On the error of extended Gaussian quadrature formulae for functions of bounded variation

We investigate the remainder R_(2n+1) pf (minimum node) extended Gaussian quadrature formulae Q_(2n+1) by means of the constants QV(R_(2n+1), which are the best possible in the error bound vertical stroke R_(2n+1)[#integral#]vertical stroke #<=#QV(R-(2n+1))Var(#integral#) for all functions #integral# of bounded variation Var(#integral#). For the most often used Gauss-Kronrod extensions QGK/2n+1, it is proved that lim\divn#->##infinity# (2n+1)QV(RGK\div2n+1) = #pi#\div2. As a consequence, we obtain that, among all extended Gaussian formulae whose additional nodes interlace with the Gaussian ones, the Gauss-Kronrod formula QGK\div2n+1 is asymptotically optimal with respect to QV. (orig.)SIGLEAvailable from TIB Hannover: RO 8347(1994,2) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman