[eigen] HessenbergDecomposition::householderCoefficients()

[Jitse Niesen <jitse@xxxxxxxxxxxxxxxxx>]

Hello,

I have some questions about the function householderCoefficients() in 
HessenbergDecomposition, which returns the vector of coefficients for the 
Householder reflections which reduce the given matrix to Hessenberg form.

Why does it return a vector of length N-1 (where N is the size of the 
matrix whose Hessenberg decomposition is being computed)? I think that you 
need only N-2 Householder reflections to reduce a matrix to Hessenberg 
form. The last two columns of the matrix are already in the correct form, 
so only N-2 columns need to be transformed. See also for instance 
Algorithm 7.4.2 in Golub & Van Loan. This should perhaps also be taken 
into account when computing the Hessenberg decomposition.

While I was thinking about this, I started to wonder whether we should 
store the coefficients at all. For real matrices, the coefficients are 
given by 2 / |v|^2 where v is the Householder vector so it is not 
necessary to store them. On the other hand, we do save some effort if we 
do not recompute the coefficients - though probably not very much - and 
the memory requirements for storing the coefficients is only O(N). In the 
complex case, the coefficients are not uniquely determined by the vectors 
IIRC, and it makes sense to use the same algorithm for both real and 
complex scalars if possible. So there seems to be a good argument for 
sticking to the current strategy of storing the coefficients.

The EigenSolver class uses its own routine for reducing the matrix to 
Hessenberg form. I assume that this is simply because nobody has had time 
to rewrite it to use HessenbergDecomposition.

Cheers,
Jitse