[petsc-users] Inaccurate Matrix Inversion using MatMatSolve

[Zhang, Hong]

Rakesh,
Can you find the condition number of your matrix A?
Hong
________________________________
From: petsc-users <petsc-users-bounces at mcs.anl.gov> on behalf of Rakesh Halder <rhalder at umich.edu>
Sent: Thursday, June 25, 2020 2:43 PM
To: petsc-users at mcs.anl.gov <petsc-users at mcs.anl.gov>
Subject: [petsc-users] Inaccurate Matrix Inversion using MatMatSolve

Hi all,

I'm using PETSc for matrix calculations as part of a model order reduction code. An algorithm I'm using requires that I compute the explicit inverse of a matrix, as it needs to be used in matrix-matrix products.

The matrix is small (I'll show a 5x5 example), and I first find the LU factorization of it as follows:

  MatGetOrdering(A,MATORDERINGNATURAL,&perm,&iperm);
  MatFactorInfoInitialize(&info);
  MatLUFactor(A,perm,iperm,&info);

I then find the inverse of it:

  MatMatSolve(A,I,B);

Where I is the identity matrix, and B is the inverse. The subroutine does calculate an inverse, although not accurately; the matrix A I looked at is:

5.23E-02        1.86E-02        2.67E-02        4.58E-02        3.55E-02
6.37E-03        5.86E-02        5.07E-03        1.64E-02        1.36E-02
-4.07E-02       3.99E-03        5.50E-02        1.77E-02        -3.21E-02
1.96E-02        -5.53E-03       4.02E-02        -5.37E-02       1.80E-02
1.51E-02        1.70E-02        -1.57E-02       -3.75E-03       -5.64E-02

And when I take the product A*B, I don't recover the identity matrix, but rather:

9.97E-01        2.77E-04        -1.66E-03       1.25E-04        -1.10E-03
-6.79E-04       9.99E-01        -1.72E-04       6.24E-04        1.16E-03
2.70E-03        -3.98E-04       9.98E-01        -1.51E-04       6.82E-04
-3.35E-04       -4.82E-04       -1.03E-03       9.99E-01        9.80E-05
-9.50E-04       -8.70E-04       8.37E-04        -3.68E-04       9.99E-01

I believe this is causing large inaccuracies in my program, as the diagonal and off-diagonal entries have large errors associated with them. I am wondering if there is a way to perhaps tighten the tolerance of the subroutine, or if there are other methods I can use. I would like to avoid using KSP and solving for the inverse vector-by-vector if possible.

Thanks,

Rakesh Halder
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