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 begin Bmatrix f 1 k p 1 f 2 k p 2 f 3 k p 3 f 4 k p 4 end Bmatrix begin bmatrix frac x k x 1 R 1 frac y k y 1 R 1 frac z k z 1 R 1 c