**3. Accurate evaluation of the influence coefficient by numerical methods**

For the correct computation of the RS by using the HMs, an accurate evaluation of the influence coefficient *aij* and *bij* (*i = 1,2 … n*, *j* = 1,2 … *i*) involved into Eq. 4–6 is necessary. As above mentioned, such coefficients are determined by using a numerical code (FEM, BEM etc) by considering the step-by step procedure for RS analysis. In detail, according to Eq. 4–6 the generic coefficient *aij* (*i = 1,2 … n*, *j* = 1,2 … *i*) is determined by considering a numerical model that represent the component to be examined with a uniform hydrostatic stress distribution applied only to the *i-th* depth increment of the geometry variation (hole or annular groove) having *j* total depth increments. The *bij* (*i = 1,2 … n*, *j* = 1,2 … *i*) coefficients, instead, can be determined numerically by considering a uniform shear stress distribution [13, 14]. Obviously, fixing the total depth, the accuracy of the in-depth RS profile determined by the HMs, increase with the number of depth increments used to reach the total depth although, as it has been demonstrated in [11, 14] if the depth increments are too small then the solution of the inverse problem represented by Eqs.(4–6), became ill-conditioned and
