  Finding the optimal parameters and functions of iterative methods is among
the main problems of the Numerical Analysis. For this aim, a technique of the
stochastic arithmetic (SA) is used to control of accuracy on Taylor-collocation
method for solving first kind weakly regular integral equations (IEs). Thus,
the CESTAC (Controle et Estimation Stochastique des Arrondis de Calculs) method
is applied and instead of usual mathematical softwares the CADNA (Control of
Accuracy and Debugging for Numerical Applications) library is used. Also, the
convergence theorem of presented method is illustrated. In order to apply the
CESTAC method we will prove a theorem that it will be our licence to use the
new termination criterion instead of traditional absolute error. By using this
theorem we can show that number of common significant digits (NCSDs) between
two successive approximations are almost equal to NCSDs between exact and
numerical solution. Finally, some examples are solved by using the
Taylor-collocation method based on the CESTAC method. Several tables of
numerical solutions based on the both arithmetics are presented. Comparison
between number of iterations are demonstrated by using the floating point
arithmetic (FPA) for different values of $\varepsilon$.
