[petsc-users] Questions on QR methods in PETSc

[Jose E. Roman]

> El 27 mar 2026, a las 15:10, Leonardo De Novellis via petsc-users <petsc-users at mcs.anl.gov> escribió:
> 
> Dear Users Support Team,
> I have a couple of questions regarding QR decompositions in PETSc.
> 
> 1) I would like to find the least-squares solution of a rectangular system of the form Ax = b, where A is a dense and tall-skinny matrix (size around 1000000 x 10).
> Since A has very bad conditioning, I want to avoid iterative methods (such as KSPLSQR), since the number of iterations can get very large, and would like to use a direct QR solving method.
> Currently, I am running my code on only 1 core, and A is of type seqaij. With this setup, the following code works fine:
> 
> call KSPSetType(ksp, KSPPREONLY, ierr)
> call KSPGetPC(ksp, pc, ierr)
> call PCSetType(pc, PCQR, ierr)
> call KSPSetOperators(ksp, A, A, ierr)
> t1 = MPI_Wtime()
> call KSPSolve(ksp, b, x, ierr)
> t2 = MPI_Wtime()
> 
> I eventually want to run this in parallel on multiple cores. Will PCQR work for an mpiaij / mpidense matrix? If not, what would you suggest as a direct solving approach for this system?

As an alternative, you can try PCSVD. If your matrix is ill-conditioned, I would trust SVD more than QR.
Both PCQR and PCSVD are sequential, but PCSVD will work in parallel by creating a redundant dense matrix replicated in all processes, so that computation is still sequential. You can obtain a similar behaviour with PCREDUNDANT + PCQR.

> 
> 2) I would also like to compute an explicit QR decomposition of A, and want to do so in parallel. Is there any way to do so in PETSc? If not, as a possible alternative, would you recommend using SLEPc BVOrthogonalize function?

PETSc does not have parallel QR. In SLEPc BVOrthogonalize() will do the job. I would try the TSQR and SVQB methods, they should work for ill-conditioned matrices. With 1 MPI process TSQR should be equivalent to PCQR. You can also use the result of BVOrthogonalize() to compute the least-squares solution as x=R^{-1}*Q^T*b.

Jose

> 
> Kind regards,
> Leonardo
> 
>