For a given matrix , its polar decomposition is
, where
and
is unitary. Let
be real symmetric and
nonsingular, than the matrix absolute value
is also nonsingular and
is the matrix sign of
, having only two distinct eigenvalues, plus
and minus one. The matrix
is the ideal symmetric
positive definite preconditioner for the linear system
, making the
preconditioned MINRES to converge in at most two steps. We call
the
absolute value preconditioner, if it is spectrally equivalent to
.
If the matrix is (block) strictly diagonally dominant, the
preconditioner
can be chosen as the inverse to the absolute value of
the (block) diagonal of
. Such a choice can, e.g., be efficient in
plain-wave electronic structure calculations.
For a model problem, where is a finite difference approximation of
the shifted negative Laplacian, we construct an efficient geometric
multigrid absolute value preconditioner, in which the smoothing is done
using the action of
and
appears only on the coarsest grid.
Our numerical tests demonstrate the effectiveness of such a
preconditioning.