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

Control of accuracy on Taylor-collocation method to solve the weakly  regular Volterra integral equations of the first kind by using the CESTAC  method