Abstract. The Jacobi matrix of the (2n+1)-point Gauss-Kronrod quadrature rule for a given measure is calculated efficiently by a five-term recurrence relation. The algorithm uses only rational operations and is therefore also useful for obtaining the Jacobi-Kronrod matrix analytically. The nodes and weights can then be computed directly by standard software for Gaussian quadrature formulas.

Abstract. An anti-Gaussian quadrature formula is an (n + 1)-point formula of degree 2n − 1 which integrates polynomials of degree up to 2n + 1 with an error equal in magnitude but of opposite sign to that of the n-point Gaussian formula. Its intended application is to estimate the error incurred in Gaussian integration by halving the difference between the results obtained from the two formulas. We show that an anti-Gaussian formula has positive weights, and that its nodes are in the integration interval and are interlaced by those of the corresponding Gaussian formula. Similar results for Gaussian formulas with respect to a positive weight are given, except that for some weight functions, at most two of the nodes may be outside the integration interval. The antiGaussian formula has only interior nodes in many cases when the Kronrod extension does not, and is as easy to compute as the (n + 1)-point Gaussian formula.

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SUMMARYThe most popular pipe network algorithms fall into three categories depending on whether node, loop or element solving equations are considered. Although node methods have some advantages over the other two methods, some authors have found the node methods to be more unreliable than the other two classes of methods. Node methods are also used in Computational Fluid Dynamics (CFD) to solve the Navier-Stokes equations. Since significant progress has been made in this field in the recent past it was felt that this should have some bearing on the development of more reliable node methods for pipe network problems. In this paper the well-known SIMPLE algorithm of Patankar and Spalding,' which is known in CFD as a segregated method, is extended to deal with pipe network problems. The method can deal with both compressible and incompressible flows. Special attention is given to the solution of the pressure correction equation, the stability of the algorithm, sensitivity to initial conditions and convergence parameters. It is shown that the present method is not very sensitive to initial conditions. The method is very reliable and it deals more effectively with compressible flows than the conventional Newton-Raphson method for incompressible flows.