Kronecker-sequences that use the fractional parts of multiples of irrationals, are well known to be one of the special type of low-discrepancy sequences that can be used for quasi-Monte Carlo integration. One simply takes the average of the function values evaluated in the points of the sequence to obtain an estimate for the integral value. In the past, it was shown that applying certain weights to the average of the function values increases the asymptotic error convergence order dramatically for certain classes of functions.
In this work, we start from the above theoretical basis and we derive algorithms for obtaining quadratic and cubic error convergence. The algorithms are `open' in the sense that extra steps in the algorithm can easily be taken in order to improve the result. The amount of work for our algorithms increases linearly with the number of steps.