Extensions of Gauss Quadrature Via Linear Programming

Abstract

Gauss quadrature is a well-known method for estimating the integral of a continuous function with respect to a given measure as a weighted sum of the function evaluated at a set of node points. Gauss quadrature is traditionally developed using orthogonal polynomials. We show that Gauss quadrature can also be obtained as the solution to an infinite-dimensional linear program (LP): minimize the nth moment among all nonnegative measures that match the 0 through n − 1 moments of the given measure. While this infinite-dimensional LP provides no computational advantage in the traditional setting of integration on the real line, it can be used to construct Gausslike quadratures in more general settings, including arbitrary domains in multiple dimensions.Keywords Gauss quadrature · Semi-infinite programming · Convex optimization Mathematics Subject Classification Gauss QuadratureWe briefly review Gauss quadrature and set up our notation. Let ⊂ R be a closed interval and q a given measure on . The standard method for approximating the definite integral of a continuous function f on isThe right-hand side is referred to as a quadrature. The coefficients w 1 , w 2 , . . . , w N are the weights and x 1 , x 2 , . . . , x N ∈ are the nodes, i.e., the locations at which the function f is sampled to form the approximation. The quadrature is said to be of order n if it is exact for polynomials up to degree n − 1, i.e.,w j x i j , i = 0, . . . , n − 1.The numbers on the left-hand side are the 0 through n − 1 moments of the measure dq. These conditions are a set of n linear equations in the N weights. For N = n (and for N ≥ n), for any choice of distinct nodes, we can always find weights that satisfy the preceding equations since the coefficient matrix for the linear equations is Vandermonde and, therefore, invertible. (However, the resulting weights are not necessarily nonnegative.) Thus, a quadrature of order n can be found by choosing an arbitrary set of distinct N = n nodes. We call a quadrature of order n with N < n nodes efficient; such a quadrature requires fewer function evaluations than its order. The linear equations for the weights of an efficient quadrature have more equations than variables; these equations are not solvable unless the nodes are chosen very carefully. In 1814 Gauss [11] discovered the first efficient quadrature, which is now called a Gauss quadrature. A Gauss quadrature of order n requires only N = n/2 nodes (for n even). Traditionally a Gauss quadrature is developed with the theory of orthogonal polynomials; such a treatment can be found in many standard texts [19,9,32]. There are efficient methods to find Gauss quadrature nodes and weights, such as the GolubWelsch algorithm [14] and the Glaser-Liu-Rokhlin algorithm [12]. Gauss Quadrature Via Linear ProgrammingAs we show in this paper, a Gauss quadrature can also be obtained as the solution of an infinite-dimensional linear program (LP) over nonnegative measures. Again, let ⊆ R be a closed (but not necessarily compact) interval. Assume that s...

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