This is definitely a fun exercise, and I'm having fun playing with it (mostly looking at intermediate results and a few variations on the theme).
That said, what I was really looking for was documentation or explorations on J's facilities (such as mt or lapack) for working with eigenvalues (and eigenvectors - in principle if you have the one the other is straightforward). Here, though, you did not use mt, nor did you use lapack. You started with a specific set of eigenvalues and demonstrated a context where you could plug them in. And I certainly appreciate the effort - and it does demonstrate some of the identities involving their use - but of course I am looking for more! Thanks, -- Raul On Sat, Oct 5, 2013 at 6:54 AM, km <[email protected]> wrote: > NB. Responding to a request from Raul in Jbeta > > > NB. A cool use of eigen values and vectors -- recursion > > > NB. Preliminaries > > clean =: * *@| NB. numbers with 2^_44 > | are replaced by 0 > > mp =: +/ . * NB. matrix product > > NB. below, square matrix x to integer power y > > mpwr =: ( 4 : 'x mp^:y =i.#x' )"2 0 > > > NB. The recursion ( (1+k) { u ) -: A mp k { u where u is a matrix > > ]A =: 1 1,.1 0 NB. used for Fibonnaci recursion > > ]lambda =: 2 %~ (1+%:5),1-%:5 NB. eigenvalues of A > > ]Lambda =: lambda * =i.# lambda NB. eigenvalues on diagonal > > ]S =: 2 2 $ lambda,1 1 NB. columns are eigenvectors > > (A mp S) -: lambda *&.|: S NB. test claimed eigen vectors, values > > ]u0 =: 1 0 NB. first row of matrix u > > ]nr =: 7 NB. number of rows in u > > ]u =: (A mpwr i. nr) mp"2 1 u0 NB. produce matrix u > > NB. Rows obey Fibonnaci recursion, observe second column > > (}. u) -: }: A mp&.|: u NB. test "A" recursion in the rows of u > > NB. You can produce u using the eigen vectors and values of A: > > u -: ( S mp"2 (Lambda mpwr i. nr) mp"2 S mpwr _1 ) mp"2 1 u0 > > NB. More simply: > > c =: (S mpwr _1) mp u0 > > u -: S mp"2 1 c *"1 lambda^"1 0 i. nr > > > NB. Above will work if n by n A has n linearly independent eigenvectors. > > > --Kip Murray > > Sent from my iPad > > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
