3. The best approximation and other polynomial approximations

  The computation of the best approximation is much more laborious than the simple computation of the Lagrange-Chebyshev polynomial. In practice, in many problems it could be not justified the extra job of computing the best approximation because of the small increase of accuracy (Clenshaw 1964). Nevertheless, in reference to planetary ephemerides we believe that it is not the case. In order to justify this assertion, in this section we present comparisons between the best approximation and other approximation methods.

First we compare the best approximation with standard Chebyshev approximation. The comparisons have been made for the Sun and navigational planets between the series provided by the "Almanac for Computers'' and the best approximation computed for the same precision and interval. In this publication the series were obtained by computing standard Chebyshev approximations to the data published in "The Astronomical Almanac''. The comparisons have been carried out using the values listed in the Astronomical Tables (Section D) of "Almanac for Computers 1989'' (hereafter termed AC89).

Table 1 (click here) presents the results for high precision series valid for intervals of 95 days. Using uniform approximation we always obtained series shorter than those provided in AC89 for the same precision. It is remarkable the case of Venus where the number of terms is drastically reduced. Analogous results are showed in Table 2 (click here) for low precision series. The column "Maximum error'' of both tables corresponds to the values given in pages A9 and A11 of the Explanation of the AC89 with two exceptions for Table 2 (click here):

For the distance of Venus we present the error obtained with the 30 terms of the best approximation, the same degree as used for the other coordinates; although that precision is slightly worse than the one listed in AC89, it is indeed enough for a low precision series. The second exception is for the declination of Saturn, where we checked that the real value of the error () is slightly larger than the estimation given in the AC89 ().

 

Body LS BA Maximum error

Sun Right Ascension 24 21

Declination 24 21 0.'' 01

Distance 24 21

Venus Right Ascension 30 20

Declination 30 20 0.'' 1

Distance 30 20

Mars Right Ascension 16 15

Declination 16 15 0.'' 2

Distance 16 15

Jupiter Right Ascension 16 14

Declination 16 14 0.'' 1

Distance 16 14

Saturn Right Ascension 12 11

Declination 12 11 0.'' 4

Distance 12 11

Table 1: High precision series. The span of validity is 95 days. We note LS for the number of terms of the polynomial approximation when using least squares fit and BA for the best approximation. "au'' stands for astronomical units

 

 

Body LS BA Maximum error

Sun Right Ascension 22 18

Declination 22 18 3''

Distance 22 18

Venus Right Ascension 50 30

Declination 50 30 3''

Distance 50 30

Mars Right Ascension 14 11

Declination 14 11 8''

Distance 14 11

Jupiter Right Ascension 14 14

Declination 14 14 0.'' 7

Distance 14 14

Saturn Right Ascension 14 14

Declination 14 14 0.'' 7

Distance 14 14

Table 2: Low precision series. The span of validity is 366 days. We note LS for the number of terms of the polynomial approximation when using least squares fit and BA for the best approximation. "au'' stands for astronomical units

 

Finally, we present a comparison between least squares fit and the best approximation. Least squares fit is not an uniform approximation: the error function oscillates with nonuniform magnitude, being normally minimal in the center of the interval. In order to illustrate this well known effect we present a concrete example using both techniques. We have computed approximating polynomials of degree 4 covering 32 days of the right ascension of Jupiter for June 1997. The comparisons have been made computing the coefficients of ordinary and Chebyshev polynomials. Figure 1 (click here) shows the error functions in both cases. In the above plot we used least squares fit with ordinary polynomials; in the other plot Chebyshev polynomials are used. Notice that the largest errors of the least squares fit occur in both cases at the ends of the interval. The least squares fit with Chebyshev polynomials resulted in a very good approximation, but notice also that one should reject both ends of the interval for providing an accuracy similar to the one reached by the best approximation. The uniform behavior of the best approximation implies that there is no need of taking slopes.

  
Figure 1: Error functions for the right ascension of Jupiter. Ordinary (above) and Chebyshev (below) polynomials of degree 4 approximating 32 days of 1997. Dashed line corresponds to least squares fit while full line is the best approximation

With the best approximation a rigorous estimate of the error valid for the entire interval is directly provided by the method. This is not the case of least squares fit, but in practical applications it is not a major problem: an estimate of the error can be always computed a posteriori making difference between source and approximation data.