4 hardware. In particular, domain-decomposition based parallelism was implemented in the code and IBM Cell BE and Power7 systems with specialized optimizations [10] were used to perform these long evolutions in a reasonable time-frame. 5. Other Enhancements: We also enhanced the code further to be able to evolve non-axisymmetric mode compu- tations (both odd and even m -modes). The even m -mode case was straightforward to implement, since these simply require some additional terms in the evolution equation. For the odd m -mode case we used the follow- ing procedure to simplify our implementation: Instead of evolving the scalar field ψ itself, we transformed the Teukolsky equation to evolve the field ψ/ sin θ . This simple transformation enabled us to reuse several of our even m -mode code algorithms (such as the higher-order angular boundary condition imposition) and also avoid the introduction of non-polynomial functions of the code’s angular grid variable, i.e., cos θ . III. NUMERICAL EVOLUTION OF GENERIC INITIAL DATA We first show for the first time numerical tests of the Barack–Ori formula [8]. Then, in Section IV we show how the Barack–Ori formula is a particular case for “up” excitation of modes. Barack and Ori noted that in the generic case of black hole perturbations, all dynamically allowed modes are present. Specifically, in the case of scalar field perturbations, all modes, starting with the monopole ( ℓ = 0) evolve. The decay rate of a multipole ℓ with azimuthal number m is given by t $_{−}$n where where q = 0 , 1 if ℓ + m is even or odd, respectively. Notably, the Barack–Ori formula depends explicitly on the azimuthal number m , and like the Hod formula includes a certain complexity that has to do with whether ℓ + m is even or odd. n$_{ℓ}$ = ℓ + | m | + 3 + q , (2) The meaning of “generic” initial data is that the relative amplitudes of the various multipoles and their radial profiles are not fine tuned. Consider, e.g., the following initial data set, φ ( r, θ ) | t =0 = 0 and Π( r, θ )$_{t}$$_{=0}$ = A sin 2 θ f ( r ), where f ( r ) is any localized radial function, say a gaussian. These initial data include non-vanishing monopole and quadrapole moments. We evolved these data numerically, and found that as expected the late–time tail includes all even multipoles, and is dominated by the monopole field that decays according to t $_{−}$3 at late times. The quadrapole projection of the late–time field decays according to t $^{$_{−}$5}$, in accordance with the Barack–Ori formula. Other choices of initial data sets and multipoles are also found to be in agreement with the Barack–Ori formula. As far as the present authors are aware, this is the first direct numerical confirmation of the Barack–Ori formula. Let us now consider the following scenario: one starts with initial data as above, and evolves the field to a certain value of time, say to t = T . The field at t = T is a complicated mixture of modes, each with its own radial profile. Now consider a different initial value problem, in which at t = 0 only a pure multipole, say the quadrapole moment, is non-vanishing, and evolve these initial data to t = T . These pure mode initial data also evolve to a complicated mixture of multipole moments at t = T , and at first look one does not notice much qualitative difference between the fields at t = T . However, treating the fields at t = T as new initial value problems, further time evolution leads to different decay rates at late times. Specifically, the generic data lead to quadrapole projection that drops off like t $^{$_{−}$5}$, and the pure data lead to quadrapole projection that drops off like t $^{$_{−}$7}$. Considering only the values at t = T as the initial data sets, what is the fundamental difference between the two sets, that leads to very different late–time evolutions? Even though either set appears at t = T to be a set of amplitudes with weighted ratios for the various multipole moments (as functions of the radial coordinate), the set describing the pure mode evolution is made at t = T of a very carefully chosen set of ( r dependent) amplitudes. In fact, it is the outcome of fine–tuned evolution of a pure mode of the original initial value problem. Examination of the data set at t = T does not reveal anything qualitative different about it: it is the specific amplitude ratios and radial profiles that make it non-generic. One may test these ideas by injecting at t = T another field, say that of a pure mode. Start at t = 0 with a pure quadrapolar field. At t = T the projection of the field is a mixture of all even multipoles. Then, at t = T , inject either a pure monopole or a pure quadrapole with some radial profile. The multipolar content of the field after the injection is changed in that the relative multipolar amplitudes changed (as functions of the radius). When the injected field is quadrapolar, the late–time decay rate of the quadrapole moment is t $^{$_{−}$7}$, as should be expected from the fact that we have a superposition of two linear pure mode evolutions (which a certain time delay between them) (Fig. 1). When the injected field is a monopole, the late–time decay rate of the quadrapole moment is t $_{−}$5 (Fig. 2). The addition of a monopole field (without fine tuning its amplitude and radial profile) makes the multipolar content of the total field generic. One may pose the following question: When the injected field is a quadrapole, its time evolution will excite a monopole. Why in the presence of this excited monopole the quadrapole projection of the field decays like t $^{$_{−}$7}$, whereas in the presence of an injected monopole the decay rate is the slower t $^{$_{−}$5}$? There is a fundamental difference