Rotation coefficients Maclaurin series
This file defines the constants needed to compute the Maclaurin series of the coefficients \(a(\varphi)\), \(b(\varphi)\) and \(c(\varphi)\), together with a few more coefficients defined in [1].
The constants are defined for LastCoeffB coefficients, with the first three corresponding to the above
mentioned \(a\), \(b\) and \(c\). The series are truncated at the LastSerTrunc+1 term.
SerExpCoeffs[i]lists the derivative of the ith coefficient;SerExpThrsh[i]defines the threshold for \(\varphi^2\) that should be used for the ith coefficient : under that value one should evaluate the coefficient using its Macliaurin series, above it the analytical expression;SerExpTrunc[i]defines the number of terms that should be used in the series in order to approximate the coefficient within the machine epsilon for the coefficient for any \(\varphi^2 < \mathrm{SerExpThrsh[i]}\)
LastCoeffB = 10
LastSerTrunc = 9
SerExpCoeffs = [[ 1., # a
-0.166666666666666657414808128124,
0.00833333333333333321768510160155,
-0.000198412698412698412526317115478,
2.75573192239858925109505932705e-06,
-2.50521083854417202238661793214e-08,
1.60590438368216133408629182949e-10,
-7.64716373181981640551384421244e-13,
2.8114572543455205981105182744e-15,
-8.22063524662432949553704004083e-18],
[ 0.5, #// b0
-0.0416666666666666643537020320309,
0.00138888888888888894189432843262,
-2.48015873015873015657896394348e-05,
2.7557319223985888275785856999e-07,
-2.08767569878681001865551494345e-09,
1.14707455977297245072965704504e-11,
-4.77947733238738525344615263278e-14,
1.56192069685862252711483983227e-16,
-4.11031762331216484406501723978e-19],
[ 0.166666666666666657414808128124, # b1
-0.00833333333333333321768510160155,
0.000198412698412698412526317115478,
-2.75573192239858925109505932705e-06,
2.50521083854417202238661793214e-08,
-1.60590438368216133408629182949e-10,
7.64716373181981640551384421244e-13,
-2.8114572543455205981105182744e-15,
8.22063524662432949553704004083e-18,
-1.95729410633912625951979573823e-20],
[ 0.0833333333333333287074040640618, # b2
-0.00277777777777777788378865686525,
4.96031746031746031315792788696e-05,
-5.51146384479717765515717139979e-07,
4.17535139757362003731102988689e-09,
-2.29414911954594490145931409007e-11,
9.55895466477477050689230526555e-14,
-3.12384139371724505422967966454e-16,
8.22063524662432968813003447955e-19,
-1.77935827849011481557783178595e-21],
[ 0.0500000000000000027755575615629, # b3
-0.00119047619047619058357811994142,
1.65343915343915355065703559623e-05,
-1.50312650312650321343197075928e-07,
9.63542630209296852150563382261e-10,
-4.58829823909188964135991479089e-12,
1.68687435260731251663849200884e-14,
-4.93238114797459800547101512646e-17,
1.17437646380347563534125591874e-19,
-2.32090210237841024059034292735e-22],
[ 0.0333333333333333328707404064062, # b4
-0.000595238095238095291789059970711,
6.61375661375661403322155293405e-06,
-5.01042167708834404477323586427e-08,
2.7529789434551340109981476195e-10,
-1.14707455977297241033997869772e-12,
3.7486096724606946567212945132e-15,
-9.86476229594919601094203025291e-18,
2.13522993418813762823012123788e-20,
-3.86817017063068412614868848635e-23],
[ 0.0238095238095238082021154468748, # b5
-0.000330687830687830669473825651039,
3.0062530062530061092265862982e-06,
-1.92708526041859378701918801982e-08,
9.1765964781837796058372563603e-11,
-3.37374870521462503327698401768e-13,
9.86476229594919625746106313448e-16,
-2.34875292760695117438601461811e-18,
4.64180420475682066925978198627e-21,
-7.73634034126136766455020156155e-24],
[ 0.0178571428571428561515865851561, # b6
-0.000198412698412698412526317115478,
1.5031265031265030546132931491e-06,
-8.25893683036540099901867716722e-09,
3.44122367931891751374768452438e-11,
-1.12458290173820830235307972744e-13,
2.95942868878475868002309263509e-16,
-6.40568980256441264394912066525e-19,
1.16045105118920516731494549657e-21,
-1.78530930952185416120588912158e-24],
[ 0.0138888888888888881179006773436, # b7
-0.000126262626262626262516747255304,
8.09375809375809410400393669599e-07,
-3.85417052083718740860225352904e-09,
1.41697445619014253417117246108e-11,
-4.14320016429866203370319390597e-14,
9.86476229594919601094203025291e-17,
-1.94955776599786480145972995843e-19,
3.24926294332977471298467236142e-22,
-4.62857969135295492663768830709e-25],
[ 0.011111111111111111535154627461, # b8
-8.41750841750841795286738888926e-05,
4.62500462500462482414825523061e-07,
-1.92708526041859370430112676452e-09,
6.29766424973396619025465886879e-12,
-1.65728006571946493969902239775e-14,
3.5871862894360713887602986745e-17,
-6.49852588665954893696369478077e-20,
9.99773213332238300887939137528e-23,
-1.32245134038655871445986836987e-25],
[ 0.00909090909090909046752493338772, # b9
-5.82750582750582750077295024482e-05,
2.77500277500277500036807154515e-07,
-1.02022160845690258582915295856e-09,
2.98310411829503676524049548059e-12,
-7.10262885308342083205542232422e-15,
1.40368159151846250297661001909e-17,
-2.33946931919743760526986796866e-20,
3.33257737777412806145791406585e-23,
-4.10415933223414743764863408539e-26],
[ 0.00757575757575757596784526981537, # b10
-4.16250416250416250055210731773e-05,
1.73437673437673437523004471572e-07,
-5.66789782476057026593166055574e-10,
1.4915520591475183826202477403e-12,
-3.22846766049246397871333263733e-15,
5.84867329799359401919320099786e-18,
-8.99795891999014451991235610618e-21,
1.19020620634790277413725941439e-23,
-1.36805311074471581254954469513e-26]]
SerExpThrsh =[ 1.e-6, # a: 1.e-6 = 0.001**2; err 1.e-15,1.e-13 // 1.e-16 = (1e-8)**2; err 1.e-15,high // 0.09 = 0.3**2; err 1.e-15,1.e-15?
0.01, # b0: 0.010 = 0.100**2; err 1.e-15,1.e-13 // 0.000225 = 0.015 **2; err 1.e-15,high // 0.16 = 0.4**2; err 1.e-15,1.e-15?
0.25, # b1: 0.25 = 0.5 **2; err 1.e-15,= //
1.44, # b2: 1.44 = 1.2 **2; err 1.e-15,= //
1.96, # b3: 1.96 = 1.4 **2; err 1.e-15,= //
4.41, # b4: 4.41 = 2.1 **2; err 1.e-15,= //
5.29, # b5: 5.29 = 2.3 **2; err 1.e-15,= //
6.76, # b6: 6.76 = 2.6 **2; err 1.e-15,= //
7.29, # b7: 7.29 = 2.7 **2; err 1.e-15,= //
12.96, # b8: 12.96 = 3.6 **2; err 1.e-15,= //
16.00, # b9: 16.00 = 4.0 **2; err 1.e-15,= //
17.64] # b10: 17.64 = 4.2 **2; err 1.e-15,= //
SerExpTrunc = [2, # a: 2 (phi^2 ); err 1.e-15,1.e-13 // 0 (phi^0); err 1.e-15,high //
4, # b0: 4 (phi^8 ); err 1.e-15,1.e-13 // 2 (phi^4); err 1.e-15,high //
5, # b1: 5 (phi^10); err 1.e-15,= //
7, # b2: 7 (phi^14); err 1.e-15,= //
7, # b3: 7 (phi^14); err 1.e-15,= //
8, # b4: 8 (phi^16); err 1.e-15,= //
8, # b5: 8 (phi^16); err 1.e-15,= //
8, # b6: 8 (phi^16); err 1.e-15,= //
8, # b7: 8 (phi^16); err 1.e-15,= //
9, # b8: 9 (phi^18); err 1.e-15,= //
9, # b9: 9 (phi^18); err 1.e-15,= //
9] # b10: 9 (phi^18); err 1.e-15,= //