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.. _band_parallelization:
====================
Band parallelization
====================
The orthogonalization can be paralleized over **k**-points, spins,
domains (see :ref:`orthogonalization`), and bands, described below.
Let's say we split the bands in five groups and give each group of
wave functions to one of five processes:
The overlap matrix contains 5x5 blocks. These are the steps::
rank: 1 2 3 4 5
A . . . . . B . . . . . C . . . . . . . . . . . .
. . . . . . A . . . . . B . . . . . C . . . . . .
S: . . . . . . . . . . . . A . . . . . B . . . . . C
C . . . . . . . . . . . . . . . . . A . . . . . B
B . . . . . C . . . . . . . . . . . . . . . . . A
A. Each process calculates its block in the diagonal and sends a copy
of its wave functions to the right (rank 5 sends to rank 1).
B. Rank 1 now has the wave functions from rank 5, so it can do the row
5, column 1 block of `\mathbf{S}`. Rank 2 can do the row 1, column
2 block and so on. Shift wave functions to the right.
C. Rank 1 now has the wave functions from rank 4, so it can do the row
4, column 1 block of `\mathbf{S}` and so on.
Since `\mathbf{S}` is symmetric, we have all we need::
A B C . .
. A B C .
. . A B C
C . . A B
B C . . A
With `B` blocks, we need `(B - 1) / 2` shifts.
Now we can calculate `\mathbf{L}^{-1}` and do the matrix product
`\tilde{\mathbf{\Psi}}_0 \mathbf{L}^{-1}` which requires `B - 1`
shifts of the wave functions.
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