|-ksp_lsqr_set_standard_error||- set standard error estimates of solution, see KSPLSQRSetComputeStandardErrorVec() and KSPLSQRGetStandardErrorVec()|
|-ksp_lsqr_exact_mat_norm||- compute exact matrix norm instead of iteratively refined estimate, see KSPLSQRSetExactMatNorm()|
|-ksp_lsqr_monitor||- monitor residual norm, norm of residual of normal equations A'*A x = A' b, and estimate of matrix norm ||A|||
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.
For least squares problems wit nonzero residual A*x - 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 preconditioner track the true norm of the residual and uses that in the convergence test.