COMPARISON BETWEEN GMRES AND THE METHOD OF CONJUGATE GRADIENTS FOR THE NORMAL EQUATIONS IN AN EFFICIENCY OF THEIR USE

We solve Ax = b, where A is an arbitrary real matrix. If A is a square invertible matrix there is possible to apply the GMRES method which is considered to be the best method for solving such problems. The use of the normal equations A(T)Ax = A(T)b is an alternative approach but it is not recommended. One of the reasons is the fact that cond(2)(A(T)A) = cond(2)(A)(2). But this reason is possible to be partly eliminated by using of a better arithmetic. A priority of the normal equations is the fact that the problem is always solvable even if A is a rectangular matrix. And it is possible to apply the classical gradient methods to this problem. Some convergent properties of both approaches are compared for various classes of matrices.