Just thinking about your application: There are good eigenvalue algorithms for almost-diagonal matrices.
If you notionally arrange your n 4x4 matrices along the principal diagonal of a 4nx4n matrix, you will have a matrix with only 5 nonzero diagonals, and the eigenvalues of this big matrix will be the same as the eigenvalues of the individual 4x4s. Of course, you wouldn't create the entire big matrix. You could convert your pentadiagonal matrix to tridigonal using Householder transformations, and then use a tridiagonal solver to get the eigenvalues. [I don't know if the eigenvalues would come out in an order that allows you to associate them with the original 4x4s. I assume it would be possible to make that happen]. Numerical Recipes discusses this. I have code that I translated from Numerical Recipes, restricted to the case of symmetric matrices, that converts a symmetric matrix to tridiagonal form and then calculates the eigenvalues. I don't have anything to perform the pentadiagonal-to-tridiagonal conversion, but one Householder transformation should do the trick, and you might be able to work it out from the general tridiagonalization code. I will send you what I have if you like. The main point is, doing many eigenvalue calculations in parallel may be a winner. Henry Rich On 6/12/2012 11:08 AM, Raul Miller wrote: > On Tue, Jun 12, 2012 at 10:41 AM, Henry Rich<henryhr...@nc.rr.com> wrote: >> That may be. But no way do you do a determinant with just one multiply >> - it takes 0 for a 1x1, 2 for a 2x2. > > You are thinking about scalars. > > -/ .(*&([smoutput)) i. 2 2 > 3 1 > 0 2 > _2 > > Here, we are computing 0 2 * 3 1 > > This is what I mean by "one multiply". > > You are correct that if I investigated at rank 0 that there would be > two multiplies. > >> The definition of DET is so quirky that it makes sense only when you >> have subtraction/ . multiplication > > All I had to do to make it behave like the implementation was terminate > recursion when derived lists contain 1 element instead of 0 elements. > >> for suitably defined operations on >> the space of interest. Even +/ . * runs in factorial time. For the >> case of finding the characteristic polynomial, why not just use the verb >> definitions I gave earlier, rather than requiring a new DET? > > Which verb? > > Anyways, one issue here is whether the dictionary and the > implementation agree. Currently, they do not. But I currently see no > value in using the dictionary definition over the actual > implementation -- they are equivalent for -/ .* and I have not found a > useful case where the dictionary DET does something meaningful that > the implementation does not. > > Another issue here is generality, and motivation. There is something > nice about being able to use the same "determinant primitive" to > compute both regular determinants and eigenvalues. And even if that > never gets into the interpreter, I find some joy in modeling things > that way. > > The "resource cost" issue you are raising can be significant, in some > contexts, but it's not going to be significant in all contexts. In > the case I am working up to dealing with (several hundred collections > 4x4 matrices where each collection typically contains of several > thousand of these 4x4 matrices), it's hard for me to see why I should > care about it. > ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
