The paper [1] deals with the modified discrete vortex method for the numerical solution of singular integral equations of the formwhere L is a closed Lyapunov contour on the complex plane, the function f (t 0 ) belongs to the Hölder class H α with exponent α ∈ (0, 1], and the function h (t 0 , t) is also assumed to be of the class H α (with respect to both variables); moreover, h (t 0 , t 0 ) ≡ 0. Note that equations of the form (1) are of interest in particular from the viewpoint of a number of applications to boundary value problems in function theory [2, p. 173; 3, p. 258]. Some more general singular integral equations can also be reduced to (1) under appropriate conditions [3, p. 194].In what follows, we use the notationJust as the classical schemes of the discrete vortex method, refined schemes of the form given in [1] are constructed on the basis of a uniform partition of L by a system of computational points and a system of monitoring points (e.g., see [4, p. 262]); moreover, in contrast to the classical scheme, where the two systems are chosen once and for all, in the scheme in [1], their roles are interchanged alternately. Thus the dimension of the system of linear algebraic equations approximating Eq. (1) is twice as large as in the classical schemes of the discrete vortex method. (Note, however, that the higher accuracy of the scheme in [1] is essentially due to the higher order of approximation of the integrals in (1) rather than to the increased dimension.) Note that for the above-mentioned partition of the contour, as well as for some other partitions, the corresponding singular integral in (1) can be approximated by various quadrature formulas with various degree of accuracy (which can be rather high). However, it is difficult to justify the corresponding schemes. In particular, this issue was considered in [1] in connection with the approximation of the integralfor t 0 = τ ν (ν = 1, . . . , n), where {τ j } 2n j=1 is the system of points splitting the contour L into 2n (n ≥ 2) parts (which is also used in this paper) and
