# Problem for computing eigenvalues with EPSSolve

Dear all,

I am facing a problem with a generalized eigenvalues problem using EPSSolve in FreeFEM of the type MASS \Phi = \lambda STIFF \Phi.

EPSSolve seems to diverge . Indeed, I have to find 14 eigenvalues. However the two first eigenvalues are big and negative while all my eigenvalues should be positive.

I compare the values with eigenvalues computed in Python. With Python, all my eigenvalues are positive. Among my 14 eigenvalues, 12 of them are similar with Python and FreeFEM.

By the way, the eigenvectors found with the two method are different.

I attach the code, so that you can reproduce the problem.
Problem_EV.zip (9.5 KB)

To compare the two method, you have just to do :

mpiexec -np 1 FreeFem++-mpi FF_COMPUTE_EV.edp
python PY_COMPUTE_EV.py

Best regards,

Loïc,

I think you should look at all the EPSSolve options in the SLEPc manual, and that will help you sort out the problem. My guess is that you need to specify -eps_gen_hermitian since you are solving a generalized eigenvalue problem.

Also, it’s perfectly normal that the eigenvectors are not the same, as long as they span the same space.

Hi,

However when I use this parameter, I get

loading matrices
[0]PETSC ERROR: --------------------- Error Message --------------------------------------------------------------
[0]PETSC ERROR: Missing or incorrect user input
[0]PETSC ERROR: The inner product is not well defined: indefinite matrix
[0]PETSC ERROR: See https://petsc.org/release/faq/ for trouble shooting.
[0]PETSC ERROR: Petsc Release Version 3.18.3, Dec 28, 2022
[0]PETSC ERROR: FreeFem++-mpi on a  named is244532 by lb265317 Wed Apr 26 10:16:06 2023
[0]PETSC ERROR: #1 BV_SafeSqrt() at /home/catB/lb265317/FreeFem-sources/3rdparty/ff-petsc/petsc-3.18.3/fr/externalpackages/git.slepc/include/slepc/private/bvimpl.h:132
[0]PETSC ERROR: #2 BV_SquareRoot_Default() at /home/catB/lb265317/FreeFem-sources/3rdparty/ff-petsc/petsc-3.18.3/fr/externalpackages/git.slepc/include/slepc/private/bvimpl.h:362
[0]PETSC ERROR: #3 BVOrthogonalizeCGS1() at /home/catB/lb265317/FreeFem-sources/3rdparty/ff-petsc/petsc-3.18.3/fr/externalpackages/git.slepc/src/sys/classes/bv/interface/bvorthog.c:101
[0]PETSC ERROR: #4 BVOrthogonalizeGS() at /home/catB/lb265317/FreeFem-sources/3rdparty/ff-petsc/petsc-3.18.3/fr/externalpackages/git.slepc/src/sys/classes/bv/interface/bvorthog.c:177
[0]PETSC ERROR: #5 BVOrthogonalizeColumn() at /home/catB/lb265317/FreeFem-sources/3rdparty/ff-petsc/petsc-3.18.3/fr/externalpackages/git.slepc/src/sys/classes/bv/interface/bvorthog.c:333
[0]PETSC ERROR: #6 EPSGetStartVector() at /home/catB/lb265317/FreeFem-sources/3rdparty/ff-petsc/petsc-3.18.3/fr/externalpackages/git.slepc/src/eps/interface/epssolve.c:803
[0]PETSC ERROR: #7 EPSSolve_KrylovSchur_Default() at /home/catB/lb265317/FreeFem-sources/3rdparty/ff-petsc/petsc-3.18.3/fr/externalpackages/git.slepc/src/eps/impls/krylov/krylovschur/krylovschur.c:242
[0]PETSC ERROR: #8 EPSSolve() at /home/catB/lb265317/FreeFem-sources/3rdparty/ff-petsc/petsc-3.18.3/fr/externalpackages/git.slepc/src/eps/interface/epssolve.c:146
0


Loïc,

I guess I could dump the matricies that have been read in and first insure that
the FF edp file read them properly or visualize them or just dump them back
out and diff the input and output txt files. I know in one case writing out
a matrix from FF I had to flush the file stream or else last value was dropped.

In phonons anyway you sometimes
see small values with the wrong sign if there is no
restoring force- the materials just falls apart.
In this case, what would large negative eignvalues mean
in your system? If you look at the eigenvectors for the negative ones

I did note some of these are nearly degenerate and the solver AFAIK has
no way to pick particular vectors to span the spaces.

BTW, if I run your FreeFEM script and dump the matrices, then with MATLAB, I can check that the results from SLEPc are correct.

>> A = PetscBinaryRead('/tmp/A0.bin');
>> eig(full(A), full(B))

ans =

-3.452024300482933e+11
-5.080069245105576e+11
1.140810058050668e-02
1.142933265808896e-02
5.482857616842862e-03
5.479991885016102e-03
2.119836942750551e-03
2.164645146026448e-03
1.522260102485258e-03
3.599930347392806e-04
5.027322264954978e-04
1.253948212699970e-03
8.058577249813228e-04
7.976357767610110e-04

>> eigs(A, B)

ans =

-5.080069245105576e+11
-3.452024300482933e+11
1.142933265808896e-02
1.140810058050668e-02
5.482857616842862e-03
5.479991885016102e-03


Thanks for testing,

It is strange that I find other eigenvalues with Python, especially since both my matrices should de positive definite.

Loïc,

Your matrix B is rank deficient, so it’s not that simple.

Are these the ev’s you got in python? I put your python coefs into
FF ,

type or paste code here
 ff-mpirun  -np 1  FF_COMPUTE_EV.edp | grep lambda
35 :     cout << "lambda[" << j << "] = " << ev(j)  << " and " << eV(:,j) <<  endl;
40 : I am facing a problem with a generalized eigenvalues problem using EPSSolve in FreeFEM of the type MASS \Phi = \lambda STIFF \Phi.
lambda[0] = 177623 and 14
lambda[1] = 177622 and 14
lambda[2] = 0.0114293 and 14
lambda[3] = 0.0114082 and 14
lambda[4] = 0.00548286 and 14
lambda[5] = 0.00547999 and 14
lambda[6] = 0.00216465 and 14
lambda[7] = 0.00211984 and 14
lambda[8] = 0.00152226 and 14
lambda[9] = 0.00125395 and 14
lambda[10] = 0.000805858 and 14
lambda[11] = 0.000797636 and 14
lambda[12] = 0.000502732 and 14
lambda[13] = 0.000359993 and 14
marchywka@happy:/home/documents/cpp/proj/freefem/play/junk/Problem_EV

I went through man solvers same result. Your python coefficients are not the same-
dod you look at the condition number or do SVD in either case? There may be
a sensivitivty problem. The eigenvalues are less than .1 and you have coefficients
as high as 13.

[https://slepc.upv.es/documentation/current/docs/manualpages/EPS/EPSType.html](https://slepc.upv.es/documentation/current/docs/manualpages/EPS/EPSType.html)

Yes, indeed, these are the values I find with Python.

And we can see that 12 of them are similar with those obtained with PETSc.

Loïc,

Ah okay, but theoretically it should not be.

Since, my operators are MASS = \int_K \Phi_i \Phi_j and STIFF= \int_K grad(\Phi_i) grad(\Phi_j). The \Phi_i define a Finite Element Basis.

Is it possible that this is due to approximations errors ?

Normally, I have the same matrices in Python.

Loïc,

STIFF= \int_K grad(\Phi_i) grad(\Phi_j)` this matrix is clearly indefinite. Everything is defined up to a constant.

You mean numerically in my case ? (Since theoretically, it should be)

Loïc,

I mean that your right-hand side matrix is indefinite, both theoretically and numerically.

To my mind, if I am not mistaken, the stiffness matrix for a Poisson problem for example is symmetric and positive definite.

Loïc,

With no boundary conditions, of course it is not, again, everything is defined up to a constant.

Ah okay, yes you are right,
Sorry for this confusion,

Loïc,

What is row 9 in the stiffness matrix? The diagonal element is large
, IIRC 6600, and some of the other terms look like they have similar coefs of opposite
signs. I guess if there is a delicate cancellation that could cause
numerical issues. It probably is of no significance but if you zero out the
terms less than a noise floor, like 1e-6 or so, what do you get?

Since all the solvers I tried gave more or less the same result, it
does not seem to be a quirk of any one of them with the original
FF coefficients

The Stiffness matrix is just a correlation matrix in norm H1. So my problem doesn’t really make physical sense. It only says that some basis functions have more “energy” than others.

Loïc,

HI @prj,

I asked around, and the stiffness matrix is at least positive semi-definite (all the eigenvalues should be greater or equal to zero). The fact that everything is defined up to a constant should not lead to negative eigenvalue, but we can have eigenvalues equal to zero which corresponds to rigid body motion.

The Mass matrix is for sure definite positive (all eigenvalues strictly greater than zero).

Consequently, all eigenvalues should be positive.

By the way when I consider the problem in the other way : STIFF \Phi = \lambda MASS \Phi, I find positive eigenvalues, and the same values than with Python.

So, we think that may be, the problem is the way we compute the eigenvalues. Indeed, when computing generalized eigenvalue problem, there is often a factorization that is is done. May be the fact that some eigenvalues of the stiffness matrix is equal to zero (when I compute STIFF \Phi = \lambda \Phi, I find two eigenvalues equal to zero) and some approximations can cause a problem.

What do you think of this ?

Loïc,