Context : This paper presents a numerical application of a self-consistent theory of partial redistribution in non-LTE conditions , developed in previous papers of the series .

Aims : The code was described in a previous paper of this series .
However , in that previous paper ( number IV of the series ) , the numerical results were unrealistic .
The present paper presents an approximation , which was able to restore the reliability of the outgoing polarization profiles .

Methods : The convergence of the results is also proved .
It is demonstrated that the step increment decreases like 1 / N ^ { \alpha } , with \alpha > 1 ,

Results : so that the results series behaves like a Riemann series , which is absolutely convergent .

Conclusions : However , agreement between the computed and observed linear polarization profiles remains qualitative only .
The discrepancy is assigned to the plane parallel atmosphere model , which is insufficient to describe the chromosphere , where these lines are formed .
As all the integrals are numerical in the code , it could probably be adapted to more realistic and higher dimensioned model atmospheres .
However , it is time consuming for lines having an hyperfine structure as the Na i D lines are .
The net linear polarization observed in Na i D _ { 1 } with the polarimeter ZIMPOL mounted on the McMath-Pierce telescope at Kitt Peak is not confirmed by the present calculations and could be an artefact of instrumental polarization .
The presence of instrumental polarization could be confirmed by the higher linear polarization degree observed by this instrument in Na i D _ { 2 } line center , with respect to the present calculation result , where the magnetic field is not accounted for , when the Hanle effect acts as a depolarizing effect in the Second Solar Spectrum .
The observed linear polarization excess is found of the same order of magnitude 0.1 % in both line centers , which is also comparable to the instrumental polarization compensation level of this experiment .