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From: charger144 on 9 Jun 2010 06:42
Hello:

I am using an iterative method that requires a matrix and vector
input (solve special Ax=b for x), such that the matrix has normalized
columns (norm = 1), and this matrix has a spectral norm (biggest
singular value) < a certain constant (for example, 1) for convergence.

For any matrix A, I can influence the spectral norm by finding the
largest singular value via power iteration and scaling by an
appropriate value (and doing same with the vector). However, given my
matrix A, the algorithm works with the matrix:

M = A*D where D is a diagonal matrix whose entries (i,i) contain 1/
column_norm(A(:,i)). This way, M is the normalized column version of
matrix A that I feed to the algorithm.
What I am unsure about is how to "precondition" A so that the
resulting M has spectral norm below a certain amount?

i.e. It seems that I can find an invertible P such that (A*P)*D has a
given spectral norm if I know the spectral norm of A.

I appreciate any input on this problem. Thanks.
From: charger144 on 10 Jun 2010 05:27
On Jun 9, 6:42 am, charger144 <charger...(a)rocketmail.com> wrote:
> Hello:
>
>  I am using an iterative method that requires a matrix and vector
> input (solve special Ax=b for x), such that the matrix has normalized
> columns (norm = 1), and this matrix has a spectral norm (biggest
> singular value) < a certain constant (for example, 1) for convergence.
>
> For any matrix A, I can influence the spectral norm by finding the
> largest singular value via power iteration and scaling by an
> appropriate value (and doing same with the vector). However, given my
> matrix A, the algorithm works with the matrix:
>
> M = A*D where D is a diagonal matrix whose entries (i,i) contain 1/
> column_norm(A(:,i)). This way, M is the normalized column version of
> matrix A that I feed to the algorithm.
> What I am unsure about is how to "precondition" A so that the
> resulting M has spectral norm below a certain amount?
>
> i.e. It seems that I can find an invertible P such that (A*P)*D has a
> given spectral norm if I know the spectral norm of A.
>
> I appreciate any input on this problem. Thanks.

Do you have any ideas on this? It seems that to influence the spectral
norm of M, you have to influence many singular eigenvalues of A
(instead of just the largest one defining the spectral norm).
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