Preconditioners generated by minimizing norms

We introduce two diagonal preconditioners, one of them is a scaling of the matrix. They are obtained by minimizing the norm of the iteration matrix using the Frobenius and ℓ∞ℓ∞ norms. We prove convergence of the corresponding iterations for the class of diagonally dominant matrices. For the ℓ∞ℓ∞ iteration, we also prove convergence for positive definite matrices. Both iterations compete very well with the Jacobi iteration.