We introduce new fast canonical local algorithms for discrete and continuous spin systems. We show that for a broad selection of spin systems they compare favorably to the known ones except for the Ising $\pm1$ spins. The new procedures use discretization scheme and the necessary information have to be stored in computer memory before the simulation. The models for testing discrete spins are the Ising $\pm1$, the general Ising $S$ or Blume-Capel model, the Potts and the clock models. The continuous spins we examine are the $O(N)$ models, including the continuous Ising model ($N=1$), the $\phi^4$ Ising model ($N=1$), the $XY$ model ($N=2$), the Heisenberg model ($N=3$), the $\phi^4$ Heisenberg model ($N=3$), the $O(4)$ model with applications to the $SU(2)$ lattice gauge theory, and the general $O(N)$ vector spins with $N\ge5$.[ pdf] [more: program ...]

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