Product integration of logarithmic singular integrands based on cubic splines (Q800686)

The authors present an investigation of some product quadrature rules \[ \int^{1}_{-1}f(x)k(x)dx\sum^{n+1}_{i=0}w_{ni}f(x_{ni}) \] based on cubic B-splines. In Section 2 they give error bounds on product quadrature rules for cubic splines when f(x) is bounded and Riemann integrable on [-1,1] and \(k(x)\in L_ p[\)-1,1], \(p\in [1,+\infty)\). In Section 3 they make a comparison between a computational procedure for product quadrature rules based on Chebyshev polynomials and a computational procedure based on cubic B-splines, when \(k(x)=\ln | x\)-\(\lambda\) \(|\), \(x\in [\)-1,1], \(\lambda\in [\)-1,1]. Finally, in Section 4, the authors give some numerical results.