Editing Global Positioning System (section)

=== Solution methods ===

==== Least squares ====
When more than four satellites are available, the calculation can use the four best, or more than four simultaneously (up to all visible satellites), depending on the number of receiver channels, processing capability, and [[Dilution of precision (GPS)|geometric dilution of precision]] (GDOP).

Using more than four involves an over-determined system of equations with no unique solution; such a system can be solved by a [[least-squares]] or weighted least squares method.<ref name=GPS_BASICS_Blewitt />

:<math>\left( \hat{x},\hat{y},\hat{z},\hat{b} \right) = \underset{\left( x,y,z,b \right)}{\arg \min} \sum_i \left( \sqrt{(x-x_i)^2 + (y-y_i)^2 + (z-z_i)^2} + bc - p_i \right)^2</math>

==== Iterative ====
Both the equations for four satellites, or the least squares equations for more than four, are non-linear and need special solution methods. A common approach is by iteration on a linearized form of the equations, such as the [[Gauss–Newton algorithm]].

The GPS was initially developed assuming use of a numerical least-squares solution method—i.e., before closed-form solutions were found.

==== Closed-form ====
One closed-form solution to the above set of equations was developed by S. Bancroft.<ref name=Bancroft /><ref name=Bancroft1985>{{cite journal |last1=Bancroft |first1=S. |date=January 1985 |title=An Algebraic Solution of the GPS Equations |journal=IEEE Transactions on Aerospace and Electronic Systems |volume=AES-21 |issue=1 |pages=56–59 |doi=10.1109/TAES.1985.310538 |bibcode=1985ITAES..21...56B|s2cid=24431129 }}</ref> Its properties are well known;<ref name="Abel1" /><ref name="Fang" /><ref name="Chaffee">Chaffee, J. and Abel, J., "On the Exact Solutions of Pseudorange Equations", ''IEEE Transactions on Aerospace and Electronic Systems'', vol:30, no:4, pp: 1021–1030, 1994</ref> in particular, proponents claim it is superior in low-[[geometric dilution of precision|GDOP]] situations, compared to iterative least squares methods.<ref name=Bancroft1985 />

Bancroft's method is algebraic, as opposed to numerical, and can be used for four or more satellites. When four satellites are used, the key steps are inversion of a 4x4 matrix and solution of a single-variable quadratic equation. Bancroft's method provides one or two solutions for the unknown quantities. When there are two (usually the case), only one is a near-Earth sensible solution.<ref name=Bancroft />

When a receiver uses more than four satellites for a solution, Bancroft uses the [[generalized inverse]] (i.e., the pseudoinverse) to find a solution. A case has been made that iterative methods, such as the Gauss–Newton algorithm approach for solving over-determined [[non-linear least squares]] problems, generally provide more accurate solutions.<ref name="Sirola2010">{{cite conference |last1=Sirola |first1=Niilo |date=March 2010 |title=Closed-form algorithms in mobile positioning: Myths and misconceptions |book-title=7th Workshop on Positioning Navigation and Communication |conference=WPNC 2010 |pages=38–44 |doi=10.1109/WPNC.2010.5653789|citeseerx=10.1.1.966.9430 }}</ref>

Leick et al. (2015) states that "Bancroft's (1985) solution is a very early, if not the first, closed-form solution."<ref>{{cite book|title=GNSS Positioning Approaches – GPS Satellite Surveying, Fourth Edition – Leick |publisher= Wiley Online Library|doi=10.1002/9781119018612.ch6|pages=257–399|chapter = GNSS Positioning Approaches|year = 2015|isbn = 9781119018612}}</ref>
Other closed-form solutions were published afterwards,<ref name="Kleus">Alfred Kleusberg, "Analytical GPS Navigation Solution", ''University of Stuttgart Research Compendium'', 1994.</ref><ref name="Oszczak">Oszczak, B., "New Algorithm for GNSS Positioning Using System of Linear Equations", ''Proceedings of the 26th International Technical Meeting of The Satellite Division of the Institute of Navigation (ION GNSS+ 2013)'', Nashville, Tennessee, September 2013, pp. 3560–3563.</ref> although their adoption in practice is unclear.