Editing Invertible matrix (section)

=== ''p''-adic approximation ===
If {{math|'''A'''}} is a matrix with [[integer]] or [[rational number|rational]] entries, and we seek a solution in [[arbitrary-precision arithmetic|arbitrary-precision]] rationals, a [[p-adic|{{mvar|p}}-adic]] approximation method converges to an exact solution in {{math|O(''n''{{sup|4}} log{{sup|2}} ''n'')}}, assuming standard {{math|O(''n''{{sup|3}})}} matrix multiplication is used.<ref>{{cite journal | doi = 10.1016/j.cam.2008.07.044 | volume=225 | title=A p-adic algorithm for computing the inverse of integer matrices | journal=Journal of Computational and Applied Mathematics | pages=320–322| year=2009 | last1=Haramoto | first1=H. | last2=Matsumoto | first2=M. | issue=1 | bibcode=2009JCoAM.225..320H | doi-access=free }}</ref> The method relies on solving {{mvar|n}} linear systems via Dixon's method of {{mvar|p}}-adic approximation (each in {{math|O(''n''{{sup|3}} log{{sup|2}} ''n'')}}) and is available as such in software specialized in arbitrary-precision matrix operations, for example, in IML.<ref>{{cite web|url=https://cs.uwaterloo.ca/~astorjoh/iml.html|title=IML - Integer Matrix Library|website=cs.uwaterloo.ca|access-date=14 April 2018}}</ref>