One of the coolest algorithms of all time is the use of Newton iterations to compute the "polar decomposition" of a matrix: http://www.cs.wisc.edu/graphics/Courses/838-s2002/Papers/polar-decomp.pdf I believe that this algorithm is used in graphics animation systems. Re quaternions: matrix algorithms can always be used, by simply using the matrix version of a quaternion (either real 4x4 or complex 2x2). I am not aware of any Newton iterations for quaternions that aren't inherited from real/complex numbers (and therefore assume commutativity), or are inherited from more general matrices. *** Challenge *** Are there any such Newton algorithms that work _only_ for quaternions? At 09:09 AM 2/13/2007, Fred lunnon wrote:

Recent discussion of Newton-Raphson iteration led me to reconsider the of the iterative algorithm for reciprocal of a scalar c --- x <- 2 x - a x^2 --- derived via N-R applied to the scalar function of a scalar f(x) = c - 1/x.

This has a matrix analogue (mentioned here a while back, in a thread discussing the shortcomings of Gauss-Seidel iteration) for the inverse of a matrix A --- X <- 2 X - X A X --- which, by a rather nice twist, can actually be employed in inverting the Jacobian matrix required to apply Newton- Raphson to a vector function of a vector variable!

But what matrix calculus framework might permit the matrix iteration to be derived in a fashion analogous to the scalar version? In particular, is it possible to restrict the universe of matrix functions --- as is done for functions of a complex variable, and attempted less successfully for a quaternion variable --- in such a way as to facilitate this?

There is indeed a recognised topic which appears relevant --- see on the web e.g. Wikipedia; or appendix D of John Dattoro "Convex Optimization & Euclidean Distance Geometry" --- but closer iinspection reveals this to be merely a reworking of tensor notation omitting the subscripts: in particular, the derivative of a matrix function of a matrix variable is a rank 4 object, in effect a Jacobian.

Fred Lunnon