# Doubt regarding GMRES(m) and preconditioned GMRES

I have the two following algorithms for GMRES(m) and left preconditioned GMRES.

GMRES(m)

Left preconditioning

I would like to know if anyone could explain why steps 10 through 12 are not used in the left preconditioned GMRES. Are they lost, or is it implicit that these need to happen?

If not, am I correct to assume that left-preconditioned works in blocks of m-vectors (number of vectors chosen before restart the krylov subspace). So, if I choose 20 vectors to store before restart,it will assemble all 20 krylov subspace basis, and only afterwards move to step 12 ?

Best Regards

It's just a difference in how the authors decided to write the algorithm. Left-preconditioned GMRES is the same as regular GMRES where $$A$$ is replaced with $$M^{-1}A$$ and $$b$$ is replaced with $$M^{-1}b$$. (I.e. you solve the preconditioned system $$M^{-1}Ax=M^{-1}b$$ instead of $$Ax=b$$.)