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- To: eigen@xxxxxxxxxxxxxxxxxxx
- Subject: Re: [eigen] cache-friendly matrix inverse
- From: Christian Mayer <mail@xxxxxxxxxxxxxxxxx>
- Date: Wed, 13 May 2009 22:43:11 +0200
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Benoit Jacob schrieb:
> It's actually something I never considered before: for me, as a
> mathematician, matrix inversion is a basic operation that "of course"
> i consider important, without even asking myself why; if you ask about
> practical uses for it, well I'm pretty sure that there must be
> situations where you want to solve Ax=b for a single given matrix A,
> but for a lot of vectors b and you don't know in advance what these
> vectors b will be, you just want to be able to solve very fast
> whenever they show up; then the best thing you can do is to precompute
> A^-1 so you only have a matrix-vector product to compute whenever a
> "b" vector shows up.
That's *exacly* what LU decomposition is for. Compute it once and use it
to solve the equation many times.
Note: the amount of operations to calculate A^-1 and an LU decomposition
is the same. *AND* using a LU to solve the equation (foreward and
backward substitution) takes also as many operations as a matrix
multiplication (i.e. exactly the same as a multiplication with the inverse)
An inverse matrix is an extremely usefull "method" in mathematics - as
long as you aren't *calculating* it in numerics :)
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