Implements LSQR

Options Database Keys#


Supports non-square (rectangular) matrices.

This variant, when applied with no preconditioning is identical to the original algorithm in exact arithmetic; however, in practice, with no preconditioning due to inexact arithmetic, it can converge differently. Hence when no preconditioner is used (PCType PCNONE) it automatically reverts to the original algorithm.

With the PETSc built-in preconditioners, such as PCICC, one should call KSPSetOperators(ksp,A,A’*A)) since the preconditioner needs to work for the normal equations A’*A.

Supports only left preconditioning.

For least squares problems with nonzero residual Ax - b, there are additional convergence tests for the residual of the normal equations, A’(b - Ax), see KSPLSQRConvergedDefault().

In exact arithmetic the LSQR method (with no preconditioning) is identical to the KSPCG algorithm applied to the normal equations. The preconditioned variant was implemented by Bas van’t Hof and is essentially a left preconditioning for the Normal Equations. It appears the implementation with preconditioning tracks the true norm of the residual and uses that in the convergence test.

Developer Note#

How is this related to the KSPCGNE implementation? One difference is that KSPCGNE applies the preconditioner transpose times the preconditioner, so one does not need to pass A’*A as the third argument to KSPSetOperators().


  • **** -*** The original unpreconditioned algorithm can be found in Paige and Saunders, ACM Transactions on Mathematical Software, Vol 8, 1982.

See Also#

KSP: Linear System Solvers, KSPCreate(), KSPSetType(), KSPType, KSP, KSPSolve(), KSPLSQRConvergedDefault(), KSPLSQRSetComputeStandardErrorVec(), KSPLSQRGetStandardErrorVec(), KSPLSQRSetExactMatNorm(), KSPLSQRMonitorResidualDrawLGCreate(), KSPLSQRMonitorResidualDrawLG(), KSPLSQRMonitorResidual()







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