Editing Global Positioning System (section)

==== 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.