Given a pair (81, e2), the corresponding values of f(el,e2) and g(81,82) are obtained by the following steps:
Step 1. For initial values of l81 and 82, compute the coordinates of vectors u and q as given by Eqs. (7.16) and (7.17). The inner products u.q and q.q can be computed using the regular inner product formula,
u.q = Uxqx + uy qy + Uzqz
and
q.q = qx2 + q2 + qz2
Step 2. Solve the reduced system of Eqs. (7.20)-(7.23) for 03 and 85.
Step 3. Compute the value of 84 from Eqs. (7.28) and (7.29) or Eqs. (7.30) and (7.31).
Step 4. Compute the inner products uL*qL and qL.qL, given by Eqs. (7.32) and (7.34), respectively, and compute the values of f and g as given by Eqs. (7.12) and (7.13). Two-Dimensional Newton-Raphson
The zeros of f and g can be iteratively computed and checked for consistency with Eq. (7.1). If a computer
program for evaluating the two functions is available, the partial derivatives of f and g with respect to el and 02, denoted fl, f2 and gl, g2 respectively, can be numerically approximated by
fl(el,e2)= 1f/ 1 =1f( +681',e2)f(e1,2)]/661, (7.35)
..