[petsc-users] Orthogonality of eigenvectors in SLEPC
Wang, Kuang-chung
kuang-chung.wang at intel.com
Thu Dec 2 14:06:21 CST 2021
Thanks Jose for your prompt reply.
I did find my matrix highly non-hermitian. By forcing the solver to be hermtian, the orthogonality was restored.
But I do need to root cause why my matrix is non-hermitian in the first place.
Along the way, I highly recommend MatIsHermitian() function or combining functions like MatHermitianTranspose () MatAXPY MatNorm to determine the hermiticity to safeguard our program.
Best,
Kuang
-----Original Message-----
From: Jose E. Roman <jroman at dsic.upv.es>
Sent: Wednesday, November 24, 2021 6:20 AM
To: Wang, Kuang-chung <kuang-chung.wang at intel.com>
Cc: petsc-users at mcs.anl.gov; Obradovic, Borna <borna.obradovic at intel.com>; Cea, Stephen M <stephen.m.cea at intel.com>
Subject: Re: [petsc-users] Orthogonality of eigenvectors in SLEPC
In Hermitian eigenproblems orthogonality of eigenvectors is guaranteed/enforced. But you are solving the problem as non-Hermitian.
If your matrix is Hermitian, make sure you solve it as a HEP, and make sure that your matrix is numerically Hermitian.
If your matrix is non-Hermitian, then you cannot expect the eigenvectors to be orthogonal. What you can do in this case is get an orthogonal basis of the computed eigenspace, see https://slepc.upv.es/documentation/current/docs/manualpages/EPS/EPSGetInvariantSubspace.html
By the way, version 3.7 is more than 5 years old, it is better if you can upgrade to a more recent version.
Jose
> El 24 nov 2021, a las 7:15, Wang, Kuang-chung <kuang-chung.wang at intel.com> escribió:
>
> Dear Jose :
> I came across this thread describing issue using krylovschur and finding eigenvectors non-orthogonal.
> https://lists.mcs.anl.gov/pipermail/petsc-users/2014-October/023360.ht
> ml
>
> I furthermore have tested by reducing the tolerance as highlighted below from 1e-12 to 1e-16 with no luck.
> Could you please suggest options/sources to try out ?
> Thanks a lot for sharing your knowledge!
>
> Sincere,
> Kuang-Chung Wang
>
> =======================================================
> Kuang-Chung Wang
> Computational and Modeling Technology
> Intel Corporation
> Hillsboro OR 97124
> =======================================================
>
> Here are more info:
> • slepc/3.7.4
> • output message from by doing EPSView(eps,PETSC_NULL):
> EPS Object: 1 MPI processes
> type: krylovschur
> Krylov-Schur: 50% of basis vectors kept after restart
> Krylov-Schur: using the locking variant
> problem type: non-hermitian eigenvalue problem
> selected portion of the spectrum: closest to target: 20.1161 (in magnitude)
> number of eigenvalues (nev): 40
> number of column vectors (ncv): 81
> maximum dimension of projected problem (mpd): 81
> maximum number of iterations: 1000
> tolerance: 1e-12
> convergence test: relative to the eigenvalue BV Object: 1 MPI
> processes
> type: svec
> 82 columns of global length 2988
> vector orthogonalization method: classical Gram-Schmidt
> orthogonalization refinement: always
> block orthogonalization method: Gram-Schmidt
> doing matmult as a single matrix-matrix product DS Object: 1 MPI
> processes
> type: nhep
> ST Object: 1 MPI processes
> type: sinvert
> shift: 20.1161
> number of matrices: 1
> KSP Object: (st_) 1 MPI processes
> type: preonly
> maximum iterations=1000, initial guess is zero
> tolerances: relative=1.12005e-09, absolute=1e-50, divergence=10000.
> left preconditioning
> using NONE norm type for convergence test
> PC Object: (st_) 1 MPI processes
> type: lu
> LU: out-of-place factorization
> tolerance for zero pivot 2.22045e-14
> matrix ordering: nd
> factor fill ratio given 0., needed 0.
> Factored matrix follows:
> Mat Object: 1 MPI processes
> type: seqaij
> rows=2988, cols=2988
> package used to perform factorization: mumps
> total: nonzeros=614160, allocated nonzeros=614160
> total number of mallocs used during MatSetValues calls =0
> MUMPS run parameters:
> SYM (matrix type): 0
> PAR (host participation): 1
> ICNTL(1) (output for error): 6
> ICNTL(2) (output of diagnostic msg): 0
> ICNTL(3) (output for global info): 0
> ICNTL(4) (level of printing): 0
> ICNTL(5) (input mat struct): 0
> ICNTL(6) (matrix prescaling): 7
> ICNTL(7) (sequential matrix ordering):7
> ICNTL(8) (scaling strategy): 77
> ICNTL(10) (max num of refinements): 0
> ICNTL(11) (error analysis): 0
> ICNTL(12) (efficiency control): 1
> ICNTL(13) (efficiency control): 0
> ICNTL(14) (percentage of estimated workspace increase): 20
> ICNTL(18) (input mat struct): 0
> ICNTL(19) (Schur complement info): 0
> ICNTL(20) (rhs sparse pattern): 0
> ICNTL(21) (solution struct): 0
> ICNTL(22) (in-core/out-of-core facility): 0
> ICNTL(23) (max size of memory can be allocated locally):0
> ICNTL(24) (detection of null pivot rows): 0
> ICNTL(25) (computation of a null space basis): 0
> ICNTL(26) (Schur options for rhs or solution): 0
> ICNTL(27) (experimental parameter): -24
> ICNTL(28) (use parallel or sequential ordering): 1
> ICNTL(29) (parallel ordering): 0
> ICNTL(30) (user-specified set of entries in inv(A)): 0
> ICNTL(31) (factors is discarded in the solve phase): 0
> ICNTL(33) (compute determinant): 0
> CNTL(1) (relative pivoting threshold): 0.01
> CNTL(2) (stopping criterion of refinement): 1.49012e-08
> CNTL(3) (absolute pivoting threshold): 0.
> CNTL(4) (value of static pivoting): -1.
> CNTL(5) (fixation for null pivots): 0.
> RINFO(1) (local estimated flops for the elimination after analysis):
> [0] 8.15668e+07
> RINFO(2) (local estimated flops for the assembly after factorization):
> [0] 892584.
> RINFO(3) (local estimated flops for the elimination after factorization):
> [0] 8.15668e+07
> INFO(15) (estimated size of (in MB) MUMPS internal data for running numerical factorization):
> [0] 16
> INFO(16) (size of (in MB) MUMPS internal data used during numerical factorization):
> [0] 16
> INFO(23) (num of pivots eliminated on this processor after factorization):
> [0] 2988
> RINFOG(1) (global estimated flops for the elimination after analysis): 8.15668e+07
> RINFOG(2) (global estimated flops for the assembly after factorization): 892584.
> RINFOG(3) (global estimated flops for the elimination after factorization): 8.15668e+07
> (RINFOG(12) RINFOG(13))*2^INFOG(34) (determinant): (0.,0.)*(2^0)
> INFOG(3) (estimated real workspace for factors on all processors after analysis): 614160
> INFOG(4) (estimated integer workspace for factors on all processors after analysis): 31971
> INFOG(5) (estimated maximum front size in the complete tree): 246
> INFOG(6) (number of nodes in the complete tree): 197
> INFOG(7) (ordering option effectively use after analysis): 2
> INFOG(8) (structural symmetry in percent of the permuted matrix after analysis): 100
> INFOG(9) (total real/complex workspace to store the matrix factors after factorization): 614160
> INFOG(10) (total integer space store the matrix factors after factorization): 31971
> INFOG(11) (order of largest frontal matrix after factorization): 246
> INFOG(12) (number of off-diagonal pivots): 0
> INFOG(13) (number of delayed pivots after factorization): 0
> INFOG(14) (number of memory compress after factorization): 0
> INFOG(15) (number of steps of iterative refinement after solution): 0
> INFOG(16) (estimated size (in MB) of all MUMPS internal data for factorization after analysis: value on the most memory consuming processor): 16
> INFOG(17) (estimated size of all MUMPS internal data for factorization after analysis: sum over all processors): 16
> INFOG(18) (size of all MUMPS internal data allocated during factorization: value on the most memory consuming processor): 16
> INFOG(19) (size of all MUMPS internal data allocated during factorization: sum over all processors): 16
> INFOG(20) (estimated number of entries in the factors): 614160
> INFOG(21) (size in MB of memory effectively used during factorization - value on the most memory consuming processor): 14
> INFOG(22) (size in MB of memory effectively used during factorization - sum over all processors): 14
> INFOG(23) (after analysis: value of ICNTL(6) effectively used): 0
> INFOG(24) (after analysis: value of ICNTL(12) effectively used): 1
> INFOG(25) (after factorization: number of pivots modified by static pivoting): 0
> INFOG(28) (after factorization: number of null pivots encountered): 0
> INFOG(29) (after factorization: effective number of entries in the factors (sum over all processors)): 614160
> INFOG(30, 31) (after solution: size in Mbytes of memory used during solution phase): 13, 13
> INFOG(32) (after analysis: type of analysis done): 1
> INFOG(33) (value used for ICNTL(8)): 7
> INFOG(34) (exponent of the determinant if determinant is requested): 0
> linear system matrix = precond matrix:
> Mat Object: 1 MPI processes
> type: seqaij
> rows=2988, cols=2988
> total: nonzeros=151488, allocated nonzeros=151488
> total number of mallocs used during MatSetValues calls =0
> using I-node routines: found 996 nodes, limit used is 5
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