In the year of the dragon. Where is the tomb of the dragon? (continued)

"...The direct analytical solution of the problem of two heights is performed either by algebraic solution of the system of equations (215) or by successive solution of three spherical triangles (see Fig. 104); one of such solutions was given by Gauss in 1808. Numerous variants of the direct analytical solution are too complicated for "manual" calculations and attracted attention only after the appearance of computers. Iterative methods of analytical solution of the problem of two (or more) heights represent solutions of equations (215) or derivatives thereof by methods of successive approximations. Two variants of solutions are usually used: the tangent method (Newton's method) and the method of successive calculation of coordinates of isoline points. The tangent method is an analytical variant of the method of lines of position. Equation (215) of the isolines is decomposed into a Taylor series and the first terms of the series are taken. Solving two such equations, one obtains φ'=φc+∆φ and λ'=λc+∆λ. With these coordinates, the problem is solved secondarily, and so on. For ordinary number errors, one approximation (rarely two) is sufficient. When the number of observations and equations is large, one applies the ANC or its generalized version and obtains the most probable φ and λ. This method is suitable for any position lines and for any number of them, which is why it is used in most of the currently operating navigation computers. In the second method - the method of sequential calculation of coordinates of isoline points - φc is substituted into the first equation U1=sinh1 and λ1 of the point on U1 is calculated, this longitude is substituted into U2 and φ1 of the point on it is calculated, and so on, until the difference of the coordinate with the previous one becomes less than the specified value", - he continued reading.
Continuation follows. Beregovoy V.I. "In the Year of the Dragon. Where is the Tomb of the Dragon?"