, 10 min read
Searching for Tendler-like formulas
1. Baseline
The formulas from Tendler from 1973 are our baseline. Clearly, we want to improve them. So here we summarize their characteristics:
- p is the order
- l is the cycle length
- α is the Widlund wedge angle
- δ is the Widlund distance
| p | l | abs(root) | α(Tendler) | δ(Tendler) | α(Tischer) | δ(Tischer) | |
|---|---|---|---|---|---|---|---|
| 1 | 3 | 0 | 90° | 0 | 90° | 0 | |
| 2 | 3 | 0.333333333 | 90° | 0 | 90° | 0 | |
| 3 | 3 | 0.55371901 | 89.427° | 0.004776 | 90° | 0 | |
| 4 | 3 | 0.35406989 | 80.882047° | 0.244157 | 90° | 0 | |
| 5 | 4 | 0.42931855 | 77.477315° | 1.421472 | 86.649352° | 0.040844 | |
| 6 | 4 | 0.52827598 | 63.245842° | 2.933167 | 76.311756° | 0.280752 | |
| 7 | 4 | 0.66669430 | 33.531759° | 10.179501 | 57.663061° | 0.959187 | |
| 8 | n/a | n/a | n/a | n/a | 22.149242° | 2.534082 |
Linear combinations for Tendler's formulas.
| p=3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|
| 1 1 0 1 -2 -2 |
1 1 0 4 -4 -5 |
1 1 0 10 0 -1 20 -51 -62 1 |
1 20 -1 20 -1 7 20 -78 -68 3 |
1 1 0 20 -31 -10 10 -20 -5 2 |
2. Specific new formulas
In light of all the options, given below, it is quite difficult to choose from all those. Here we focus on Widlund wedge angle and root magnitude.
| p | l | abs(root) | α(new) | δ(new) | Comment |
|---|---|---|---|---|---|
| 3 | 3 | 0.70756795 | 89.72423° | -0.00164 | Worse root modulus |
| 4 | 3 | 0.28351644 | 84.91216° | -0.07106 | Better α, better δ |
| 5 | 3 | 0.48870093 | 77.81321° | -0.42370 | shorter cycle length, better δ |
| 6 | 4 | 0.29026688 | 71.63809° | -1.03854 | better modulus, better α, better δ |
| 7 | 4 | 0.57300425 | 55.13529° | -3.87902 | better α, better δ |
| 8 | 4 | 0.61600197 | 0.09835° | -15.05503 | no other formula with α>0.1 found |
| 9 | 5 | 0.76270334 | 0° | -38.22753 | cycle length > 4, seems to be required |
Linear combinations for specific new formulas.
| p=3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|
| 1 56 -41 44 -63 -64 |
1 3 -7 2 -5 4 |
1 3 -10 4 -9 9 |
1 3 -4 12 -10 5 3 -10 -4 5 |
1 3 -10 3 -7 5 6 -7 8 -4 |
1 91 -67 29 -77 30 97 -97 3 25 |
1 14 -39 47 -25 -32 91 14 -29 -16 86 68 68 -67 -96 |
3. Searching across grids
Order 3:
$ stabregion2 -bBase3 -I3:-10:10:99500 -r600 -W89.7
W= 89.71433, xmin= -0.00148, parasitic=0.7271 1.000000 1.014789 -0.862211 3.382612 -6.186519 -2.667767 1
W= 89.72158, xmin= -0.00164, parasitic=0.7078 1.000000 7.147972 -4.843552 4.402910 -6.810172 -6.294276 2
W= 89.75083, xmin= -0.00138, parasitic=0.7295 1.000000 7.281818 -6.302012 -5.499220 7.713734 7.144969 3
W= 89.70637, xmin= -0.00179, parasitic=0.6956 1.000000 5.669366 -4.084860 -4.445988 6.326142 6.429647 4
W= 89.74108, xmin= -0.00145, parasitic=0.7320 1.000000 -6.943684 3.030275 -4.845535 9.701816 7.428335 5
W= 89.71300, xmin= -0.00176, parasitic=0.7200 1.000000 -4.329238 5.730175 6.691008 -5.083019 -8.718852 6
W= 89.74998, xmin= -0.00134, parasitic=0.7373 1.000000 -5.493899 6.535201 7.225032 -8.391933 -7.634557 7
W= 89.72105, xmin= -0.00166, parasitic=0.7250 1.000000 -6.606269 8.847401 -4.787458 3.865795 5.918802 8
W= 89.70360, xmin= -0.00186, parasitic=0.7016 1.000000 -5.828525 6.328744 6.625133 -6.116368 -9.324740 9
W= 89.70019, xmin= -0.00171, parasitic=0.7020 1.000000 -2.899801 2.550235 6.601982 -9.871467 -7.061955 10
W= 89.77457, xmin= -0.00123, parasitic=0.7466 1.000000 9.380576 -5.098564 5.465408 -9.779463 -8.651397 11
W= 89.71456, xmin= -0.00172, parasitic=0.7018 1.000000 7.267734 -6.059593 4.815367 -6.221483 -6.753425 12
W= 89.72979, xmin= -0.00152, parasitic=0.7339 1.000000 -9.724850 3.321305 -2.928986 6.448823 4.544496 13
W= 89.73788, xmin= -0.00126, parasitic=0.7428 1.000000 -5.691011 7.230664 -2.726001 4.042016 1.481291 14
W= 89.73339, xmin= -0.00157, parasitic=0.7217 1.000000 7.884026 -8.786478 -5.021069 5.073325 6.621230 15
W= 89.76737, xmin= -0.00125, parasitic=0.7423 1.000000 6.167498 -5.282941 2.274352 -3.341372 -2.900454 16
W= 89.71597, xmin= -0.00178, parasitic=0.6976 1.000000 1.478147 -0.672833 4.995894 -8.158646 -9.015963 17
W= 89.71418, xmin= -0.00181, parasitic=0.6992 1.000000 -8.179874 2.592054 4.946106 -9.048167 -9.613549 18
W= 89.74502, xmin= -0.00139, parasitic=0.7499 1.000000 -5.142131 7.961104 -5.770687 4.643132 5.899515 19
W= 89.73222, xmin= -0.00155, parasitic=0.7190 1.000000 7.320413 -3.741168 -3.960796 7.161449 6.076927 20
W= 89.72708, xmin= -0.00157, parasitic=0.7298 1.000000 5.025061 -6.897571 -7.713340 6.601699 8.814953 21
W= 89.76083, xmin= -0.00121, parasitic=0.7479 1.000000 7.278887 -6.149731 -5.172102 8.858583 5.203890 22
W= 89.71201, xmin= -0.00175, parasitic=0.6985 1.000000 -9.549363 6.355211 5.531987 -8.161459 -8.346913 23
W= 89.70198, xmin= -0.00177, parasitic=0.6972 1.000000 -7.463809 5.325495 2.906285 -4.471238 -3.853365 24
W= 89.74603, xmin= -0.00129, parasitic=0.7432 1.000000 -6.293695 5.014876 -2.445970 4.481771 2.292100 25
W= 89.70520, xmin= -0.00145, parasitic=0.7370 1.000000 -3.900991 6.798960 -9.103732 9.830910 3.768878 26
W= 89.70724, xmin= -0.00170, parasitic=0.7161 1.000000 5.620052 -7.572185 -5.497718 5.080016 5.810091 27
W= 89.70072, xmin= -0.00169, parasitic=0.7047 1.000000 -9.218341 7.369757 -6.012441 9.770176 6.445294 28
W= 89.76968, xmin= -0.00129, parasitic=0.7379 1.000000 -9.958895 6.495552 4.953943 -7.660258 -7.920589 29
W= 89.72249, xmin= -0.00143, parasitic=0.7268 1.000000 4.756662 -5.504313 6.145669 -8.928870 -4.402545 30
W= 89.70722, xmin= -0.00144, parasitic=0.7381 1.000000 -3.321221 2.875147 -3.589631 7.244469 1.971108 31
W= 89.71331, xmin= -0.00180, parasitic=0.7323 1.000000 -1.778073 2.627731 -7.387278 4.307442 9.925928 32
W= 89.75530, xmin= -0.00139, parasitic=0.7298 1.000000 -2.599186 2.255055 5.764773 -7.500226 -8.206446 33
W= 89.73312, xmin= -0.00146, parasitic=0.7225 1.000000 -7.523940 7.031001 6.674883 -9.608448 -7.344268 34
W= 89.71894, xmin= -0.00162, parasitic=0.7248 1.000000 3.526626 -4.857279 -4.940031 4.345027 5.401786 35
W= 89.70496, xmin= -0.00194, parasitic=0.7083 1.000000 -6.147503 7.140553 -5.975592 4.322352 9.278829 36
W= 89.75080, xmin= -0.00134, parasitic=0.7339 1.000000 -9.228658 7.075027 -5.533824 9.012720 6.799435 37
Order 4:
echo 1 5 0 5 0 0 | stabregion2 -b Base4 -l- -i-4:4:2 -r500 -W80
W= 80.04684, xmin= -0.24413, parasitic=0.3323 1.000 1.000 -2.000 3.000 -2.000 0.000 1
W= 81.81252, xmin= -0.19378, parasitic=0.6228 1.000 1.000 -2.000 5.000 -4.000 -2.000 2
W= 81.87757, xmin= -0.19662, parasitic=0.6932 1.000 1.000 0.000 1.000 -2.000 -2.000 3
W= 80.97462, xmin= -0.23266, parasitic=0.4090 1.000 1.000 0.000 3.000 -4.000 -4.000 4
W= 83.14511, xmin= -0.15639, parasitic=0.7482 1.000 3.000 -4.000 3.000 -4.000 -2.000 5
W= 81.99847, xmin= -0.18558, parasitic=0.4194 1.000 3.000 -4.000 3.000 -4.000 0.000 6
W= 82.70187, xmin= -0.17225, parasitic=0.6961 1.000 3.000 -2.000 3.000 -4.000 -4.000 7
W= 81.52919, xmin= -0.20730, parasitic=0.4432 1.000 3.000 -2.000 3.000 -4.000 -2.000 8
W= 82.00772, xmin= -0.19780, parasitic=0.5314 1.000 3.000 -2.000 3.000 -2.000 -4.000 9
W= 80.63234, xmin= -0.24699, parasitic=0.3050 1.000 3.000 -2.000 3.000 -2.000 -2.000 10
W= 80.86939, xmin= -0.24514, parasitic=0.3967 1.000 3.000 -2.000 3.000 0.000 -4.000 11
W= 80.14505, xmin= -0.26250, parasitic=0.2417 1.000 3.000 -2.000 5.000 -4.000 -2.000 12
W= 81.87757, xmin= -0.19662, parasitic=0.6932 1.000 3.000 0.000 1.000 -2.000 -2.000 13
W= 81.04040, xmin= -0.24819, parasitic=0.3482 1.000 3.000 0.000 3.000 -2.000 -4.000 14
W= 81.90431, xmin= -0.19478, parasitic=0.4978 1.000 5.000 -4.000 3.000 -4.000 -2.000 15
W= 80.82624, xmin= -0.23705, parasitic=0.3353 1.000 5.000 -4.000 3.000 -2.000 -2.000 16
W= 81.37712, xmin= -0.21642, parasitic=0.4129 1.000 5.000 -4.000 5.000 -4.000 -4.000 17
W= 82.11563, xmin= -0.19145, parasitic=0.5729 1.000 5.000 -2.000 3.000 -4.000 -4.000 18
W= 80.66048, xmin= -0.23783, parasitic=0.3441 1.000 5.000 -2.000 3.000 -4.000 -2.000 19
W= 81.73483, xmin= -0.21116, parasitic=0.4539 1.000 5.000 -2.000 3.000 -2.000 -4.000 20
W= 80.11193, xmin= -0.27484, parasitic=0.2493 1.000 5.000 -2.000 3.000 -2.000 -2.000 21
W= 80.83151, xmin= -0.42500, parasitic=0.3546 1.000 5.000 -2.000 3.000 0.000 -4.000 22
W= 80.65907, xmin= -0.24934, parasitic=0.3071 1.000 5.000 -2.000 5.000 -4.000 -4.000 23
W= 80.06829, xmin= -0.28517, parasitic=0.2540 1.000 5.000 -2.000 5.000 -2.000 -4.000 24
W= 81.87757, xmin= -0.19662, parasitic=0.6932 1.000 5.000 0.000 1.000 -2.000 -2.000 25
W= 80.03662, xmin= -0.25657, parasitic=0.4627 1.000 5.000 2.000 1.000 -2.000 -2.000 26
Searching for order 4 on a wider grid.
$ echo 1 10 0 10 0 0 | stabregion2 -b Base4 -l- -i-10:10:1 -r500 -W80 > stabregion2-base4-10101.txt
. . .
W= 84.91228, xmin= -0.07106, parasitic=0.2835 1.000 3.000 -7.000 2.000 -5.000 4.000 1783
W= 84.93410, xmin= -0.03582, parasitic=0.6720 1.000 2.000 -10.000 4.000 -6.000 9.000 585
W= 85.01008, xmin= -0.08115, parasitic=0.5897 1.000 1.000 -2.000 2.000 -5.000 3.000 70
W= 85.01008, xmin= -0.08115, parasitic=0.5897 1.000 1.000 -2.000 4.000 -10.000 6.000 84
W= 85.01008, xmin= -0.08115, parasitic=0.5897 1.000 2.000 -4.000 2.000 -5.000 3.000 788
W= 85.01008, xmin= -0.08115, parasitic=0.5897 1.000 3.000 -6.000 4.000 -10.000 6.000 1885
W= 85.01008, xmin= -0.08115, parasitic=0.5897 1.000 4.000 -8.000 2.000 -5.000 3.000 3461
W= 85.09335, xmin= -0.06884, parasitic=0.3658 1.000 3.000 -10.000 6.000 -10.000 10.000 1619
W= 85.09878, xmin= -0.07695, parasitic=0.6078 1.000 3.000 -10.000 6.000 -9.000 8.000 1620
. . .
W= 86.34152, xmin= -0.03875, parasitic=0.4773 1.000 2.000 -5.000 3.000 -9.000 8.000 698
W= 86.73807, xmin= -0.03074, parasitic=0.6250 1.000 2.000 -10.000 5.000 -7.000 10.000 586
W= 86.76246, xmin= -0.03807, parasitic=0.6497 1.000 3.000 -10.000 2.000 -4.000 4.000 1610
W= 86.76246, xmin= -0.03807, parasitic=0.6497 1.000 3.000 -10.000 3.000 -6.000 6.000 1613
W= 86.76647, xmin= -0.03296, parasitic=0.4816 1.000 1.000 -3.000 2.000 -5.000 5.000 25
W= 86.76647, xmin= -0.03296, parasitic=0.4816 1.000 1.000 -3.000 4.000 -10.000 10.000 29
W= 87.22705, xmin= -0.02278, parasitic=0.5960 1.000 1.000 -4.000 2.000 -4.000 5.000 4
W= 87.22705, xmin= -0.02278, parasitic=0.5960 1.000 2.000 -8.000 2.000 -4.000 5.000 600
W= 87.22705, xmin= -0.02278, parasitic=0.5960 1.000 2.000 -8.000 4.000 -8.000 10.000 602
Order 5: Searching a order 5 method with cycle length 3:
$ echo 1 10 0 10 0 0 | stabregion2 -b Base5 -l- -i-10:10:3 -r300 -W70
W= 78.68142, xmin= -0.35173, parasitic=0.5851 1.000 3.000 -10.000 3.000 -7.000 8.000 1
W= 76.42265, xmin= -0.49321, parasitic=0.4927 1.000 3.000 -7.000 3.000 -10.000 8.000 2
W= 72.98742, xmin= -0.46278, parasitic=0.7403 1.000 3.000 -7.000 3.000 -10.000 11.000 3
W= 70.12967, xmin= -0.75292, parasitic=0.6413 1.000 6.000 -10.000 3.000 -10.000 5.000 4
W= 70.28984, xmin= -0.71158, parasitic=0.7206 1.000 6.000 -10.000 3.000 -7.000 2.000 5
$ echo 1 10 0 10 0 0 | stabregion2 -b Base5 -l- -i-10:10:2 -r300 -W70
W= 72.33897, xmin= -0.68035, parasitic=0.7201 1.000 2.000 -8.000 4.000 -6.000 6.000 1
W= 70.39636, xmin= -0.58691, parasitic=0.6383 1.000 2.000 -8.000 4.000 -6.000 10.000 2
W= 79.68307, xmin= -0.54250, parasitic=0.6281 1.000 2.000 -6.000 2.000 -6.000 6.000 3
W= 74.98124, xmin= -0.53677, parasitic=0.3954 1.000 2.000 -6.000 2.000 -4.000 4.000 4
W= 77.71542, xmin= -0.33596, parasitic=0.6514 1.000 2.000 -6.000 4.000 -10.000 12.000 5
W= 74.98124, xmin= -0.53677, parasitic=0.3954 1.000 2.000 -6.000 4.000 -8.000 8.000 6
W= 70.70604, xmin= -0.69569, parasitic=0.2687 1.000 2.000 -6.000 4.000 -6.000 6.000 7
W= 72.30615, xmin= -0.64015, parasitic=0.3087 1.000 2.000 -6.000 6.000 -10.000 10.000 8
W= 73.51722, xmin= -0.65303, parasitic=0.5149 1.000 2.000 -4.000 2.000 -8.000 6.000 9
W= 71.33672, xmin= -0.66564, parasitic=0.2951 1.000 2.000 -4.000 4.000 -10.000 8.000 10
W= 70.14594, xmin= -0.70882, parasitic=0.7294 1.000 2.000 -4.000 6.000 -10.000 2.000 11
W= 75.07268, xmin= -0.54117, parasitic=0.3784 1.000 4.000 -10.000 4.000 -10.000 8.000 12
W= 72.65486, xmin= -0.61600, parasitic=0.3577 1.000 4.000 -10.000 4.000 -8.000 8.000 13
W= 71.33672, xmin= -0.66564, parasitic=0.2951 1.000 4.000 -8.000 4.000 -10.000 8.000 14
W= 70.76203, xmin= -0.69721, parasitic=0.4510 1.000 4.000 -8.000 4.000 -8.000 4.000 15
W= 70.19851, xmin= -0.71794, parasitic=0.2348 1.000 4.000 -8.000 4.000 -8.000 6.000 16
In intervals of one:
$ echo 1 10 0 10 0 0 | stabregion2 -b Base5 -l- -i-10:10:1 -r120 -W70
. . .
W= 77.81402, xmin= -0.42360, parasitic=0.4887 1.000 3.000 -10.000 4.000 -9.000 9.000 184
W= 77.83497, xmin= -0.41206, parasitic=0.5767 1.000 2.000 -5.000 3.000 -10.000 9.000 135
W= 77.83497, xmin= -0.41206, parasitic=0.5767 1.000 4.000 -10.000 3.000 -10.000 9.000 296
W= 77.95536, xmin= -0.40394, parasitic=0.5161 1.000 1.000 -3.000 2.000 -5.000 5.000 19
W= 77.95536, xmin= -0.40394, parasitic=0.5161 1.000 3.000 -9.000 4.000 -10.000 10.000 203
W= 77.98148, xmin= -0.36473, parasitic=0.6923 1.000 2.000 -5.000 2.000 -7.000 7.000 130
W= 78.03001, xmin= -0.36923, parasitic=0.5794 1.000 2.000 -6.000 4.000 -10.000 11.000 116
W= 78.12242, xmin= -0.29649, parasitic=0.7324 1.000 2.000 -6.000 3.000 -8.000 10.000 108
W= 78.47928, xmin= -0.30160, parasitic=0.7461 1.000 1.000 -4.000 1.000 -2.000 3.000 4
W= 78.47928, xmin= -0.30160, parasitic=0.7461 1.000 1.000 -4.000 3.000 -6.000 9.000 7
W= 78.47928, xmin= -0.30160, parasitic=0.7461 1.000 2.000 -8.000 3.000 -6.000 9.000 71
W= 78.68359, xmin= -0.35172, parasitic=0.5851 1.000 3.000 -10.000 3.000 -7.000 8.000 179
W= 78.70712, xmin= -0.34502, parasitic=0.6422 1.000 2.000 -8.000 3.000 -6.000 8.000 70
W= 79.11288, xmin= -0.32809, parasitic=0.6703 1.000 3.000 -8.000 3.000 -10.000 10.000 225
W= 79.59837, xmin= -0.30917, parasitic=0.7056 1.000 3.000 -8.000 2.000 -7.000 7.000 221
W= 79.65979, xmin= -0.28701, parasitic=0.6690 1.000 3.000 -10.000 2.000 -5.000 6.000 176
W= 79.68307, xmin= -0.53978, parasitic=0.6281 1.000 1.000 -3.000 2.000 -6.000 6.000 18
W= 79.68307, xmin= -0.53978, parasitic=0.6281 1.000 1.000 -3.000 3.000 -9.000 9.000 24
W= 79.68307, xmin= -0.53978, parasitic=0.6281 1.000 2.000 -6.000 2.000 -6.000 6.000 103
W= 80.37267, xmin= -0.26824, parasitic=0.6803 1.000 2.000 -7.000 2.000 -5.000 6.000 84
W= 80.44412, xmin= -0.33195, parasitic=0.6620 1.000 3.000 -10.000 3.000 -8.000 9.000 178
Searching random for cycle length 3:
$ stabregion2 -bBase5 -I3:-20:20:99500 -r500 -W75
W= 77.94955, xmin= -0.33293, parasitic=0.7045 1.000000 -3.760775 15.031899 5.419764 -10.502505 15.452839 1
W= 75.95454, xmin= -0.42902, parasitic=0.6907 1.000000 -8.070655 19.002001 4.965984 -17.415622 17.529791 2
W= 79.18427, xmin= -0.33883, parasitic=0.6144 1.000000 -3.548929 13.668851 6.447070 -13.718679 16.890842 3
W= 75.31512, xmin= -0.53099, parasitic=0.3871 1.000000 -5.520428 14.051470 -5.100269 12.714603 -10.366682 4
W= 77.87613, xmin= -0.39038, parasitic=0.6025 1.000000 -7.235772 18.481448 5.738588 -18.413200 17.660706 5
W= 75.90191, xmin= -0.48294, parasitic=0.6256 1.000000 -6.314060 14.299655 -5.067656 18.516146 -16.911631 6
W= 78.54997, xmin= -0.32077, parasitic=0.7452 1.000000 1.801247 -4.684607 -5.274913 18.165317 -19.336120 7
W= 77.51294, xmin= -0.43919, parasitic=0.4782 1.000000 -1.582809 5.403277 -7.709742 16.677003 -16.787055 8
W= 77.97953, xmin= -0.41706, parasitic=0.5671 1.000000 2.471351 -10.277907 -5.605553 10.665238 -13.191398 9
W= 75.45818, xmin= -0.46437, parasitic=0.5298 1.000000 -6.404706 18.826354 6.310424 -13.455846 15.748183 10
Order 6: Searching for cycle length 5 methods.
. . .
W= 63.86917, xmin= -1.67741, parasitic=0.4163 1.000 7.000 -10.000 7.000 -10.000 4.000 7.000 -10.000 4.000 4.000 7.000 -10.000 -10.000 11.000 -3.000 2045
W= 63.89942, xmin= -1.92062, parasitic=0.3161 1.000 7.000 -10.000 14.000 -10.000 -3.000 7.000 -10.000 -3.000 11.000 7.000 -10.000 -3.000 11.000 -10.000 4689
W= 64.15015, xmin= -5.10995, parasitic=0.2626 1.000 7.000 -10.000 7.000 -10.000 -10.000 14.000 -10.000 4.000 11.000 14.000 -3.000 -10.000 11.000 -3.000 264
W= 64.18964, xmin= -5.04406, parasitic=0.3019 1.000 14.000 -3.000 14.000 -10.000 -3.000 7.000 -10.000 -3.000 4.000 7.000 -10.000 -3.000 11.000 -10.000 16767
W= 64.23779, xmin= -3.73262, parasitic=0.3314 1.000 14.000 -3.000 7.000 -10.000 -10.000 7.000 -10.000 -3.000 11.000 7.000 -10.000 4.000 11.000 -10.000 14849
W= 64.38508, xmin= -3.89165, parasitic=0.2881 1.000 14.000 -10.000 7.000 -10.000 -10.000 7.000 -10.000 4.000 11.000 14.000 -10.000 -10.000 11.000 -3.000 11010
W= 64.73986, xmin= -3.85047, parasitic=0.2923 1.000 7.000 -10.000 7.000 -10.000 -10.000 14.000 -10.000 4.000 11.000 7.000 -3.000 -3.000 4.000 -3.000 245
W= 65.03750, xmin= -3.62875, parasitic=0.2820 1.000 7.000 -10.000 7.000 -10.000 -10.000 14.000 -3.000 -10.000 11.000 7.000 -10.000 -3.000 11.000 -3.000 356
W= 65.06248, xmin= -2.36716, parasitic=0.2628 1.000 14.000 -3.000 7.000 -10.000 -10.000 7.000 -10.000 4.000 11.000 7.000 -10.000 -3.000 4.000 -3.000 14893
W= 65.08957, xmin= -1.95196, parasitic=0.3167 1.000 7.000 -10.000 7.000 -10.000 -10.000 14.000 -10.000 -3.000 11.000 7.000 -10.000 -3.000 11.000 4.000 182
W= 65.20361, xmin= -2.95019, parasitic=0.1769 1.000 7.000 -3.000 7.000 -10.000 -3.000 7.000 -10.000 -3.000 11.000 7.000 -10.000 4.000 11.000 -10.000 6813
W= 65.22085, xmin= -4.36243, parasitic=0.2452 1.000 14.000 4.000 7.000 -10.000 -10.000 7.000 -10.000 4.000 11.000 7.000 -10.000 -3.000 4.000 -10.000 17607
W= 65.37364, xmin= -3.74950, parasitic=0.2284 1.000 7.000 4.000 7.000 -10.000 -10.000 7.000 -10.000 4.000 11.000 7.000 -10.000 -3.000 4.000 -10.000 9500
W= 65.38512, xmin= -2.09260, parasitic=0.2990 1.000 7.000 -10.000 7.000 -10.000 -10.000 7.000 -10.000 4.000 11.000 14.000 -10.000 -3.000 11.000 -3.000 20
W= 65.39266, xmin= -3.44317, parasitic=0.2216 1.000 7.000 -3.000 7.000 -10.000 -3.000 7.000 -10.000 4.000 4.000 7.000 -3.000 -10.000 11.000 -3.000 6888
W= 66.12236, xmin= -4.30166, parasitic=0.3350 1.000 7.000 -10.000 7.000 -10.000 -3.000 7.000 -10.000 4.000 4.000 7.000 -3.000 -10.000 11.000 -3.000 665
W= 69.19280, xmin= -2.47883, parasitic=0.2498 1.000 7.000 -10.000 7.000 -10.000 -10.000 7.000 -10.000 4.000 11.000 14.000 -10.000 -3.000 11.000 -10.000 19
Order 7: Searching with flag -i and already improving on Tendler's 7th order method:
$ echo 1 10 0 10 0 0 10 0 0 0 | stabregion2 -b Base7 -l- -i-10:10:7 -r400 -W30
W= 37.02937, xmin= -6.70664, parasitic=0.6365 1.000 7.000 -10.000 7.000 -10.000 -3.000 7.000 -10.000 4.000 4.000 1
W= 31.19765, xmin=-11.93934, parasitic=0.5073 1.000 7.000 -10.000 7.000 -10.000 4.000 7.000 -10.000 -10.000 11.000 2
W= 35.81634, xmin= -9.41862, parasitic=0.5793 1.000 14.000 -10.000 7.000 -10.000 -10.000 7.000 -10.000 4.000 11.000 3
W= 31.75908, xmin=-11.19023, parasitic=0.5000 1.000 14.000 -10.000 7.000 -10.000 -3.000 7.000 -10.000 -3.000 11.000 4
Finer grid in searching for order 7 and cycle length 4:
$ echo 1 10 0 10 0 0 10 0 0 0 | stabregion2 -b Base7 -l- -i-10:10:3 -r400 -W30
. . .
W= 48.97241, xmin= -3.91137, parasitic=0.6017 1.000 3.000 -10.000 6.000 -10.000 2.000 6.000 -7.000 5.000 2.000 122
W= 49.01681, xmin= -4.95756, parasitic=0.5478 1.000 12.000 -10.000 3.000 -10.000 2.000 6.000 -10.000 8.000 -4.000 3909
W= 49.02201, xmin= -4.12402, parasitic=0.5892 1.000 3.000 -4.000 6.000 -10.000 5.000 3.000 -10.000 2.000 2.000 1525
W= 49.23966, xmin= -6.41786, parasitic=0.6282 1.000 3.000 -7.000 3.000 -7.000 2.000 12.000 -10.000 8.000 2.000 430
W= 49.34231, xmin= -5.17057, parasitic=0.7009 1.000 3.000 -7.000 3.000 -7.000 2.000 12.000 -10.000 8.000 -7.000 428
W= 49.43582, xmin= -6.07999, parasitic=0.7144 1.000 3.000 -10.000 3.000 -7.000 5.000 12.000 -7.000 5.000 -4.000 16
W= 49.69150, xmin= -6.75076, parasitic=0.6604 1.000 3.000 -7.000 3.000 -7.000 2.000 9.000 -7.000 5.000 2.000 422
W= 49.69246, xmin= -5.58407, parasitic=0.6504 1.000 3.000 -10.000 6.000 -10.000 -1.000 6.000 -7.000 5.000 2.000 114
W= 49.82197, xmin= -5.58658, parasitic=0.6464 1.000 3.000 -7.000 3.000 -7.000 2.000 9.000 -7.000 5.000 -4.000 420
W= 49.89612, xmin= -4.04658, parasitic=0.7261 1.000 3.000 -10.000 3.000 -7.000 5.000 9.000 -7.000 8.000 -10.000 9
W= 49.90940, xmin= -5.72627, parasitic=0.5532 1.000 6.000 -10.000 3.000 -7.000 5.000 3.000 -7.000 2.000 -1.000 1958
W= 50.02874, xmin= -6.28973, parasitic=0.4765 1.000 3.000 -10.000 6.000 -10.000 -4.000 6.000 -10.000 8.000 8.000 109
W= 50.22562, xmin= -6.44879, parasitic=0.7026 1.000 3.000 -7.000 3.000 -7.000 2.000 6.000 -4.000 2.000 -1.000 411
W= 50.23991, xmin= -5.84158, parasitic=0.5545 1.000 9.000 -7.000 3.000 -10.000 -1.000 6.000 -10.000 11.000 -1.000 3453
W= 50.25552, xmin= -6.85592, parasitic=0.6824 1.000 9.000 -4.000 3.000 -10.000 -4.000 6.000 -10.000 11.000 -1.000 3660
W= 50.36004, xmin= -7.40042, parasitic=0.6057 1.000 9.000 -4.000 3.000 -10.000 -4.000 6.000 -10.000 11.000 2.000 3661
W= 50.37801, xmin= -5.59422, parasitic=0.6417 1.000 3.000 -7.000 3.000 -7.000 2.000 9.000 -10.000 8.000 -1.000 415
W= 50.50615, xmin= -5.87301, parasitic=0.6260 1.000 3.000 -7.000 3.000 -7.000 2.000 12.000 -10.000 8.000 -1.000 429
W= 51.38923, xmin= -6.05126, parasitic=0.6540 1.000 3.000 -7.000 3.000 -7.000 2.000 9.000 -7.000 5.000 -1.000 421
W= 51.89876, xmin= -3.69142, parasitic=0.5493 1.000 3.000 -10.000 3.000 -7.000 5.000 9.000 -10.000 11.000 -7.000 7
W= 52.02594, xmin= -7.74860, parasitic=0.6336 1.000 3.000 -7.000 3.000 -7.000 -1.000 6.000 -10.000 11.000 2.000 403
W= 53.38128, xmin= -6.73675, parasitic=0.6485 1.000 6.000 -10.000 3.000 -7.000 5.000 6.000 -10.000 -1.000 -1.000 1963
W= 55.13527, xmin= -3.87834, parasitic=0.5730 1.000 3.000 -10.000 3.000 -7.000 5.000 6.000 -7.000 8.000 -4.000 5
Random search:
$ stabregion2 -bBase7 -I4:-120:120:48900 -r2960 -W50
W= 50.23919, xmin= -6.26976, parasitic=0.6525 1.000000 -88.759541 95.674195 31.729317 -96.859122 26.162025 81.615113 -119.764931 52.733653 -30.670380 1
$ stabregion2 -bBase7 -I4:-70:70:9995100 -W51 -r600
W= 51.42985, xmin= -6.84906, parasitic=0.6407 1.000000 34.194642 -57.338007 -17.001913 52.394410 -13.620043 45.979845 -48.756102 40.782862 -11.033654 1
4. Various new formulas
Below output shows the various possibilities.
Order 3: slightly better α, better δ
$ echo 1 11 -9 7 -10 -10 | stabregion2 -b Base3 -l- | perl -ne 'print if 1..4 or eof'
Parasitic roots of Linear combination
0 1.00000000 0.00000000 1.00000000
1 0.74921933 -0.00000000 0.74921933
2 0.00436052 -0.00000000 0.00436052
19 -0.00014033 -0.10488920, -0.00117 89.79150
Order 4: better α, better δ, slightly better root
echo 1.000 -5.542 14.498 -18.122 42.367 -37.117 | stabregion2 -bBase4 -l- -r500 | perl -ne 'print if 1..5 or eof'
Parasitic roots of Linear combination
0 1.00000000 0.00000000 1.00000000
1 0.26408559 0.19557790 0.32862123
2 0.26408559 -0.19557790 0.32862123
3 0.16860831 0.00000000 0.16860831
499 1.26751184 -0.01658136, -0.06096 85.35055
Order 5: slightly better α, better δ, slightly worse root magnitude
echo 1 3 -10 4 -9 9 | stabregion2 -b Base5 -l- -r500 | perl -ne 'print if 1..6 or eof'
Parasitic roots of Linear combination
0 1.00000000 -0.00000000 1.00000000
1 0.45985170 0.16542372 0.48870093
2 0.45985170 -0.16542372 0.48870093
3 -0.08860554 0.10430695 0.13686081
4 -0.08860554 -0.10430695 0.13686081
499 0.83703643 -0.02068931, -0.42370 77.81331
Order 6: better α, slightly better δ, better root, but one higher cycle length!
$ echo 1 20 -9 20 -10 -12 10 -25 -15 18 | ~/c/stabregion2 -r400 -b Base6 -l- | perl -ne 'print if 1..6 or eof'
0 1.00000000 0.00000000 1.00000000
1 -0.13526122 -0.32183715 0.34910564
2 -0.13526122 0.32183715 0.34910564
3 0.05457185 -0.14009988 0.15035313
4 0.05457185 0.14009988 0.15035313
5 0.01265647 0.00000000 0.01265647
399 -1.01316343 2.07749451, -3.81061 59.92844
$ echo 1 19.9 -9.75 20.105 -10 -11.58 9.6105 -25.83 -14.6 19.9 | ~/c/stabregion2 -r300 -b Base6 -l- | perl -ne 'print if 1..6 or eof'
0 1.00000000 0.00000000 1.00000000
1 -0.19432134 -0.24905256 0.31589232
2 -0.19432134 0.24905256 0.31589232
3 0.05240366 0.18042794 0.18788397
4 0.05240366 -0.18042794 0.18788397
5 0.01021887 -0.00000000 0.01021887
299 -1.05960992 2.13752343, -3.63381 63.09781
$ echo 1 7 -10 7 -10 -10 7 -10 4 11 14 -10 -3 11 -10 | stabregion2 -b Base6 -l- -r700 | perl -ne 'print if 1..7 or eof'
Parasitic roots of Linear combination
0 1.00000000 -0.00000000 1.00000000
1 0.24980571 -0.00000000 0.24980571
2 -0.14328484 0.00000000 0.14328484
3 0.05318928 0.06825981 0.08653612
4 0.05318928 -0.06825981 0.08653612
5 -0.01171677 0.00000000 0.01171677
699 -0.49908435 1.32693994, -2.47902 69.19266
Order 7: better α, better δ, better root
$ echo 1 2 1 25 -10 -40 2 -3 1 3 20 10 -40 -30 -10 | ~/c/stabregion2 -r500 -b Base7 -l- | perl -ne 'print if 1..7 or eof'
0 1.00000000 0.00000000 1.00000000
1 -0.08617372 0.55436985 0.56102748
2 -0.08617372 -0.55436985 0.56102748
3 0.38639102 -0.00000000 0.38639102
4 -0.12271706 -0.00000000 0.12271706
5 -0.01311481 0.00000000 0.01311481
6 0.00233200 0.00000000 0.00233200
499 0.07850982 0.95177010, -11.72180 0.00000
$ echo 1 3 -1 1 0 -2 3 0 -3 4 3 0 -3 4 -4 | ~/c/stabregion2 -r500 -b Base7 -l- | perl -ne 'print if 1..7 or eof'
0 1.00000000 0.00000000 1.00000000
1 0.11913730 0.47735727 0.49199966
2 0.11913730 -0.47735727 0.49199966
3 0.28233408 -0.00000000 0.28233408
4 -0.05854063 -0.00000000 0.05854063
5 -0.03078690 -0.00000000 0.03078690
6 0.00246362 -0.00000000 0.00246362
499 0.08631458 -0.91461418, -82.23638 0.45520
$ echo 1 3 -10 3 -7 5 6 -7 8 -4 | stabregion2 -b Base7 -l- -r600 | perl -ne 'print if 1..8 or eof'
Parasitic roots of Linear combination
0 1.00000000 0.00000000 1.00000000
1 -0.57300425 0.00000000 0.57300425
2 0.44845186 -0.00000000 0.44845186
3 -0.36924474 -0.00000000 0.36924474
4 -0.08390465 -0.07829099 0.11475831
5 -0.08390465 0.07829099 0.11475831
6 0.03519651 0.00000000 0.03519651
599 0.00000000 -0.00261799, -3.87902 55.13542
$ echo 1 -503 199 859 214 -593 -241 393 -52 -387 -409 624 22 -277 574 -393 834 -410 -379 861 -6 | stabregion2 -bBase7 -l- -r500 | perl -ne 'print if 1..8 or eof'
Parasitic roots of Linear combination
0 1.00000000 0.00000000 1.00000000
1 0.57974563 -0.00000000 0.57974563
2 0.32478676 -0.09621522 0.33873856
3 0.32478676 0.09621522 0.33873856
4 -0.00833937 0.04158511 0.04241305
5 -0.00833937 -0.04158511 0.04241305
6 0.00050814 -0.00000000 0.00050814
499 -0.00000000 -0.00209440, -9.31106 34.84493
Order 8:
$ echo 1 36 -43 102 -221 -72 567 -684 587 378 | stabregion2 -bBase8 -l- -r76200 | perl -ne 'print if 1..9 or eof'
Parasitic roots of Linear combination
0 1.00000000 -0.00000000 1.00000000
1 0.45444538 0.14823495 0.47801067
2 0.45444538 -0.14823495 0.47801067
3 0.02754920 -0.44960947 0.45045270
4 0.02754920 0.44960947 0.45045270
5 -0.11768977 0.00000000 0.11768977
6 -0.05579669 0.00000000 0.05579669
7 0.03286398 -0.00000000 0.03286398
76199 -0.50977423 0.00009225, -21.18735 0.00493
Cycle length 5:
$ echo 1 -13.5 31 -19.5 29 -12 -47.5 60 14.25 -76.74 73 17.6 23 8.9 -88.7 | stabregion2 -bBase8 -l- -r176200 | perl -ne 'print if 1..9 or eof'
Parasitic roots of Linear combination
0 1.00000000 -0.00000000 1.00000000
1 -0.55384672 -0.00000000 0.55384672
2 0.20514106 0.50119562 0.54155324
3 0.20514106 -0.50119562 0.54155324
4 0.50676510 0.00000000 0.50676510
5 -0.04784472 -0.00000000 0.04784472
6 -0.00631625 -0.00863555 0.01069896
7 -0.00631625 0.00863555 0.01069896
176199 -0.00000000 -0.00000713, -16.97737 6.71800
$ echo 1 10.413 -23.757 -15 22.5 -9.3 -36.6 46.1 10.97 -59 56.1 13.5 17.7 6.84 -68.316038 | stabregion2 -bBase8 -l- -r6900 | perl -ne 'print if 1..9 or eof'
Parasitic roots of Linear combination
0 1.00000000 0.00000000 1.00000000
1 -0.55764352 -0.00000000 0.55764352
2 0.21168088 0.50209167 0.54488975
3 0.21168088 -0.50209167 0.54488975
4 0.50305785 -0.00000000 0.50305785
5 -0.04747804 -0.00000000 0.04747804
6 -0.00630592 -0.00861259 0.01067433
7 -0.00630592 0.00861259 0.01067433
6899 0.00000000 -0.00018212, -17.07914 6.84418
Order 9: cycle length=4
$ echo 1 3.65 -11 38 -70 68 -46 -23 50 -13.5 | stabregion2 -bBase9 -l- -r500 | perl -ne 'print if 1..10 or eof'
Parasitic roots of Linear combination
0 1.00000000 0.00000000 1.00000000
1 0.69341362 0.23294720 0.73149630
2 0.69341362 -0.23294720 0.73149630
3 0.11882161 -0.29545535 0.31845319
4 0.11882161 0.29545535 0.31845319
5 -0.14232428 0.08346739 0.16499396
6 -0.14232428 -0.08346739 0.16499396
7 0.02232496 -0.05087849 0.05556100
8 0.02232496 0.05087849 0.05556100
499 -0.00000000 -0.00314159, -36.25919 0.00000
$ echo 1.000000 27.094825 -139.893728 67.030957 -128.854947 224.182932 191.776479 -95.370077 26.617924 -67.920828 | stabregion2 -bBase9 -l- -r59900 | perl -ne 'print if 1..10 or eof'
Parasitic roots of LinearCombination
0 1.00000000 0.00000000 1.00000000
1 -0.43729107 0.59023631 0.73457633
2 -0.43729107 -0.59023631 0.73457633
3 0.34487625 -0.50834727 0.61429356
4 0.34487625 0.50834727 0.61429356
5 -0.16234653 0.00000000 0.16234653
6 -0.10492970 0.00000000 0.10492970
7 0.02817343 0.03911910 0.04820836
8 0.02817343 -0.03911910 0.04820836
59899 0.00000000 -0.00002622, -13.57947 0.00281