Algorithm 691: Improving QUADPACK automatic integration routines

Abstract

Two automatic adaptive integrators from QUADPACK (namely, QAG, and QAGS) are modified by substituting the Gauss-Kronrod rules used for local quadrature with recursive monotone stable (RMS) formulas. Extensive numerical tests, both for one-dimensional and two-dimensional integrals, show that the resulting programs are faster, perform less functional evaluations, and are more suitable

An extended doubly-adaptive quadrature method based on the combination of the Ninomiya and the FLR schemes

“…The case KEY= 2 in the routine QXG in FLR [5] uses a family of four formulas in the RMS formulas, namely, 13-point (I 1 ), 19-point (I 2 ), 27-point (I 3 ), 41-point (I 4 ) rules. Including Ninomiya's stable 11-point rule (I 0 ) yields a sequence of rules, I 0 , I 1 , .…”

“…In the FLR method [5] below P = 0.16 is chosen. In summary, if hint given by (7) is smaller than 0.2, we choose to apply the higher order rule (the 13-point rule I 1 of the RMS family), see Section 4.2 below.…”

“…This scheme chooses either to apply the order adaptive method to the current subinterval or to further split the subinterval, by detecting the local regularity of the integrand. An improved version QXG (QXGS) due to Favati, Lotti and Romani [5] (we call FLR henceforth) of QAG (QAGS) is a doublyadaptive algorithm based on recursive monotone stable (RMS) formulas [4,6]. Ninomiya's method is less effective for oscillatory integrands than the FLR method.…”

Theoretical and practical efficiency measures for symmetric interpolatory quadrature formulas

“…In [3] it is shown how to generate all these formulas algorithmically and an implicit definition is given for the nodes of a RMS formula. The first few elements of a family of RMS formulas is given in Table 5.1; RMS formulas with 13, 19, 27, and 41 nodes have been used in [4]. Figure 5.1 shows the values of R for some Newton-Cotes, Gauss-Kronrod, Clenshaw-Curtis, Gauss-Legendre and RMS formulas.…”

“…Moreover, for functions with internal difficulties and error estimators based on the available points, open rules are potentially less reliable than closed ones (see Fig. 9 in [4]). Such effects cannot be detected in our tests since we use the exact error.…”