mathematics of computation, volume 28, number 125, January, 1974 A Note on the Optimal Addition of Abscissas to Quadrature Formulas of Gauss and Lobatto Type By Robert Piessens and Maria Branders Abstract. An improved method for the optimal addition of abscissas to quadrature for- mulas of Gauss and Lobatto type is given. 1. Introduction. We consider the quadrature formula (1) f f(x) rf^t akf(xk) + £ &/(&), J-i *-i t-i where the xk's are the abscissas of the Appoint Gaussian quadrature formula. We want to determine the additional abscissas £t and the weights ak and ßk so that the degree of exactness of (1) is maximal. This problem has already been discussed by Kronrod [1] and Patterson [2] and it is well known that the abscissas £* must be the zeros of the polynomial 4>n+i(x) which satisfies (2) j PN(x)d>N+i(x)xk dx = 0, * - 0, 1. • • • , N, where PN(x) is the Legendre polynomial of degree A'. Thus, <p¿v+1(x) must be an ortho- gonal polynomial with respect to the weight function Piv(x). Then, the weights ak and ßk can be determined so that the degree of exactness of (1) is 3N + 1 if N is even and 3^ + 2 if TV is odd. Szegö [3] proved that the zeros of <p,v+i(x) and PN(x) are distinct and alternate on the interval [— 1, +1]. Kronrod [1] gave a simple method for the computation of the coefficients of <pN+1(x). This method requires the solution of a triangular system of linear equations, which is, unfortunately, very ill-conditioned. Patterson [2] expanded <Pn+i(x) in terms of Legendre polynomials. The coefficients of this expansion satisfy a linear system of equations which is well-conditioned, although its construction requires a certain amount of computing time. The present note proposes the expansion of <pN+1(x) in a series of Chebyshev polynomials. We also give explicit formulas for the weights ak and ßk. Finally, we consider the optimal addition of abscissas to Lobatto rules. As compared with Patterson's method, our method has three advantages: (i) It leads to a considerable saving in computing time since the formulas are much simpler. (ii) The loss of significant figures through cancellation and round-off is slightly reduced, as we verified experimentally. This is in agreement with some theoretical results given by Gautschi [4]. (iii) It is applicable for every value of N, while Patterson's method fails in the Received May 15, 1972,revised March 2, 1973. AMS (MOS) subject classifications (1970). Primary 65D30; Secondary 33A65. Copyright © 1974, American Mathematical Society 135 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use