Generalized quadrature rules of Gaussian type for numerical evaluation of singular integrals

Generalized Gaussian quadratures for integrals with logarithmic singularity

“…where {ϕ ν } 2m ν=1 is a system of orthogonal functions on (0, 1) with respect to the weight function t → w( √ t)/ √ t = t β (1 − t) α , β = (γ − 1)/2, and δ ν,1 is the Kronecker's delta. Evidently, this system of equations gives a characterization for the quadrature formula (24) to be Gaussian on (0, 1).…”

“…Recently, Milovanović, Igić an Turnić [24] have developed an efficient method for constructing a class of generalized quadrature formulae of Gaussian type on (−1, 1) for integrals (the logarithmic space), where 1 ≤ ≤ n. The construction of such quadratures is based on solving systems of nonlinear equations, using orthogonal system of basis functions. The last provides the well conditioned matrices in the corresponding iterative procedure.…”

“…we start with Lemma 4.2 and construct first the quadrature rule (24) with m nodes. In this case (24) should be a weighted quadrature rule of Gaussian type, with respect to the weight function…”

“…. , m, in the Gaussian quadrature formula (24), by our Mathematica package OrthogonalPolynomials (see [4] and [23]), because Mathematica evaluates ψ(z) to arbitrary numerical precision, using the function PolyGamma[z]. Then, by (23) we obtain the parameters x L k and A L k in the rule (13).…”

“…Now, we can calculate Gaussian parameters (nodes and weights) in (24) for each m ≤ 20, as well as the parameters in the rule (13). For example, for m = 10 we print the parameters x L k and A L k in the rule (13), with 20 decimal digits.…”