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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 \