The region examined here is the Spherical Equilateral Right triangle under Neumann Boundary conditions. Side lengths are pi/2. This triangle was one of our test cases to determine the accuracy of MATLAB's finite element solver in spherical space and whether we could accurately assume that we could treat the remainder of the error as a portion of the geometric series. This is because this triangle is one of the regions for which the eigenvalue spectrum is known: the ith distinct eigenvalue is equal to 2(2i-1)(i-1), and has multiplicity i. It confirmed both that the MATLAB solver was very accurate in Spherical Space, and that our estimate of the remaining error resulted in even more accurate values. Together with our other known cases, this triangle also helped us to determine that the accuracy of the built in solver begins to deteriorate after the first approximately 250 eigenvalues.
Here are the counting graphs for the eigenvalue
spectrum, using the true values since the spectrum is known. The first approximately 1250
values were used here.
Here are the difference graphs for the eigenvalue
spectrum, using the true values since the spectrum is known. The first approximately 1250
values were used here.
Below is the Eigenvalue spectrum for the Spherical Equilateral Right
triangle. These are the calculated and predicted eigenvalues, accompanied by their true values. This also demonstrates
that more than three refinements of MATLAB's auto-generated mesh should be done for regions where the true value
of the spectrum is not known.
| Spherical Equilateral Right Neumann Triangle | ||||||
|---|---|---|---|---|---|---|
| Refinements | Predicted | True Value | ||||
| 0 | 0 | 0 | 0 | -0.027021132 | 0 | |
| 6.022473935 | 6.005761702 | 6.001453884 | 6.000364586 | 5.999981864 | 6 | |
| 6.051407985 | 6.013012945 | 6.003271028 | 6.000819345 | 5.999957952 | 6 | |
| 20.25318555 | 20.06484722 | 20.01635702 | 20.00410129 | 19.99979528 | 20 | |
| 20.47498097 | 20.11960043 | 20.030015 | 20.00751455 | 19.99960907 | 20 | |
| 20.52762853 | 20.13329245 | 20.03349413 | 20.00838938 | 19.99956889 | 20 | |
| 43.1316214 | 42.28999988 | 42.07323929 | 42.01837384 | 41.99909701 | 42 | |
| 43.8757926 | 42.47345759 | 42.11895085 | 42.02979249 | 41.99846694 | 42 | |
| 44.1253972 | 42.53550969 | 42.13450571 | 42.03368731 | 41.99826502 | 42 | |
| 44.38017811 | 42.59630266 | 42.1495303 | 42.03743369 | 41.99804884 | 42 | |
| 75.29896205 | 72.84139801 | 72.21209383 | 72.05317485 | 71.99733907 | 72 | |
| 77.07626547 | 73.28449968 | 72.32306057 | 72.0809453 | 71.99587872 | 72 | |
| 78.14288355 | 73.53850448 | 72.38575485 | 72.09656718 | 71.99496184 | 72 | |
| 78.54672934 | 73.6367273 | 72.41000264 | 72.10261061 | 71.99460921 | 72 | |
| 79.24744987 | 73.80032674 | 72.44995673 | 72.11252443 | 71.99396846 | 72 | |
| 117.6122072 | 111.938759 | 110.4883701 | 110.1224134 | 109.9938355 | 110 | |
| 121.1982679 | 112.8246661 | 110.7097894 | 110.1778047 | 109.9908932 | 110 | |
| 123.7572266 | 113.4586929 | 110.866706 | 110.2169071 | 109.988602 | 110 | |
| 124.7378116 | 113.6908614 | 110.9248748 | 110.2315033 | 109.987889 | 110 | |
| 125.063357 | 113.7537964 | 110.9393762 | 110.235044 | 109.9875787 | 110 | |
| 126.7967412 | 114.2085195 | 111.0529406 | 110.2634203 | 109.9860245 | 110 | |
| 171.1135306 | 159.8340868 | 156.9653951 | 156.2420025 | 155.9878404 | 156 | |
| 177.4731258 | 161.4175636 | 157.3601667 | 156.3406479 | 155.9824426 | 156 | |
| 181.629507 | 162.4753912 | 157.6243247 | 156.4066538 | 155.9788282 | 156 | |
| 184.8327994 | 163.216984 | 157.8083992 | 156.4527059 | 155.9763865 | 156 | |
| 185.1158149 | 163.3631838 | 157.8442251 | 156.4615284 | 155.9757215 | 156 | |
| 186.9522811 | 163.8063933 | 157.9531464 | 156.4885394 | 155.9739535 | 156 | |
| 190.7003731 | 164.6431926 | 158.1497681 | 156.5368743 | 155.9701882 | 156 | |
| 237.9921033 | 216.9952988 | 211.7543671 | 210.4392823 | 209.9772306 | 210 | |
| 247.8197193 | 219.6017761 | 212.4111204 | 210.6039141 | 209.9689569 | 210 | |
| 254.1559756 | 221.2806061 | 212.8282727 | 210.7078883 | 209.9628966 | 210 | |
| 259.8683006 | 222.6887228 | 213.1707578 | 210.7929047 | 209.9574521 | 210 | |
| 261.3727012 | 223.1607139 | 213.301111 | 210.8264278 | 209.9569542 | 210 | |
| 263.5480284 | 223.8695359 | 213.4683447 | 210.8673221 | 209.9534596 | 210 | |
| 265.6450881 | 224.5501125 | 213.6254491 | 210.9055662 | 209.9499424 | 210 | |
| 268.0263411 | 224.9512275 | 213.7443604 | 210.9370309 | 209.9506829 | 210 | |
| 318.3159965 | 283.625032 | 274.9131603 | 272.7292702 | 271.961966 | 272 | |
| 332.8021854 | 287.6803215 | 275.9356711 | 272.985558 | 271.9490433 | 272 | |
| 341.0452905 | 290.0566677 | 276.5345206 | 273.1355559 | 271.9413385 | 272 | |
| 346.9598872 | 292.652752 | 277.1761862 | 273.2941265 | 271.9301748 | 272 | |
| 352.342888 | 293.6227812 | 277.4419496 | 273.3640034 | 271.9312276 | 272 | |
| 355.4787413 | 294.2476044 | 277.5344185 | 273.3825485 | 271.9237996 | 272 | |
| 360.5137388 | 295.8075453 | 277.9349557 | 273.4816536 | 271.9169974 | 272 | |
| 364.2046668 | 295.9258686 | 277.961401 | 273.4902485 | 271.9193206 | 272 | |
| 365.304924 | 298.4459868 | 278.6067504 | 273.6509143 | 271.909694 | 272 | |
| 417.4546348 | 360.7771282 | 346.6789392 | 343.1695511 | 341.9365367 | 342 | |
| 434.7783981 | 366.2874707 | 348.0800322 | 343.5213092 | 341.9196137 | 342 | |
| 438.4027501 | 370.1472881 | 349.0318599 | 343.7587294 | 341.9060285 | 342 | |
| 448.8529086 | 372.6974969 | 349.6881483 | 343.9232409 | 341.8977556 | 342 | |
| 452.7876779 | 374.7350528 | 350.2047097 | 344.0522724 | 341.8906294 | 342 | |
| 462.2846611 | 376.7052282 | 350.6976902 | 344.1752998 | 341.8836747 | 342 | |
| 464.2266032 | 378.6852663 | 351.111677 | 344.2729679 | 341.870205 | 342 | |
| 473.5177878 | 379.194184 | 351.2857514 | 344.3218416 | 341.8750898 | 342 | |
| 477.8248314 | 380.5401207 | 351.713115 | 344.4320108 | 341.8738136 | 342 | |
| 491.306572 | 383.8083331 | 352.4821954 | 344.6214732 | 341.8596287 | 342 | |
| 527.9555909 | 447.2686654 | 426.8062077 | 421.7019998 | 419.9086495 | 420 | |
| 544.7527604 | 454.7905019 | 428.8063015 | 422.2098506 | 419.8922045 | 420 | |
| 556.16824 | 459.7510277 | 430.0256193 | 422.5127261 | 419.8730904 | 420 | |
| 565.6607968 | 465.1336374 | 431.3560815 | 422.8430549 | 419.8520249 | 420 | |
| 571.9647613 | 468.9432575 | 432.3703342 | 423.1002631 | 419.8432475 | 420 | |
| 575.5765583 | 470.3852505 | 432.7508952 | 423.1948267 | 419.8373267 | 420 | |
| 578.9785834 | 473.1292163 | 433.37341 | 423.3453069 | 419.8219588 | 420 | |
| 588.3124441 | 475.1989406 | 433.8421301 | 423.4622249 | 419.8152719 | 420 | |
| 603.5863143 | 478.095471 | 434.3943071 | 423.5884819 | 419.7918831 | 420 | |
| 619.0324783 | 482.2507201 | 435.3482487 | 423.8207906 | 419.7706479 | 420 | |
| 654.6850222 | 485.5053342 | 436.2113916 | 424.0407706 | 419.7646543 | 420 | |
| 657.9245216 | 545.3577806 | 515.8249794 | 508.4566667 | 505.8678291 | 506 | |
| 675.1084658 | 560.2760293 | 519.5306892 | 509.3811531 | 505.8151398 | 506 | |
| 690.3040127 | 563.1620968 | 520.3421836 | 509.590109 | 505.8123953 | 506 | |
| 701.6791494 | 568.5512383 | 521.7982082 | 509.9590219 | 505.7993543 | 506 | |
| 706.8464495 | 576.4346726 | 523.747045 | 510.442935 | 505.7685702 | 506 | |
| 712.7145366 | 580.3222304 | 524.5201831 | 510.6197337 | 505.7358466 | 506 | |
| 718.1572101 | 581.9660267 | 525.0621194 | 510.7659397 | 505.7430138 | 506 | |
| 738.9153091 | 585.3452526 | 525.7317981 | 510.9178592 | 505.7130199 | 506 | |
| 762.7755784 | 588.6917268 | 526.5090861 | 511.1144349 | 505.7055639 | 506 | |
| 772.8486914 | 590.6161457 | 527.0890198 | 511.2571013 | 505.6945975 | 506 | |
| 786.2187106 | 594.7247954 | 527.9825322 | 511.4887596 | 505.6937152 | 506 | |
| 809.6756322 | 602.8204807 | 530.193181 | 512.0329345 | 505.6523788 | 506 | |
| 822.7214113 | 655.5586811 | 613.7884521 | 603.4416981 | 599.8063927 | 600 | |
| 839.087962 | 672.814485 | 618.152937 | 604.5369241 | 599.7529731 | 600 | |
| 852.6884094 | 681.6366196 | 620.2290584 | 605.0466049 | 599.712289 | 600 | |
| 862.0856209 | 690.0072198 | 622.5270006 | 605.6322052 | 599.6962624 | 600 | |
| 880.5949464 | 694.4555727 | 623.7350001 | 605.9386329 | 599.6859252 | 600 | |
| 894.8034842 | 701.6932216 | 625.4115554 | 606.3370167 | 599.6352267 | 600 | |
| 906.5816063 | 707.8939957 | 626.877938 | 606.7069187 | 599.6198831 | 600 | |
| 919.9836068 | 711.7723659 | 627.7837739 | 606.9091085 | 599.5748487 | 600 | |
| 931.4419853 | 712.4955564 | 628.1227664 | 607.0193139 | 599.6046703 | 600 | |
| 945.3014297 | 715.0167014 | 628.7523923 | 607.1748298 | 599.5936089 | 600 | |
| 962.0036604 | 718.6445839 | 629.6526928 | 607.3934049 | 599.5726616 | 600 | |
| 968.5987905 | 723.6517661 | 631.0033571 | 607.7493676 | 599.5791384 | 600 | |
| 981.7565754 | 736.3247094 | 633.8057237 | 608.4206679 | 599.501694 | 600 | |
| 993.9730951 | 776.7076088 | 720.6168233 | 706.6508755 | 701.7439757 | 702 | |
| 1022.02989 | 795.3604556 | 725.6614239 | 707.9386778 | 701.7118367 | 702 | |
| 1035.53577 | 812.2785944 | 729.2022989 | 708.7798367 | 701.6044573 | 702 | |
| 1062.464401 | 825.0624992 | 732.5867472 | 709.6263604 | 701.5592877 | 702 | |
| 1078.490786 | 830.7928873 | 734.1242196 | 710.0206518 | 701.5519254 | 702 | |
| 1098.819412 | 841.2754739 | 736.6684454 | 710.6288897 | 701.4799589 | 702 | |
| 1118.809243 | 842.7454391 | 737.1722213 | 710.7916457 | 701.5228985 | 702 | |
| 1169.711118 | 850.5704467 | 739.5285926 | 711.3317587 | 701.4248738 | 702 | |
| 1185.916501 | 854.356565 | 739.6598777 | 711.3949882 | 701.4641921 | 702 | |
| 1192.436173 | 855.5472307 | 740.6544702 | 711.6493925 | 701.4585332 | 702 | |
| 1199.560106 | 858.0488865 | 741.3868004 | 711.8472879 | 701.4686564 | 702 | |
| 1205.226507 | 865.1910513 | 742.8585939 | 712.164162 | 701.3797526 | 702 | |
| 1223.424448 | 872.8907201 | 745.0901305 | 712.7512682 | 701.3890923 | 702 | |
| 1252.760949 | 886.0588238 | 748.3527672 | 713.5546723 | 701.3284514 | 702 | |
| 1257.499632 | 917.2351432 | 837.8340696 | 818.4294337 | 811.6116649 | 812 | |
| 1290.204458 | 939.0340854 | 844.06219 | 820.0382881 | 811.5975522 | 812 | |
| 1303.892437 | 959.446227 | 848.1841487 | 820.9940034 | 811.4408159 | 812 | |
| 1331.365988 | 970.5906931 | 851.6646644 | 821.9029591 | 811.4462606 | 812 | |
| 1338.297089 | 982.8376731 | 853.8487388 | 822.4042907 | 811.356365 | 812 | |
| 1347.661194 | 991.7758761 | 857.2305366 | 823.2824549 | 811.354884 | 812 | |
| 1355.378862 | 996.3409894 | 858.3332989 | 823.5605611 | 811.3432493 | 812 | |
| 1388.437734 | 1009.409013 | 861.3984248 | 824.3155327 | 811.2865542 | 812 | |
| 1422.827057 | 1015.412447 | 862.1364774 | 824.406597 | 811.1503009 | 812 | |
| 1451.453947 | 1018.857355 | 863.5361963 | 824.8138403 | 811.2088404 | 812 | |
| 1482.263832 | 1020.442609 | 864.6308903 | 825.1017541 | 811.213294 | 812 | |
| 1508.140124 | 1023.617948 | 865.8157049 | 825.4299734 | 811.2405508 | 812 | |
| 1526.087992 | 1027.291174 | 866.4042314 | 825.5572628 | 811.2057857 | 812 | |
| 1545.610201 | 1036.84911 | 869.481435 | 826.3975253 | 811.2601047 | 812 | |
| 1554.510307 | 1064.273033 | 874.3665404 | 827.4807187 | 811.0075062 | 812 | |
| 1590.581181 | 1065.820289 | 963.3727127 | 938.3084831 | 929.5022307 | 930 | |
| 1605.153359 | 1096.261605 | 971.4127774 | 940.3456634 | 929.430313 | 930 | |
| 1627.64583 | 1126.250162 | 977.5004273 | 941.7646995 | 929.2090437 | 930 | |
| 1669.866042 | 1132.603485 | 980.9094357 | 942.7213959 | 929.3041265 | 930 | |
| 1680.931813 | 1143.193005 | 983.5349376 | 943.3572048 | 929.240862 | 930 | |
| 1732.221546 | 1156.758669 | 986.9183407 | 944.221829 | 929.2205197 | 930 | |
| 1747.756289 | 1159.638738 | 989.2103894 | 944.8063833 | 929.2051503 | 930 | |
| 1822.972453 | 1177.143929 | 992.542855 | 945.6117224 | 929.1225899 | 930 | |
| 1867.189269 | 1185.260158 | 994.0531497 | 945.9303939 | 929.0225877 | 930 | |
| 1883.739776 | 1191.139278 | 996.1050268 | 946.4762859 | 929.0393558 | 930 | |
| 1905.116364 | 1194.998333 | 997.339493 | 946.7738475 | 929.0077383 | 930 | |
| 1925.629737 | 1202.96774 | 999.3816401 | 947.2418848 | 928.9227162 | 930 | |
| 1944.868788 | 1206.065194 | 1000.797833 | 947.6109298 | 928.9238487 | 930 | |
| 1975.021914 | 1211.060838 | 1001.676215 | 947.8512186 | 928.939945 | 930 | |
| 2046.242111 | 1221.472042 | 1006.194703 | 949.053404 | 928.976956 | 930 | |
| 2134.000566 | 1228.050355 | 1012.019668 | 950.43107 | 928.7920748 | 930 | |
| 2162.739618 | 1244.971987 | 1098.561308 | 1066.595149 | 1055.363921 | 1056 | |
| 2214.31636 | 1265.874886 | 1108.582123 | 1069.161834 | 1055.311617 | 1056 | |
| 0 | 1301.316945 | 1116.393584 | 1070.983785 | 1055.029169 | 1056 | |
| 0 | 1310.18152 | 1119.789499 | 1071.942154 | 1055.131113 | 1056 | |
| 0 | 1330.745542 | 1125.046521 | 1073.200927 | 1054.985111 | 1056 | |
| 0 | 1335.34437 | 1127.208422 | 1073.831024 | 1055.077013 | 1056 | |
| 0 | 1356.449498 | 1133.426857 | 1075.328132 | 1054.915295 | 1056 | |
| 0 | 1357.947579 | 1134.27894 | 1075.581596 | 1054.958436 | 1056 | |
| 0 | 1372.66299 | 1138.121156 | 1076.37123 | 1054.675553 | 1056 | |
| 0 | 1377.698335 | 1139.195435 | 1076.712476 | 1054.759249 | 1056 | |
| 0 | 1384.504938 | 1141.759568 | 1077.31713 | 1054.675446 | 1056 | |
| 0 | 1395.52906 | 1142.470164 | 1077.542865 | 1054.730825 | 1056 | |
| 0 | 1399.509728 | 1145.853789 | 1078.3652 | 1054.653259 | 1056 | |
| 0 | 1409.198542 | 1148.149675 | 1078.869146 | 1054.52761 | 1056 | |
| 0 | 1413.738535 | 1148.808562 | 1079.093911 | 1054.599848 | 1056 | |
| 0 | 1419.953086 | 1153.773007 | 1080.46782 | 1054.712231 | 1056 | |
| 0 | 1420.442179 | 1162.195739 | 1082.484037 | 1054.477536 | 1056 | |
| 0 | 1449.296201 | 1245.370005 | 1203.731689 | 1189.102174 | 1190 | |
| 0 | 1454.335566 | 1255.700237 | 1206.425527 | 1189.112985 | 1190 | |
| 0 | 1502.852855 | 1267.587086 | 1209.184546 | 1188.664964 | 1190 | |
| 0 | 1509.006901 | 1269.131111 | 1209.796006 | 1188.94877 | 1190 | |
| 0 | 1526.729413 | 1276.754161 | 1211.642809 | 1188.766103 | 1190 | |
| 0 | 1542.348293 | 1279.356516 | 1212.259963 | 1188.685762 | 1190 | |
| 0 | 1554.082383 | 1285.831654 | 1214.009978 | 1188.775618 | 1190 | |
| 0 | 1562.804731 | 1288.265683 | 1214.37258 | 1188.410429 | 1190 | |
| 0 | 1569.811764 | 1292.907406 | 1215.682048 | 1188.549118 | 1190 | |
| 0 | 1573.334672 | 1294.520018 | 1216.006039 | 1188.420355 | 1190 | |
| 0 | 1590.524199 | 1296.914923 | 1216.617016 | 1188.404554 | 1190 | |
| 0 | 1598.429969 | 1298.077768 | 1216.901715 | 1188.380718 | 1190 | |
| 0 | 1606.912246 | 1301.391736 | 1217.807641 | 1188.440584 | 1190 | |
| 0 | 1612.046483 | 1303.903209 | 1218.365384 | 1188.311889 | 1190 | |
| 0 | 1617.90302 | 1308.072709 | 1219.196957 | 1187.970691 | 1190 | |
| 0 | 1627.115306 | 1308.622782 | 1219.525324 | 1188.221162 | 1190 | |
| 0 | 1634.899685 | 1314.002555 | 1221.21552 | 1188.615034 | 1190 | |
| 0 | 1652.674099 | 1323.973254 | 1223.462557 | 1188.148383 | 1190 | |
| 0 | 1662.679293 | 1401.156739 | 1349.115508 | 1330.830955 | 1332 | |
| 0 | 1676.228567 | 1414.241522 | 1352.53321 | 1330.852154 | 1332 | |
| 0 | 1723.14215 | 1428.447059 | 1355.852576 | 1330.346691 | 1332 | |
| 0 | 1726.775947 | 1430.05838 | 1356.386102 | 1330.501537 | 1332 | |
| 0 | 1743.241394 | 1440.206826 | 1358.962978 | 1330.418161 | 1332 | |
| 0 | 1756.128839 | 1441.100225 | 1359.167511 | 1330.380664 | 1332 | |
| 0 | 1761.30986 | 1448.850155 | 1361.209761 | 1330.417535 | 1332 | |
| 0 | 1772.926917 | 1454.635994 | 1362.598335 | 1330.261141 | 1332 | |
| 0 | 1787.003051 | 1457.094895 | 1363.313557 | 1330.363726 | 1332 | |
| 0 | 1791.17388 | 1463.675958 | 1364.709018 | 1329.937239 | 1332 | |
| 0 | 1804.33865 | 1464.56982 | 1365.134756 | 1330.198502 | 1332 | |
| 0 | 1811.241273 | 1467.714923 | 1365.577534 | 1329.691826 | 1332 | |
| 0 | 1821.375296 | 1469.565903 | 1366.053017 | 1329.684032 | 1332 | |
| 0 | 1836.643952 | 1472.828422 | 1367.100284 | 1329.952976 | 1332 | |
| 0 | 1843.265243 | 1476.416379 | 1367.972759 | 1329.871373 | 1332 | |
| 0 | 1848.966736 | 1479.843139 | 1368.582896 | 1329.491897 | 1332 | |
| 0 | 1866.450756 | 1480.878738 | 1368.95986 | 1329.63745 | 1332 | |
| 0 | 1868.709315 | 1491.11741 | 1372.025294 | 1330.182587 | 1332 | |
| 0 | 1892.407376 | 1503.299399 | 1374.76406 | 1329.6035 | 1332 | |
| 0 | 1900.882887 | 1568.030518 | 1503.274407 | 1480.522515 | 1482 | |
| 0 | 1935.753146 | 1582.535779 | 1507.031546 | 1480.503329 | 1482 | |
| 0 | 1947.624936 | 1599.79851 | 1511.251753 | 1480.141079 | 1482 | |
| 0 | 1957.186462 | 1601.331366 | 1511.571592 | 1480.034726 | 1482 | |
| 0 | 1979.709135 | 1612.769014 | 1514.553963 | 1480.046358 | 1482 | |
| 0 | 1984.215485 | 1618.304436 | 1515.976061 | 1480.023251 | 1482 | |
| 0 | 1991.042384 | 1624.094417 | 1517.538121 | 1480.099842 | 1482 | |
| 0 | 1998.265043 | 1633.379194 | 1519.697343 | 1479.755518 | 1482 | |
| 0 | 2005.239543 | 1633.828892 | 1520.029221 | 1480.046001 | 1482 | |
| 0 | 2019.664364 | 1641.995084 | 1521.83583 | 1479.618187 | 1482 | |
| 0 | 2026.991655 | 1645.725298 | 1522.523041 | 1479.236245 | 1482 | |
| 0 | 2038.237575 | 1647.862903 | 1523.314586 | 1479.554856 | 1482 | |
| 0 | 2058.381425 | 1649.93194 | 1523.686698 | 1479.330758 | 1482 | |
| 0 | 2070.045235 | 1651.977715 | 1524.22466 | 1479.338954 | 1482 | |
| 0 | 2072.268731 | 1659.444429 | 1526.058038 | 1479.193074 | 1482 | |
| 0 | 2093.469398 | 1660.69365 | 1526.835138 | 1479.804295 | 1482 | |
| 0 | 2095.096242 | 1664.112832 | 1527.211013 | 1479.110912 | 1482 | |
| 0 | 2102.835422 | 1667.137331 | 1528.129533 | 1479.289501 | 1482 | |
| 0 | 2120.714175 | 1679.166448 | 1531.507719 | 1479.628206 | 1482 | |
| 0 | 2132.967851 | 1694.482979 | 1534.958247 | 1478.909643 | 1482 | |
| 0 | 2149.456987 | 1744.635427 | 1665.838743 | 1638.153731 | 1640 | |
| 0 | 2180.968556 | 1758.695827 | 1669.707659 | 1638.441895 | 1640 | |
| 0 | 2189.742622 | 1780.619471 | 1675.057629 | 1637.968748 | 1640 | |
| 0 | 2203.710834 | 1784.604052 | 1675.913328 | 1637.725123 | 1640 | |
| 0 | 2215.499171 | 1797.226047 | 1679.159673 | 1637.677356 | 1640 | |
| 0 | 2224.8071 | 1806.310202 | 1681.403729 | 1637.518162 | 1640 | |
| 0 | 2234.460573 | 1810.158274 | 1682.465592 | 1637.601097 | 1640 | |
| 0 | 0 | 1821.901512 | 1685.534095 | 1637.621754 | 1640 | |
| 0 | 0 | 1825.094115 | 1686.134773 | 1637.311766 | 1640 | |
| 0 | 0 | 1834.463239 | 1688.325108 | 1636.979853 | 1640 | |
| 0 | 0 | 1837.747163 | 1689.213699 | 1637.026849 | 1640 | |
| 0 | 0 | 1840.561745 | 1690.118382 | 1637.260495 | 1640 | |
| 0 | 0 | 1846.221629 | 1691.347527 | 1636.93291 | 1640 | |
| 0 | 0 | 1848.844477 | 1691.629137 | 1636.391933 | 1640 | |
| 0 | 0 | 1849.575859 | 1692.072889 | 1636.734626 | 1640 | |
| 0 | 0 | 1855.755602 | 1694.134904 | 1637.349888 | 1640 | |
| 0 | 0 | 1860.933554 | 1694.927914 | 1636.602261 | 1640 | |
| 0 | 0 | 1864.024889 | 1695.661635 | 1636.50764 | 1640 | |
| 0 | 0 | 1868.939553 | 1696.752751 | 1636.255362 | 1640 | |
| 0 | 0 | 1882.296134 | 1701.077413 | 1637.406682 | 1640 | |
| 0 | 0 | 1904.064425 | 1705.671326 | 1635.966422 | 1640 | |
| 0 | 0 | 1934.390626 | 1837.615967 | 1803.61444 | 1806 | |
| 0 | 0 | 1952.25589 | 1842.374185 | 1803.767531 | 1806 | |
| 0 | 0 | 1976.247395 | 1848.360403 | 1803.427639 | 1806 | |
| 0 | 0 | 1983.195565 | 1849.72132 | 1802.825488 | 1806 | |
| 0 | 0 | 1993.80211 | 1852.587807 | 1802.972526 | 1806 | |
| 0 | 0 | 2005.968081 | 1855.829291 | 1803.078414 | 1806 | |
| 0 | 0 | 2006.738837 | 1856.361285 | 1803.52652 | 1806 | |
| 0 | 0 | 2022.11571 | 1860.092568 | 1803.166154 | 1806 | |
| 0 | 0 | 2030.267056 | 1861.885365 | 1802.724892 | 1806 | |
| 0 | 0 | 2036.730623 | 1863.586096 | 1802.752213 | 1806 | |
| 0 | 0 | 2046.0735 | 1865.249828 | 1801.717897 | 1806 | |
| 0 | 0 | 2048.668098 | 1866.258781 | 1802.169738 | 1806 | |
| 0 | 0 | 2049.678367 | 1866.796544 | 1802.541488 | 1806 | |
| 0 | 0 | 2057.217499 | 1867.943539 | 1801.442621 | 1806 | |
| 0 | 0 | 2060.223511 | 1868.996273 | 1801.809076 | 1806 | |
| 0 | 0 | 2062.219062 | 1870.049734 | 1802.531537 | 1806 | |
| 0 | 0 | 2071.833658 | 1872.254463 | 1802.132828 | 1806 | |
| 0 | 0 | 2074.052606 | 1872.853576 | 1802.162816 | 1806 | |
| 0 | 0 | 2078.909423 | 1873.816482 | 1801.757607 | 1806 | |
| 0 | 0 | 2085.843475 | 1875.224738 | 1801.224388 | 1806 | |
| 0 | 0 | 2097.962723 | 1880.534391 | 1804.141508 | 1806 | |
| 0 | 0 | 2124.355279 | 1885.906096 | 1802.127591 | 1806 | |
| 0 | 0 | 2135.518187 | 2018.192524 | 1976.970455 | 1980 | |
| 0 | 0 | 2151.459214 | 2022.711127 | 1977.475819 | 1980 | |
| 0 | 0 | 2183.481162 | 2030.416182 | 1976.637195 | 1980 | |
| 0 | 0 | 2193.361587 | 2032.795513 | 1976.381037 | 1980 | |
| 0 | 0 | 2201.706883 | 2035.269583 | 1976.792267 | 1980 | |
| 0 | 0 | 2216.834352 | 2039.50005 | 1977.194101 | 1980 | |
| 0 | 0 | 2218.954599 | 2039.688643 | 1976.704011 | 1980 | |
| 0 | 0 | 2233.464786 | 2043.497385 | 1976.752829 | 1980 | |
| 0 | 0 | 0 | 2047.053337 | 1980 | ||
| 0 | 0 | 0 | 2047.812831 | 1980 | ||
| 0 | 0 | 0 | 2050.967382 | 1980 | ||
| 0 | 0 | 0 | 2051.780115 | 1980 | ||
| 0 | 0 | 0 | 2052.376745 | 1980 | ||
| 0 | 0 | 0 | 2053.69863 | 1980 | ||
| 0 | 0 | 0 | 2054.833817 | 1980 | ||
| 0 | 0 | 0 | 2056.697869 | 1980 | ||
| 0 | 0 | 0 | 2057.310221 | 1980 | ||
| 0 | 0 | 0 | 2059.85662 | 1980 | ||
| 0 | 0 | 0 | 2060.526148 | 1980 | ||
| 0 | 0 | 0 | 2061.421791 | 1980 | ||
| 0 | 0 | 0 | 2063.296968 | 1980 | ||
| 0 | 0 | 0 | 2069.763492 | 1980 | ||
| 0 | 0 | 0 | 2075.892584 | 1980 | ||
| 0 | 0 | 0 | 2207.334989 | 2162 | ||
| 0 | 0 | 0 | 2213.258818 | 2162 | ||
| 0 | 0 | 0 | 2221.079148 | 2162 | ||
| 0 | 0 | 0 | 2224.770503 | 2162 | ||
| 0 | 0 | 0 | 2226.270544 | 2162 | ||
| 0 | 0 | 0 | 2232.393472 | 2162 | ||
| 0 | 0 | 0 | 2233.403867 | 2162 | ||
| 0 | 0 | 0 | 2237.043679 | 2162 |