N

Aiv(xk + Axk)Wvdv(xk + Axk) = 0, i = 1,..., n. (4.3.4)

Taylor-series expansion up to first-order derivatives gives

(A(xk) + AxTG) W (d(xk) + AT(xk) ■ Axk) = 0, or with the Hessian matrix G^ [interpret this as an M(n, n) matrix at each data point indexed by v]

5 Many photometric light curve data have been analyzed with the Wilson-Devinney program (Wilson & Devinney, 1971) who were the first to use the method of Differential Corrections with a physical light curve model.

d id

Multiplication (and neglecting second-order terms in Axk ) finally leads to

A(xk) ■ W ■ d(xk) + [dT(Xk) ■ W ■ G(xk) + A(xk) ■ W ■ AT(x^)] Axk = 0, (4.3.6) or component-wise for i = 1,..., n [see, for instance, Powell (1964a)]

Gjxk )w dv (xk) + Ai „ (xk) Ajv (xk) Under the "small residual assumption"

||dT(xk) ■ W ■ G(xk)Axk | « ||A(xk) ■ W ■ AT(xk)Axk || , (4.3.8) we get the equations

[A(xk) ■ W ■ AT(xk)] ■ Axk = -A(xk) ■ Wd(xk), (4.3.9) which is the normal equation of the linear least-squares problem min xk

Note the similarity between the method described here and the formal procedure in Appendix A.3.3 with J = AT. Iterations are continued with xk+1 = xk + Axk (4.3.11)

until one of the stopping criteria described in Sect. 4.4.2 terminates the algorithm. To compute the correction vector Axk in the kth iteration

Axk = -C ■ A(xk) ■ Wd(xk), C := C-1 = (£,,), (4.3.12)

we need the inverse of the matrix

The inverse of C is just the covariance matrix C. Under the assumptions discussed, for example, by Press et al. (1992, pp. 690-693), the diagonal elements of the covariance matrix C provide a measure for the probable error 8 (see Sect. 4.4.3).

In summary, the method of Differential Corrections belongs to a class of Newtontype methods without second derivatives. It makes use of the fact that the gradient

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