Editing Invertible matrix (section)

=== Newton's method ===
A generalization of [[Newton's method]] as used for a [[Multiplicative inverse#Algorithms|multiplicative inverse algorithm]] may be convenient if it is convenient to find a suitable starting seed:
: <math>X_{k+1} = 2X_k - X_k A X_k.</math>

[[Victor Pan]] and [[John Reif]] have done work that includes ways of generating a starting seed.<ref>{{Citation
 | first1 = Victor
 | last1 = Pan
 | first2 = John
 | last2 = Reif
 | title = Efficient Parallel Solution of Linear Systems
 | series = Proceedings of the 17th Annual ACM Symposium on Theory of Computing
 | year = 1985
 | place = Providence
 | publisher = [[Association for Computing Machinery|ACM]]
}}</ref><ref>
{{Citation
 | first1 = Victor
 | last1 = Pan
 | first2 = John
 | last2 = Reif
 | title = Harvard University Center for Research in Computing Technology Report TR-02-85
 | year = 1985
 | place = Cambridge, MA
 | publisher = [[Aiken Computation Laboratory]]
}}</ref>

Newton's method is particularly useful when dealing with [[family (set theory)|families]] of related matrices that behave enough like the sequence manufactured for the [[homotopy]] above: sometimes a good starting point for refining an approximation for the new inverse can be the already obtained inverse of a previous matrix that nearly matches the current matrix. For example, the pair of sequences of inverse matrices used in obtaining [[Matrix square root#By Denman–Beavers iteration|matrix square roots by Denman–Beavers iteration]]. That may need more than one pass of the iteration at each new matrix, if they are not close enough together for just one to be enough. Newton's method is also useful for "touch up" corrections to the Gauss–Jordan algorithm which has been contaminated by small errors from [[round-off error|imperfect computer arithmetic]].