[Trilinos-Users] condition number of an Epetra_FECrsMatrix

[Nate Roberts]

Hello all,

I'm interested in studying the condition number of our global system matrix, which we store as an Epetra_FECrsMatrix.   My approach thus far doesn't give accurate estimates, and I'd like advice on getting better ones.

Based on some sample code I found in the AztecOO documentation, I wrote the following method:

double Solution::conditionNumberEstimate( Epetra_LinearProblem & problem ) {
  AztecOOConditionNumber conditionEstimator;
  conditionEstimator.initialize(*problem.GetOperator());
    
  int maxIters = 40000;
  double tol = 1e-10;
  int status = conditionEstimator.computeConditionNumber(maxIters, tol);
  if (status!=0)
    cout << "status result from computeConditionNumber(): " << status << endl;
  double condest = conditionEstimator.getConditionNumber();
  
  return condest;
}

(The input Epetra_LinearProblem is initialized with the global system matrix.)

For some smaller problems, I also tried exporting the matrix to file, and loading it into MATLAB, and using rcond() and/or condest() there.  What I found was that the method above significantly underreports the condition number.  For example, for a fairly dense 243x243 matrix (corresponding to a 2x2 finite element mesh), the method above reports 2611.05 as the condition number.  MATLAB's condest and rcond both report 16716.  (Decreasing the tolerance to 1e-12 and increasing maxIters to 100000, I instead get 2267.47 from the method above.)

(It may be worth noting that the Epetra_FECrsMatrix ends up storing quite a few zeros.  Of the 19237 entries stored, 5024 of them are zeros.)

I'd really like to be able to get high-fidelity condition numbers for this matrix without going to MATLAB.  I don't mind if it takes some computational effort—I'm not using the condition number during my real solves, but studying the effect of various parameters (e.g. choice of basis functions) on conditioning.

Any advice would be greatly appreciated!

Thanks,
Nate
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