Table 1
Taylor expansion of the ωM,O,P coefficients for small optical path intervals.
| Coefficient | Maxt | Expansion |
|
|
||
| ω M | 0.14 | (t(t(t(t(t(t((140 − 18t)t − 945) + 5400) − 25 200) + 90 720) − 226 800) + 302 400))/907 200 |
| ω O | 0.18 | (t(t(t(t(t(t((10 − t)t − 90) + 720) − 5040) + 30 240) − 151 200) + 604 800))/1 814 400 |
| ω P | 0.18 | (t(t(t(t(t(t((35 − 4t)t − 270) + 1800) − 10 080) + 45 360) − 151 200) + 302 400))/907 200 |
Notes. Taylor expansion of the interpolation coefficients of the BESSER method for small τMO values (for the sake of notational simplicity we use t ≡ τMO). Column 2 of the table indicates the approximate maximum value of τMO for which this expansion is more accurate, using double precision arithmetics, than the expressions given by Eqs. (11)–(13). We use the Horner rule in Col. 3, which provides a better numerical accuracy and also reduces the number of multiplications in comparison to the explicit expansion of the Taylor power series.
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