, Another useful form of these equations is in terms of the pseudoranges, which are simply the ranges approximated based on GPS receiver clock's indicated (i.e. uncorrected) time so that . Then the equations becomes:
. Two of the most important methods of computing GPS receiver position and clock bias are (1) trilateration followed by one dimensional numerical root finding and (2) multidimensional Newton-Raphson calculations. These two methods along with their advantages are discussed.
- The receiver can solve by trilateration followed by one dimensional numerical root finding[. This method involves using Trilateration to determine the intersection of the surfaces of three spheres. It is clearly shown in trilateration that the surfaces of three spheres intersect in 0, 1, or 2 points. In the usual case of two intersections, the solution which is nearest the surface of the sphere corresponding to the fourth satellite is chosen. The surface of the earth can also sometimes be used instead, especially in the case of civilian GPS receivers since it is illegal in the United States to track vehicles of more than 60,000 feet in altitude. The bias, is then computed based on the distance from the solution to the surface of the sphere corresponding to the fourth satellite. Using an updated received time based on this bias, new spheres are computed and the process is repeated. One advantage of this method is that it involves one dimensional as opposed to multidimensional numerical root finding.
- The receiver can utilize multidimensional Newton-Raphson calculations. Linearize around an approximate solution, say from iteration k, then solve four linear equations derived from the quadratic equations above to obtain . The radii are large and so the sphere surfaces are close to flat. This near flatness may cause the iterative procedure to converge rapidly in the case where is near the correct value and the primary change is in the values of , since in this case the problem is merely to find the intersection of nearly flat surfaces and thus close to a linear problem. However when is changing significantly, this near flatness does not appear to be advantageous in producing rapid convergence, since in this case these near flat surfaces will be moving as the spheres expand and contract. This possible fast convergence is an advantage of this method. Also it has been claimed that this method is the "typical" method used by GPS receivers. A disadvantage of this method is that according to , "There are no good general methods for solving systems of more than one nonlinear equations." For a more detailed description of the mathematics see Multidimensional Newton Raphson.
- Other methods include:
- Solving for the intersection of the expanding signals form light cones in 4-space cones
- Solving for the intersection of hyperboloids determined by the time difference of signals received from satellites utilizing multilateration,
- Solving the equations in accordance with.
- More than four satellites should be used, if available. This results in an over-determined system of equations with no unique solution, which must be solved by least-squares or a similar technique. If all visible satellites are used, the results are always at least as good as using the four best, and usually better. Also the errors in results can be estimated through the residuals. With each combination of four or more satellites, a geometric dilution of precision (GDOP) vector can be calculated, based on the relative sky positions of the satellites used. As more satellites are picked up, pseudoranges from more combinations of four satellites can be processed to add more estimates to the location and clock offset. The receiver then determines which combinations to use and how to calculate the estimated position by determining the weighted average of these positions and clock offsets. After the final location and time are calculated, the location is expressed in a specific coordinate system such as latitude and longitude, using the WGS 84 geodetic datum or a local system specific to a country.
Finally, results from other positioning systems such as GLONASS or the upcoming Galileo can be used in the fit, or used to double check the result. (By design, these systems use the same bands, so much of the receiver circuitry can be shared, though the decoding is different.)