## Fx x q Fqx d E

This nonlinear equation (E.12.4) is solved with the Newton-Raphson algorithm. Therefore, the derivative f ' (x ) is needed:

Now the Newton-Raphson procedure proceeds as x («+1) = x« + ^x, 4x = -f (x (n))/f '(x(n)) (E.12.6)

with the initial value x(Q) = d/2. If sufficient accuracy is achieved, viz., if \Ax | < 10-6, the iteration is halted and xLp is set to xlp := x(«+1). (E.12.7)

With known xLp it is easy to compute ^j3"' as

The case to compute xLp and for Lp is somewhat more difficult. The computation of xLp and is only valid for F1 = F2 = 1 and d = 1. For nonsynchronous x or noncircular cases these quantities are not needed. In the valid cases, the inequality d < xLp (E.12.9)

holds; Wilson uses the same function f (x) as defined in (E.12.4) but computation is performed after explicit setting of Fj = 1 and d = 1. Furthermore, this time, the initial value x(0) = 1 + M1/3 + 3 M2/3 + I M3/3 + (E.12.10)

with

1 Q ^ \ q, 0 < q < 1, ,cn,n a := 1-, Q := i 1 , (E.12.11)

is used. If convergence is completed, xL2p follows from xLp := x(n+1). (E.12.12)

For completeness we note for d = 1 and F1 = 1

^ = 1 + q( . 1 o - ^ + x2 (E.12.13) 2 x * V V1 - 2x + x2 / 2

with

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