Looks like the algorithm will need some modification or the polynomial expanded/re-written. The polynomial as written has the proton radius, and obviously, by inspection, if one tries to iterate for the proton radius, it comes out NEGATIVE due to the other constants being so far off:
error, dx, $r_p$
-0.0005446185946568205, 1e-16, 8.41235640479985e-16
-0.0009587998163527178, 9.999999999999999e-18, 8e-16
-0.0009523258036489013, 9.999999999999999e-18, 7.9e-16
-0.0009458517909451958, 9.999999999999999e-18, 7.800000000000001e-16
-0.0009393777782413792, 9.999999999999999e-18, 7.700000000000001e-16
-0.0009329037655375627, 9.999999999999999e-18, 7.600000000000001e-16
-0.0009264297528338572, 9.999999999999999e-18, 7.500000000000002e-16
-0.0009199557401300407, 9.999999999999999e-18, 7.400000000000002e-16
-0.0009134817274263352, 9.999999999999999e-18, 7.300000000000002e-16
-0.0009070077147225186, 9.999999999999999e-18, 7.200000000000003e-16
-0.0009005337020188131, 9.999999999999999e-18, 7.100000000000003e-16
-0.0008940596893149966, 9.999999999999999e-18, 7.000000000000003e-16
-0.0008875856766112911, 9.999999999999999e-18, 6.900000000000004e-16
-0.0008811116639074745, 9.999999999999999e-18, 6.800000000000004e-16
-0.000874637651203658, 9.999999999999999e-18, 6.700000000000004e-16
-0.0008681636384999525, 9.999999999999999e-18, 6.600000000000005e-16
-0.000861689625796136, 9.999999999999999e-18, 6.500000000000005e-16
-0.0008552156130924304, 9.999999999999999e-18, 6.400000000000005e-16
-0.0008487416003886139, 9.999999999999999e-18, 6.300000000000006e-16
-0.0008422675876849084, 9.999999999999999e-18, 6.200000000000006e-16
-0.0008357935749810919, 9.999999999999999e-18, 6.100000000000006e-16
-0.0008293195622773863, 9.999999999999999e-18, 6.000000000000007e-16
-0.0008228455495735698, 9.999999999999999e-18, 5.900000000000007e-16
-0.0008163715368697533, 9.999999999999999e-18, 5.800000000000007e-16
-0.0008098975241660478, 9.999999999999999e-18, 5.700000000000008e-16
-0.0008034235114622312, 9.999999999999999e-18, 5.600000000000008e-16
-0.0007969494987585257, 9.999999999999999e-18, 5.500000000000009e-16
-0.0007904754860547092, 9.999999999999999e-18, 5.400000000000009e-16
-0.0007840014733510037, 9.999999999999999e-18, 5.300000000000009e-16
-0.0007775274606471871, 9.999999999999999e-18, 5.20000000000001e-16
-0.0007710534479434816, 9.999999999999999e-18, 5.10000000000001e-16
-0.0007645794352396651, 9.999999999999999e-18, 5.00000000000001e-16
-0.0007581054225358486, 9.999999999999999e-18, 4.900000000000011e-16
-0.000751631409832143, 9.999999999999999e-18, 4.800000000000011e-16
-0.0007451573971283265, 9.999999999999999e-18, 4.700000000000011e-16
-0.000738683384424621, 9.999999999999999e-18, 4.600000000000012e-16
-0.0007322093717208045, 9.999999999999999e-18, 4.500000000000012e-16
-0.000725735359017099, 9.999999999999999e-18, 4.400000000000012e-16
-0.0007192613463132824, 9.999999999999999e-18, 4.3000000000000117e-16
-0.0007127873336094659, 9.999999999999999e-18, 4.2000000000000115e-16
-0.0007063133209057604, 9.999999999999999e-18, 4.1000000000000114e-16
-0.0006998393082019438, 9.999999999999999e-18, 4.0000000000000113e-16
-0.0006933652954982383, 9.999999999999999e-18, 3.900000000000011e-16
-0.0006868912827944218, 9.999999999999999e-18, 3.800000000000011e-16
-0.0006804172700907163, 9.999999999999999e-18, 3.700000000000011e-16
-0.0006739432573868998, 9.999999999999999e-18, 3.6000000000000107e-16
-0.0006674692446831942, 9.999999999999999e-18, 3.5000000000000105e-16
-0.0006609952319793777, 9.999999999999999e-18, 3.4000000000000104e-16
-0.0006545212192755612, 9.999999999999999e-18, 3.30000000000001e-16
-0.0006480472065718557, 9.999999999999999e-18, 3.20000000000001e-16
-0.0006415731938680391, 9.999999999999999e-18, 3.10000000000001e-16
-0.0006350991811643336, 9.999999999999999e-18, 3.00000000000001e-16
-0.0006286251684605171, 9.999999999999999e-18, 2.9000000000000097e-16
-0.0006221511557568116, 9.999999999999999e-18, 2.8000000000000095e-16
-0.000615677143052995, 9.999999999999999e-18, 2.7000000000000094e-16
-0.0006092031303492895, 9.999999999999999e-18, 2.600000000000009e-16
-0.000602729117645473, 9.999999999999999e-18, 2.500000000000009e-16
-0.0005962551049416565, 9.999999999999999e-18, 2.400000000000009e-16
-0.0005897810922379509, 9.999999999999999e-18, 2.300000000000009e-16
-0.0005833070795341344, 9.999999999999999e-18, 2.200000000000009e-16
-0.0005768330668304289, 9.999999999999999e-18, 2.100000000000009e-16
-0.0005703590541266124, 9.999999999999999e-18, 2.000000000000009e-16
-0.0005638850414229069, 9.999999999999999e-18, 1.9000000000000092e-16
-0.0005574110287190903, 9.999999999999999e-18, 1.8000000000000093e-16
-0.0005509370160153848, 9.999999999999999e-18, 1.7000000000000094e-16
-0.0005444630033115683, 9.999999999999999e-18, 1.6000000000000095e-16
-0.0005379889906077517, 9.999999999999999e-18, 1.5000000000000096e-16
-0.0005315149779040462, 9.999999999999999e-18, 1.4000000000000097e-16
-0.0005250409652002297, 9.999999999999999e-18, 1.3000000000000098e-16
-0.0005185669524965242, 9.999999999999999e-18, 1.20000000000001e-16
-0.0005120929397927076, 9.999999999999999e-18, 1.1000000000000099e-16
-0.0005056189270890021, 9.999999999999999e-18, 1.0000000000000098e-16
-0.0004991449143851856, 9.999999999999999e-18, 9.000000000000098e-17
-0.0004926709016813691, 9.999999999999999e-18, 8.000000000000098e-17
-0.00048619688897766355, 9.999999999999999e-18, 7.000000000000098e-17
-0.000479722876273847, 9.999999999999999e-18, 6.000000000000097e-17
-0.0004732488635701415, 9.999999999999999e-18, 5.000000000000097e-17
-0.000466774850866325, 9.999999999999999e-18, 4.000000000000097e-17
-0.00046030083816261946, 9.999999999999999e-18, 3.000000000000097e-17
-0.00045382682545880293, 9.999999999999999e-18, 2.000000000000097e-17
-0.0004473528127550974, 9.999999999999999e-18, 1.000000000000097e-17
-0.0004408788000512809, 9.999999999999999e-18, 9.706686919711669e-31
-0.00043440478734746435, 9.999999999999999e-18, -9.999999999999029e-18
-0.00042793077464375884, 9.999999999999999e-18, -1.9999999999999028e-17
-0.0004214567619399423, 9.999999999999999e-18, -2.9999999999999027e-17
-0.0004149827492362368, 9.999999999999999e-18, -3.999999999999902e-17
-0.00040850873653242026, 9.999999999999999e-18, -4.9999999999999025e-17
-0.00040203472382871475, 9.999999999999999e-18, -5.999999999999903e-17
-0.0003955607111248982, 9.999999999999999e-18, -6.999999999999903e-17
-0.0003890866984211927, 9.999999999999999e-18, -7.999999999999903e-17
-0.00038261268571737617, 9.999999999999999e-18, -8.999999999999903e-17
-0.00037613867301355963, 9.999999999999999e-18, -9.999999999999904e-17
-0.0003696646603098541, 9.999999999999999e-18, -1.0999999999999904e-16
-0.0003631906476060376, 9.999999999999999e-18, -1.1999999999999904e-16
-0.0003567166349023321, 9.999999999999999e-18, -1.2999999999999903e-16
-0.00035024262219851554, 9.999999999999999e-18, -1.3999999999999902e-16
-0.00034376860949481003, 9.999999999999999e-18, -1.49999999999999e-16
-0.0003372945967909935, 9.999999999999999e-18, -1.59999999999999e-16
-0.000330820584087288, 9.999999999999999e-18, -1.69999999999999e-16
-0.00032434657138347145, 9.999999999999999e-18, -1.7999999999999898e-16
-0.0003178725586796549, 9.999999999999999e-18, -1.8999999999999897e-16
-0.0003113985459759494, 9.999999999999999e-18, -1.9999999999999896e-16
-0.00030492453327213287, 9.999999999999999e-18, -2.0999999999999895e-16
-0.00029845052056842736, 9.999999999999999e-18, -2.1999999999999894e-16
-0.0002919765078646108, 9.999999999999999e-18, -2.2999999999999895e-16
-0.0002855024951609053, 9.999999999999999e-18, -2.3999999999999897e-16
-0.0002790284824570888, 9.999999999999999e-18, -2.49999999999999e-16
-0.00027255446975327224, 9.999999999999999e-18, -2.59999999999999e-16
-0.00026608045704956673, 9.999999999999999e-18, -2.69999999999999e-16
-0.0002596064443457502, 9.999999999999999e-18, -2.7999999999999903e-16
-0.0002531324316420447, 9.999999999999999e-18, -2.8999999999999904e-16
-0.00024665841893822815, 9.999999999999999e-18, -2.9999999999999906e-16
-0.00024018440623452264, 9.999999999999999e-18, -3.0999999999999907e-16
-0.0002337103935307061, 9.999999999999999e-18, -3.199999999999991e-16
-0.0002272363808270006, 9.999999999999999e-18, -3.299999999999991e-16
-0.00022076236812318406, 9.999999999999999e-18, -3.399999999999991e-16
-0.00021428835541936753, 9.999999999999999e-18, -3.4999999999999913e-16
-0.00020781434271566201, 9.999999999999999e-18, -3.5999999999999914e-16
-0.00020134033001184548, 9.999999999999999e-18, -3.6999999999999916e-16
-0.00019486631730813997, 9.999999999999999e-18, -3.7999999999999917e-16
-0.00018839230460432344, 9.999999999999999e-18, -3.899999999999992e-16
-0.00018191829190061792, 9.999999999999999e-18, -3.999999999999992e-16
-0.0001754442791968014, 9.999999999999999e-18, -4.099999999999992e-16
-0.00016897026649309588, 9.999999999999999e-18, -4.1999999999999923e-16
-0.00016249625378927934, 9.999999999999999e-18, -4.2999999999999925e-16
-0.0001560222410854628, 9.999999999999999e-18, -4.3999999999999926e-16
-0.0001495482283817573, 9.999999999999999e-18, -4.499999999999992e-16
-0.00014307421567794076, 9.999999999999999e-18, -4.599999999999992e-16
-0.00013660020297423525, 9.999999999999999e-18, -4.699999999999992e-16
-0.00013012619027041872, 9.999999999999999e-18, -4.799999999999991e-16
-0.0001236521775667132, 9.999999999999999e-18, -4.899999999999991e-16
-0.00011717816486289667, 9.999999999999999e-18, -4.999999999999991e-16
-0.00011070415215919116, 9.999999999999999e-18, -5.09999999999999e-16
-0.00010423013945537463, 9.999999999999999e-18, -5.19999999999999e-16
-0.00009775612675155809, 9.999999999999999e-18, -5.299999999999989e-16
-0.00009128211404785258, 9.999999999999999e-18, -5.399999999999989e-16
-0.00008480810134403605, 9.999999999999999e-18, -5.499999999999989e-16
-0.00007833408864033053, 9.999999999999999e-18, -5.599999999999988e-16
-0.000071860075936514, 9.999999999999999e-18, -5.699999999999988e-16
-0.00006538606323280849, 9.999999999999999e-18, -5.799999999999988e-16
-0.000058912050528991955, 9.999999999999999e-18, -5.899999999999987e-16
-0.00005243803782517542, 9.999999999999999e-18, -5.999999999999987e-16
-0.00004596402512146991, 9.999999999999999e-18, -6.099999999999987e-16
-0.000039490012417653375, 9.999999999999999e-18, -6.199999999999986e-16
-0.00003301599971394786, 9.999999999999999e-18, -6.299999999999986e-16
-0.00002654198701013133, 9.999999999999999e-18, -6.399999999999986e-16
-0.000020067974306425818, 9.999999999999999e-18, -6.499999999999985e-16
-0.000013593961602609284, 9.999999999999999e-18, -6.599999999999985e-16
-0.000007119948898903772, 9.999999999999999e-18, -6.699999999999985e-16
-6.459361950872378e-7, 9.999999999999999e-18, -6.799999999999984e-16
-6.459361950872378e-7, 9.999999999999999e-19, -6.799999999999984e-16
1.465075394335713e-9, 9.999999999999999e-19, -6.809999999999984e-16
1.465075394335713e-9, 9.999999999999999e-20, -6.809999999999984e-16
1.465075394335713e-9, 9.999999999999998e-21, -6.809999999999984e-16
1.465075394335713e-9, 9.999999999999997e-22, -6.809999999999984e-16
8.176739285659096e-10, 9.999999999999997e-22, -6.809989999999984e-16
1.70272684840711e-10, 9.999999999999997e-22, -6.809979999999984e-16
1.70272684840711e-10, 9.999999999999997e-23, -6.809979999999984e-16
1.055326936949541e-10, 9.999999999999997e-23, -6.809978999999984e-16
4.0792480504592277e-11, 9.999999999999997e-23, -6.809977999999984e-16
Likely will have to iterate ALL constants that are in the original limited Rydberg equation EXCEPT for proton radius until they are corrected enough to allow for the proton radius to be a positive number closer to expected value.
updates incoming...
For the Rydberg Constant $(R_H or R_{\infty})$
error, dx, $R_H$
-0.0005446185946568205, 10000000, 10973731.5685083
-0.0005446185946568205, 1000000, 10973731.5685083
-0.0005446185946568205, 100000, 10973731.5685083
-0.0005446185946568205, 10000, 10973731.5685083
0.0003679759110610803, 10000, 10963731.5685083
0.0003679759110610803, 1000, 10963731.5685083
0.0002766415942339062, 1000, 10964731.5685083
0.00018532392648995533, 1000, 10965731.5685083
0.00009402290327442664, 1000, 10966731.5685083
0.0000027385200349616667, 1000, 10967731.5685083
0.0000027385200349616667, 100, 10967731.5685083
0.0000027385200349616667, 10, 10967731.5685083
0.0000018257602190097089, 10, 10967741.5685083
9.130020666159311e-7, 10, 10967751.5685083
2.4557778033340583e-10, 10, 10967761.5685083
2.4557778033340583e-10, 1, 10967761.5685083
2.4557778033340583e-10, 0.1, 10967761.5685083
2.4557778033340583e-10, 0.01, 10967761.5685083
2.4557778033340583e-10, 0.001, 10967761.5685083
1.5430212663147813e-10, 0.001, 10967761.569508301
6.302669497415536e-11, 0.001, 10967761.570508301
Will continue to check all constants to uncover other issues and
reformulate approach.
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