In article <1174438438.819699.201...@l75g2000hse.googlegroups.com>,
car...@colorado.edu> wrote:
I have an undetermined linear system A x = b where A
is m x n with m < n. A is real and has full rank m.
The well known general solution is
x = Y b + Z q (*) where
Y n x m Penrose pseudoinverse of A,
Z n x (n-m) basis for null space (kernel) of A,
q an array of n-m undetermined coefficients.
Y and Z are computed and saved while processing (*).
Suppose later I have to solve the overdetermined dual system
A' y = c where A' is the transpose of A. Can Y and Z be
of use here?
The scenario arises in the guidance and trajectory control
simulation of a discrete dynamical model. Typically
m = 10K to 1M, n approx 2m to 3m, but A is very sparse.
P_inverse(A^T) = p_inverse(A)^T. so you need to compute
the right and left nullspaces of A, so you have x = Y b + Zq
and y = Y^T c + W^T w where W is a basis of the left null space of
A ( {x | xA = 0}) if I read this correctly. I don't know of any
relationships between Y,Z and W and not A [or a decomposition of A]
