Re: invert many matrices


In article <drnu27$32v$1@xxxxxxxxxxxxxxxx>,
Erik van Zwet <evanzwet@xxxxxxxxxxxxxxxxxx> writes:
>Hi all,
>
>Let m<n and A a real mxn matrix of full rank. I keep A fixed, and need
>to invert ADA' for many different nxn diagonal matrices D. Can anyone
>think of a trick to do this efficiently?
>
>Best regards,
>Erik van Zwet
>

aaah, interior point once again: no. if you change all of D, then there is
little to save. but if you use an iterative solver, say conjugate gradients,
and some components of D vary slowly and others are neglectible small
compared with the largest ones, then you could fix a decomposition of
the previous case and use this as a preconditioner , which makes cg fast
changing the preconditioner only if cg slows down. this is an old trick
used already by monteiro and others and is quite useful.
also you never should really invert those matrices, rather decompose them
by Cholesky or L*D*L' (this is another D here of course)
the savings you think about apply only if you have low rank changes in the
matrix, i.e. only some components of D change. there are updating formulas
for Cholesky and LU in this case. (for the inverse matrix this is
sherman-morrison-woodbury, but again, never invert these matrices explicitly)
hth
peter


.

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