Abstract Gauss quadrature rules associated with a nonnegative measure with support on (part of) the real axis find many applications in Scientific Computing. It is important to be able to estimate the quadrature error when replacing an integral by an l-node Gauss quadrature rule in order to choose a suitable number of nodes. A classical approach to estimate this error is to evaluate the associated ( 2 l + 1 ) -node Gauss–Kronrod rule. However, Gauss–Kronrod rules with 2 l + 1 real nodes might not exist. The ( 2 l + 1 ) -node generalized averaged Gauss formula associated with the l-node Gauss rule described in [M. M. Spalevic, On generalized averaged Gaussian formulas, Math. Comp., 76 (2007), pp. 1483–1492] is guaranteed to exist and provides an attractive alternative to the ( 2 l + 1 ) -node Gauss–Kronrod rule. This paper describes a new representation of generalized averaged Gauss formulas that is cheaper to evaluate than the available representation.

中文翻译：

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广义平均高斯求积规则的一种新表示

摘要 与支持实轴（部分）的非负测度相关的高斯求积规则在科学计算中有许多应用。在用 l 节点高斯正交规则替换积分时，能够估计正交误差是很重要的，以便选择合适数量的节点。估计此误差的经典方法是评估相关的 (2 l + 1 ) 节点 Gauss-Kronrod 规则。然而，具有 2 l + 1 个实节点的 Gauss-Kronrod 规则可能不存在。( 2 l + 1 ) - 节点广义平均高斯公式与 [MM Spalevic，关于广义平均高斯公式，数学。Comp., 76 (2007), pp. 1483–1492] 保证存在，并为 (2 l + 1 ) 节点 Gauss–Kronrod 规则提供了一个有吸引力的替代方案。