An exploration of matrix equilibration

We review three algorithms that scale the in nity-norm of each row and column in a matrix to 1. The rst algorithm applies to unsymmetric matrices, and uses techniques from graph theory to scale the matrix. The second algorithm applies to symmetric matrices, and is notable for being asymptotically optimal in terms of operation counts. The third is an iterative algorithm that works for both symmetric and unsymmetric matrices. Experimental evidence of the e ectiveness of all three algorithms in improving the conditioning of the matrix is demonstrated.