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Thread Subject:
Regularized weighted linear system solution

Subject: Regularized weighted linear system solution

From: Michal Kolaj

Date: 16 Feb, 2012 20:23:12

Message: 1 of 2

I would appreciate some guidance for the following problem,

I am looking to solve a linear system of the form:
x = inv(A'*inv(V)*A+E+S)*A'*inv(V)*b
where Ax=b is the linear system I am solving, V is the covariance matrix (from the error in b) and E and S are various regularization matrices.

Previously, I have been solving such systems by using x=[A;E;S]/[b;zeros;zeros] but this did not include the errror matrix V. Could someone point me in the right direction as to how to include this? I believe it could be implemented into the above simple equation by Cholesky decomposition of V but I am not certain as to how to do this.

I noted that lscov has an option for including a covariance matrix but I am not certain how to then include my reguarlization matrices; would I just append them to A and b like I did above i.e. lscov([A;E;S],[b;zeros;zeros],V)? Would there be a difference if I did it this way versus incorporating V into matlabs backslash operator solver?

Thank you for any suggestions/help

Subject: Regularized weighted linear system solution

From: Michal Kolaj

Date: 16 Feb, 2012 21:59:23

Message: 2 of 2

"Michal Kolaj" wrote in message <jhjojg$ala$1@newscl01ah.mathworks.com>...
> I would appreciate some guidance for the following problem,
>
> I am looking to solve a linear system of the form:
> x = inv(A'*inv(V)*A+E+S)*A'*inv(V)*b
> where Ax=b is the linear system I am solving, V is the covariance matrix (from the error in b) and E and S are various regularization matrices.
>
> Previously, I have been solving such systems by using x=[A;E;S]/[b;zeros;zeros] but this did not include the errror matrix V. Could someone point me in the right direction as to how to include this? I believe it could be implemented into the above simple equation by Cholesky decomposition of V but I am not certain as to how to do this.
>
> I noted that lscov has an option for including a covariance matrix but I am not certain how to then include my reguarlization matrices; would I just append them to A and b like I did above i.e. lscov([A;E;S],[b;zeros;zeros],V)? Would there be a difference if I did it this way versus incorporating V into matlabs backslash operator solver?
>
> Thank you for any suggestions/help


if V=1/error, could I do this simply by just computing
x=[diag(error)*A;E;S]/[diag(error)*b;zeros;zeros]

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