Re: matrix size

On Jun 8, 2009, at 4:50 AM, matt wrote:
> In MPB, eigenvalues are solved for using a hermitian positive  
> definite matrix.  Is there a way to deduce the number of degrees of  
> freedom or the matrix size as a function of the geometry resolution  
> chosen?  Also, under what conditions is this matrix sparse or dense?

The number of degrees of freedom is

	= (# grid points) * 2

where the number of grid points is the product of the resolution times  
each dimension of the unit cell (the grid is also printed out when you  
run MPB). The "* 2" is for 2 polarizations; if you are in two  
dimensions and only want the TE or TM polarization, it is "* 1"  
instead.  These degrees of freedom are complex numbers; if you have  
inversion symmetry (mpbi) they are real numbers.

The matrix is never sparse.  However, the matrix need not be stored  
explicitly; the key point is that the matrix-vector products can be  
computed in O(N log N) time and O(N) storage using FFTs, as described  
in the MPB paper (and also in appendix D of our book online, ab- 
initio.mit.edu/book).

In linear algebra terms, the reason we can do this is because the  
matrix can be factorized into a product of sparse matrices and  
circulant matrices. Also, we only want a few of the eigenvalues so we  
can use iterative methods.

Steven