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
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