The inner-products u.q and q.q can then be easily computed by


 u.q = uxqx + uyqy + Uzqz

and

 q.q = qx2 + qy2 + qz2


when the numeric values of the components of u and q have been obtained. These inner products can be obtained from Eqs. (6.5) and (6.6) as well,


u.q = (R1-1 t) (R-1 (p 11)) = t (p 11), u.q = tx (Px al Cl) + ty (py al Sl) (6.13)

and

q.q = R1-(p 1i) R1-1(p 11) = (p 11).(p 11), q.q = p.p + 11.11 2(p.11)

q.q = p.p + a12 2 (pxal C1 + pyal Sl). (6.14)


 Equations (6.11)-(6.14) clearly show that all components of u and q, and the inner-products u.q and q.q are linear functions of S1 and C1, a result that will prove useful in the next section.

 To summarize, f(e1) can be computed for a given value of e1 according to the following steps:

 Step 1. From the current estimate of e1, Compute the components of u and q and the inner products u.q and q.q as shown in Eqs. (6.11)-(6.14).

..