## Group Okunbor

Group Description ```Acoustic radiation around aft duct fan. Walter Eversman and Daniel Okunbor. These matrices are generated by a finite element code. (C)1997 Copyright by Walter Eversman and Daniel Okunbor University of Missouri-Rolla okunbor :at the domain: cs.umr.edu -------------------------------------------------------------------------------- NOTE (added Feb 9, 2012): The aft01 matrix is numerically rank deficient. It is tagged in the UF Sparse Matrix Collection as 'positive definite' because chol works on it, but some eigensolvers report negative eigenvalues. See below for details. -------------------------------------------------------------------------------- On 01/26/2012 11:37 AM, Alexander Andrianov wrote: Just a quick note somewhat related to the previous discussion. It looks like aft01 instance is not PD: its smallest eigenvalue is about -2.0312E-03. Just confirmed that by running dense eigensolvers including one from MKL library (LAPACK’s DSYEV with 'N' option). A modified Cholesky factorization however does not choke and factorizes it nicely, with normalized residual of about 1.0e-17. Not sure about its numerical rank though. Best regards, Alex -------------------------------------------------------------------------------- From: Tim Davis Sent: Friday, January 27, 2012 5:50 PM To: Alexander Andrianov Cc: Leslie Foster Subject: Re: UF collection matrices info This is a wierd matrix. Its numerical rank is one. chol in MATLAB succeeds, unmodified, with a low residual [L,p,s]=chol(A,'lower','vector'); norm (L*L'-A(s,s),1) the diagonal of L has elements in the range 0.1325 to 3.16e7 Maybe the eigenvalue solver is the one giving the 'wrong' answer. This problem might be close to PD, and chol and eig are giving different results. I tried this: e = eig (full(A)) ; and got a largest eigenvalue of 1e15, and a smallest one as 1.493e-4. I don't see the -2e-3 that you got. So eig in MATLAB (also uses LAPACK), says the matrix (barely) PD. Numerically, the positive definiteness of this matrix looks like a toss up. Thanks, Tim -------------------------------------------------------------------------------- From: Alexander Andrianov at SAS.com Subject: RE: UF collection matrices info Date: January 30, 2012 9:52:55 AM EST To: Davis Tim Cc: Leslie Foster You may be right. I just ran it under various dense MKL solvers and got the following: Smallest DSYEV -2.0311917e-3 eigenvalues only ('N' option) DYSEV -2.3789104e-1 complete eigendecomposition ('V' option) DSYEVR 1.5329287e-4 complete eigendecomposition DSYEVD -2.2634604e-3 complete eigendecomposition Also ran with my version of DSYEVD and got: 1.5259751e-4 for eigenvalues only 9.0656595e-3 for complete eigendecomposition The normalized maximum residual was 6.423e-4 (excellent) but the normalized orthogonality loss was 1.347e11 (very bad). This is probably due to the fact that it is rank-deficient. I don't know if this helps: the LAPACK team labels DSYEV and DSYEVD as being in general more accurate (see LAPACK working note 183, page 16) but it doesn't imply that they will beat DSYEVR all the time. I guess this instance can be labeled as 'numerically PD' due to the rounding error during Cholesky factorization which helps make it PD after all. Thank you for running MATLAB on it, Alex```
Displaying all 2 collection matrices
Id Name Group Rows Cols Nonzeros
792 aft02 Okunbor 8,184 8,184 127,762
791 aft01 Okunbor 8,205 8,205 125,567