0000-108800637.0000
141.0000 507576.0000 2376.0000-1127179512.0000
0.0000 59301.0000 1656.0000-1155484872.0000
0.0000 0.0000 141.0000 100081848.0000
0.0000 0.0000 0.0000 44607033.0000
rho_0 0.000000000 0.000000000 0.000000000 0.000000000
rho_1 0.000000000 0.000000000 0.000000000 0.000000000
rho_2 0.000000000 0.000000000 0.000000000 0.000000000
rho_3 0.000000000 0.000000000 0.000000000 0.000000000
rho_4 0.000000000 0.000000000 0.000000000 0.000000000
rho_5 0.000000000 0.000000000 0.000000000 0.000000000
rho_6 0.000000000 0.000000000 0.000000000 0.000000000
rho_7 0.000000000 0.000000000 0.000000000 0.000000000
rho_8 -0.000974026 -0.001794744 -0.000974026 -0.003714201 <-----
Roots at zero and infinity.
parasitic roots of Hansen7
nr real imag abs 4-th root
0 1.00000001 0.00000000 1.00000001 1.00000000
1 -0.00000034 -0.00000016 0.00000038 0.02477842
2 0.00000034 0.00000016 0.00000037 0.02469975
3 0.00000000 0.00000000 0.00000000 0.00000000
radius at infinity of Hansen7
nr real imag abs 4-th root
0 2937.95103433 0.00000000 2937.95103433 7.36225979
1 -3.21757016 0.00000000 3.21757016 1.33931276
2 -0.02534791 0.00000000 0.02534791 0.39901161
3 0.00003567 0.00000000 0.00003567 0.07728341
2. Stability regions
Using stabregion3 to graph the stability regions.
stabregion3 -f Hansen5 -oj -r300
stabregion3 -f Hansen7 -oj -r300
3. Stability mountains
Below is the output of:
stabregion3 -f Hansen5 -o3
Hansen5 stability mountain. So visually it is obvious that the method is not $A_\infty^0[\alpha]$-stable. It was not designed to be $A_\infty^0[\alpha]$-stable.
Output for the order 7 method.
stabregion3 -f Hansen7 -o3 -L30:-2:1:2 -r300
This method is not $A_\infty^0[\alpha]$-stable. It was not designed to be so.