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It is known that for a given matrix A of rank r, and a set D of positive diagonal matrices, _WD||(W^12 A)^†W^12 ||_2=(_i 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 W^12 A)^†W^12 is close to a multi-level constrained weighted pseudoinverse therefore ||(W^12 A)^†W^12 ||_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. , number=3, journal=Journal of Computational Mathematics, year=2004, month=June, pages=427–436