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Scaling transformations are used in numerical calculations of the properties of the two-dimensional Ising model near its critical point. When compared with the exact Onsager solution, the best approximation is seen to lead to 1 part in 10^4 accuracy for the two largest scaling indices. This rather accurate calculation is obtained by utilizing a scaling transformation which depends upon a parameter. The parameter is set by demanding that two different evaluations of the magnetic eigenvalue agree with one another. One evaluation is found via the standard eigenvalue method; the other comes from a consistency condition for the spin-spin correlation function. This condition may also be used to distinguish scaling eigenvalues from other eigenvalues.
Kadanoff et al. (1975) studied this question.