Ij pd p sin f

with cos y 2

The characteristic function has been replaced by appropriate boundaries or integration limits for longitude (p and colatitude 0. <ps (0 ) is the starting longitude and (pn (0 ) is the ending longitude on a given colatitude 0. 0/ and 0u represent the lower and upper limits for colatitude. The limits depend on phase 0 and the geometrical parameters. According to Leibniz's rule, the derivatives are now given by di,

-¡T (0 ) = f2(6u d p d6u d6l r p)^ - ¡2(6l, p)^1 +

Since no analytic expression exists for derivatives of <ps(0), (0), 0l, and 0u w.r.t. V p e P2, these derivatives need to be computed numerically. Thus, three numerical derivatives have to be computed instead of one. It is not so obvious whether some advantage in accuracy is really achieved.

4.5.5 Accurate Finite Difference Approximation

When approximating derivatives f ' (x) of a function f (x) by asymmetric finite differences f '(x) = f (X + AX) - f (X), (4.5.25)

Ax or symmetric finite differences

Ax the question arises as how to choose the size of the increment Ax. As discussed by Press et al. (1992), the optimal choice depends on the curvature, i.e., on the second derivative of f (x). As in most practical cases, generally, as well as in light curve analysis, this information is not available. Instead, for symmetric differences, we may use the heuristic approach

described by Press et al. (1992, p. 180) where ef is the fractional accuracy with which f (x) is computed. For simple functions, this may be comparable to the machine accuracy, ef « em, but for complicated calculations, in which the functions eventually yield the EB observables with additional sources of inaccuracy, e f is certainly larger. For this reason, it is wise to use those values suggested by the light curve program developers, who presumably know best the accuracy properties of their program.

4.6 Selected Bibliography

This section is intended to guide the reader to recommended books or articles on least-squares techniques and their statistical foundations.

• Statistical and Computational Methods in Data Analysis by Brandt (1976) is a useful resource on basic statistical concepts, e.g., maximum likelihood estimators and least-squares problems.

• Eichhorn's (1993) paper is an excellent review of the methods of least-squares as known and used in astronomy.

• Solving Least-Squares Problems by Lawson & Hanson (1974). A classic work worthwhile reading.

• Scientific Data Analysis by Branham (1990) is an introduction to overdetermined systems on an elementary level.

• Numerical Recipes - The Art of Scientific Computing by Press et al. (1992) is already a classic work and a useful source for efficient numerical calculations.

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