P.S. Make that f(Q) -: V mp (f E) mp %. V
(-: replaces =) Kip Kip Murray wrote: > There is a topic called analytic functions of a matrix that could be > applied here, if you represent the quaternion a0 + a1 i + b0 j + b1 k by > the matrix > > Q =: 2 2 $ a , b, (- + b), - + a > > where a means a0 j. a1 and b means b0 j. b1 > > When Q has distinct eigenvalues e0 and e1 with corresponding > eigenvectors v0 and v1, then > > f(Q) = V mp (f E) mp %. V > > when f is an analytic function, V is the matrix v0 ,. v1 and E is the > diagonal matrix 2 2 % e0 , 0, 0, e1 and f E is the diagonal matrix 2 2 $ > (f e0),0,0,f e1 . Function f has to be analytic in neighborhoods of e0 > and e1. mp is the matrix product +/ . * > > Analytic functions include monadic ^ and ^. and functions like those > found on the dictionary page for o. . They make this topic about as > hard as a typical engineering course in complex variables for which I > recommend Fundamentals of Complex Analysis by E. B. Saff and A. D. > Snider, Third Edition, Prentice Hall 2003. > > You can have x^y when quaternion y is a complex number but thinking of > z^y as an analytic function of z has the difficulties presented in a > complex variables course like Saff and Snider. > > Kip Murray > > > Raul Miller wrote: >> I have in the past thought that quaternions would be a nice addition >> to J. >> >> They are very useful, for example, for expressing rotations of >> geometric models. >> >> However, J of course needs its implementation to be generally correct >> and not just correct for some specific application. >> >> So I have been thinking of how to model J's primitives that would need >> to interpret quaternion values. >> >> This leads to questions such as: what does x^y mean when x and y are >> quaternions? >> >> Apparently, this is a hard problem: >> >> http://www.zipcon.net/~swhite/docs/math/quaternions/analysis.html >> >> I have been trying to find a definition of quaternion exponentials >> that that does not do something silly (like assume that quaternion >> multiplication is commutative). So far, I have not found anything I >> feel I could implement a J model for transcendental functions from. >> Also, x^y for quaternion x and quaternion y would often enough have an >> infinite set of valid results, which complicates things. >> >> So... does anyone have an good ideas about how to handle >> transcendental functions that work with quaternions? >> >> Thanks, >> > ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
