solves an upper triangular system of linear equations.
The rhs is the last column of the N by N+1 matrix A. The solve starts
in the process column owning the Nth column of A, so the rhs b may
need to be moved one process column to the left at the beginning. The
routine therefore needs a column vector in every process column but
the one owning b. The result is replicated in all process rows, and
returned in XR, i.e. XR is of size nq = LOCq( N ) in all processes.
The algorithm uses decreasing one-ring broadcast in process rows and
columns implemented in terms of synchronous communication point to
point primitives. The lookahead of depth 1 is used to minimize the
critical path. This entire operation is essentially latency bound
and an estimate of its running time is given by:
(move rhs) lat + N / ( P bdwth ) +
(solve) ((N / NB)-1) 2 (lat + NB / bdwth) +
gam2 N^2 / ( P Q ),
where gam2 is an estimate of the Level 2 BLAS rate of execution.
There are N / NB diagonal blocks. One must exchange 2 messages of
length NB to compute the next NB entries of the vector solution, as
well as performing a total of N^2 floating point operations.