`===  The equations used in multidimensional Newton-Raphson for GPS are stated below.  For another derivation of these equations see Langley, R. B., ``The Mathematics of GPS,`` GPS World, Vol. 2, No. 7, July/August 1991, pp. 45-50 or for an online derivation see Navigation Solution in GPS World by Torralbas, Roberto and Alvarez Jose Manuel.  For an understanding of multidimensional Newton-Raphson as a general numerical method, see .  Let  and  denote the true receiver position coordinates. Let  denote the unknown clock error or bias. Let the coordinates of each satellite, and the time the message was sent, be , let the GPS clock's indicated received time be  and c be the speed of light. The pseudorange is computed as: : Assume the message travels at the speed of light, then the pseudorange satisfies the equation: :   When an approximate solution,  rather than the exact solution,  is used in equation 1, there is a residual, . Transforming  to the right hand side of the equation produces,  : :  A solution will have been obtained when  is zero or sufficiently close to zero for .  In order to linearize equation 2, the partial derivatives are computed as: : :  where :.  Linearizing the right hand side of equation 2 about the approximate solution,  there results  : :  where  is the residual due to linearization, which is in addition to the residual, , due to an approximate solution.  In order to drive  closer to zero choose the values  such that  : :  That is choose the values : such that the residual in equation 2 changes by approximately  as shown below.  Let : :  Substituting and transposing  to the left hand side of the equation, there results  :   Equations 6 provide four linear equations in four unknowns, the delta terms. They are in a form for solution. Using the values of  and  determined by this linear equation solution, :  is evaluated using: : :  Then set  in equations 2 through 6, plug the terms :  from equations 7 into equations 2, set  in equations 7, and reevaluate the residuals in equations 2. This procedure is repeated until the residuals are sufficiently small in magnitude.    === Multidimensional Case for more than four satellites`