- #1
vibe3
- 46
- 1
I have a matrix for which I know its QR decomposition: [itex]A = QR[/itex]. I want to estimate the largest and smallest singular values of [itex]A[/itex] ([itex]\sigma_1[/itex] and [itex]\sigma_n[/itex]) however in my application it is too expensive to compute the full SVD of [itex]A[/itex].
Is it possible to estimate the largest/smallest singular values from the QR decomposition? The only result I've been able to find so far is
[tex]
\left| \prod_i r_{ii} \right| = \prod_i \sigma_i
[/tex]
where [itex]r_{ii}[/itex] are the diagonal entries of [itex]R[/itex]. I'm not sure if this implies that the singular values of [itex]R[/itex] are the same as the singular values of [itex]A[/itex]. If that's true, it might be possible and less expensive for my application to compute [itex]SVD(R)[/itex] rather than [itex]SVD(A)[/itex].
Is it possible to estimate the largest/smallest singular values from the QR decomposition? The only result I've been able to find so far is
[tex]
\left| \prod_i r_{ii} \right| = \prod_i \sigma_i
[/tex]
where [itex]r_{ii}[/itex] are the diagonal entries of [itex]R[/itex]. I'm not sure if this implies that the singular values of [itex]R[/itex] are the same as the singular values of [itex]A[/itex]. If that's true, it might be possible and less expensive for my application to compute [itex]SVD(R)[/itex] rather than [itex]SVD(A)[/itex].