It is known that for a given matrix A of rank r, and a set D of positive diagonal matrices, supw∈D ||(W1/2A)+W1/2||2 = (mini σ+(A(i)))-1, in which (A(i)) is a submatrix of A formed with r = (rank(A)) rows of A, such that (A(i)) has full row rank r. In many practical applications this value is too large to be used.In this paper we consider the case that both A and W(∈ D) are fixed with W severely stiff. We show that in this case the weighted pseudoinverse (W1/2 A)+W1/2 is close to a multi level constrained weighted pseudoinverse therefore ||(W1/2A)+W1/2 ||2 is uniformly bounded.We also prove that in this case the solution set the stiffly weighted least squares problem is close to that of corresponding multi-level constrained least squares problem.