Subject: Re: [ublas] Question/request for matrix powers and other stuff
From: Jens Seidel (jensseidel_at_[hidden])
Date: 2008-10-22 04:02:08
On Tue, Oct 21, 2008 at 04:18:52PM -0400, Daryle Walker wrote:
> On Oct 21, 2008, at 3:25 PM, Gunter Winkler wrote:
>> In general you can use the following procedure:
>> given: square matrix A
>> compute a the jacobi matrix A = X^T J X
"jacobi" matrix" should be probably replaced by "Jordan". See
A Jordan block has the great advantage that it is trivial
to compute powers of it and every square matrix can be transferred
into such a type (by a base transformation). But this is generally
not numerically stable IIRC.
>> compute J^n
>> compute A^n = X^T J^n X
>> The main problem is the first step which requires the solution of a
>> generalized eigenvalue problem ...
> The code I'm adapting sometimes enters a LOT of zero-values in a row.
> The original author realized that, since inserting a zero can be
> expressed as a linear transformation, using the matrix form and raising
> it to a power can _save_ time over inserting each zero manually. (The
> original author used a bit-packing scheme for the GF(2) matrices
> involved, which I'll add later.) The manual method is by definition
> linear on the number of zeros added. Using matrices and powers,
> especially with square-and-multiply, should represent a savings when the
> length gets long enough.
I don't understand this :-)