Numer. Math. 38, 263-278 (1981) Numerische MathemalJk 9 Springer-Verlag 1981 Analysis of General Quadrature Methods for Integral Equations of the Second Kind Ian H. Sloan Institute for Physical Science and Technology, and Department of Physics and Astronomy University of Maryland, College Park, Maryland 20742, USA Summary. This paper is concerned with a class of approximation methods b for integral equations of the form y(t)=f(t)+~k(t,s)y(s)ds, where a and b a are finite, f and y are continuous, and the kernel k may be weakly singular. The methods are characterized by approximate equations of the form y,(t) =f(t)+ ~" W,i(t)y,(s,i); such methods include the Nystr6m method and a i=1 variety of product-integration methods. A general convergence theory is de- veloped for methods of this type. In suitable cases it has the feature that its application to a specific method depends only on a knowledge of conver- gence properties of the underlying quadrature rule. The theory is used to deduce convergence results, some of them new, for a number of specific methods. Subject Classifications: AMS(MOS) 65R05, 45B05; CR 5.18. 1. Introduction We consider the integral equation of the second kind b y(t) = f (t) + ~k(t, s) y(s) ds, (1.1) a where a and b are finite. The inhomogeneous term f and the solution y are as- sumed to be continuous, whereas the kernel k is allowed to be weakly singular. The detailed discussion is restricted to the one-dimensional equation (1.1), but the main results extend easily to analogous integral equations over compact re- Work supported by the U.S. Department of Energy Permanent address." School of Mathematics, University of New South Wales, Sydney, N.S.W. 2033, Australia 0029 - 599 X/81/0038/0263/$03.20