# How does matrix scaling influence linear solvers?

For instance, in MUMPS there is option to scale matrix s.t. all rows/columns have the same norm. This claims to decrease condition number and improve numerical properties of the matrix: ftp://cuter.rl.ac.uk/pub/reports/adruRAL2008013.pdf

Can anybody explain whether it can be used as a preconditioner and how efficient would it be?

• Isn't this simply the Jacobi preconditioner? – shuhalo Mar 18 '12 at 16:24

• Jacobi is the simplest preconditioner and it either puts $1$ on the diagonal or makes the absolute value row sums equal to $1$. It's not a matter of how big the jumps are, it's a matter of whether the preconditioner can be constructed stably without pre-scaling first. If the preconditioner can be constructed stably, it will generally correct the bad scaling, except in cases that use auxiliary information like geometric multigrid with rediscretized coarse operators or Schur complement methods. – Jed Brown Mar 18 '12 at 15:36