|-ksp_lsqr_set_standard_error||- Set Standard Error Estimates of Solution see KSPLSQRSetStandardErrorVec()|
|-ksp_lsqr_monitor||- Monitor residual norm and norm of residual of normal equations|
This varient, when applied with no preconditioning is identical to the original algorithm in exact arithematic; however, in practice, with no preconditioning due to inexact arithematic, 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 ICC, one should call KSPSetOperators(ksp,A,A'*A)) since the preconditioner needs to work for the normal equations A'*A.
Supports only left preconditioning.
In exact arithmetic the LSQR method (with no preconditioning) is identical to the KSPCG algorithm applied to the normal equations. The preconditioned varient was implemented by Bas van't Hof and is essentially a left preconditioning for the Normal Equations. It appears the implementation with preconditioner track the true norm of the residual and uses that in the convergence test.
Developer Notes: 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().
For least squares problems without a zero to A*x = b, there are additional convergence tests for the residual of the normal equations, A'*(b - Ax), see KSPLSQRDefaultConverged()