Hi,
I'm exploring ways to improve the ability of my GluCat library [ http://glucat.sf.net ] to evaluate functions in Clifford algebras. Since Clifford algebras have Real (and Complex) matrix representations, GluCat can use functions of matrices to do this.

I've been reading Nick Higham's book, Functions of Matrices: Theory and Computation [ http://www.maths.manchester.ac.uk/~higham/fm/ ], and see that for GluCat to evaluate matrix functions, it would need to first perform a Schur decomposition. LAPACK does this in two steps, by first reducing to upper Hessenberg form and then performing the Schur factorization [ http://www.netlib.org/lapack/lug/node50.html ] but LAPACK also provides simple driver routines [ http://www.netlib.org/lapack/lug/node31.html ].

GluCat currently uses uBLAS [ http://www.boost.org/doc/libs/1_39_0/libs/numeric/ublas/doc/index.htm ] but to perform the Schur decomposition, it would also need to use Boost Numeric Bindings [ http://mathema.tician.de/node/391 ], which is not yet released into Boost.

So, finally, questions. What is the status of Schur decomposition within eigen? Is it usable now? Is it comparable in speed to LAPACK?

We don't have a general schur decomposition yet. However we have a general eigensolver (for real matrices only, afaicr) and a selfadjoint eigensolver (does handle complex, afaicr). Since you're talking about functional calculus on matrices, what you need is an eigensolver, not a Schur decomposition. Although in the normal case, that amounts to the same. Are your elements normal (do they commute with their own adjoint)?

I'm updating a quite general library, and the matrices are real, square, but not necessarily normal.

An algorithm for matrix functions which uses an eigensolver which does not do Schur decomposition will suffer from numerical instability. Besides Higham's book, see Philip I. Davies, Nicholas J. Higham, "A Schur-Parlett Algorithm for Computing Matrix Functions" http://www.maths.manchester.ac.uk/~nareports/narep404.pdf.

ah ok i didn't know about the numerical stability issue!

well, all I can say is that we're adding Schur decomposition to our TODO.

FYI, Gael just merged a new general complex eigensolver and Schur decomposition. (In the devel branch).

See class ComplexSchur.(currently in QR module)

This is very recently merged code so zero guarantees....