On the error of quadrature formulae for Cauchy principal value integrals based on piecewise interpolation

We consider the computation of the Cauchy principal value integral by quadrature formulae Q _n ^F f] of compound type, which are obtained by replacing f by a piecewise defined function F_nf]. The behaviour of the constants k_{i, n} in the estimates R _n ^F f]] |⩽K _i,n ‖f ⁽ⁱ⁾‖_∞ (where R _n ^F f] is the quadrature error) is determined for fixed i and n→∞, which means that not only the order, but also the coefficient of the main term of k_{i, n} is determined. The behaviour of these error constants k_{i, n} is compared with the corresponding ones obtained for the method of subtraction of the singularity. As it turns out, these error constants have, in general, the same asymptotic behaviour.