Favati P., Fiorentino G., Lotti G., Romani F. Local error estimates and singularity test in double and triple adaptive quadrature. Internal note IEI-B4-43, 1994. |

Abstract (English) |
We consider the problem of computing an approximation of the definite integral (a,b) f(x) dx, using an automatic integration routine. In a recent work [5] two new algorithms, well suited for high precision adaptive automatic quadrature, have been introduced. The basic idea of the first algorithm (Double Adaptive Quadrature, DAQ in the following) consists of combining, in a general adaptive scheme, the two main strategies for improving the approximation of an integral: interval subdivision and application of more accurate formulas. In practice, the active subintervals are stored into a queue and sorted according to decreasing error estimates, at each step the subinterval with the larger error estimate is either bisected or processed with a higher degree formula. The choice between the two alternatives is determined by the presence of difficulties in the subinterval. Obviously, since the location of the singularities is not known a priori, an empirical test (which can sometimes fail) needs to be performed. The second algorithm (Triple Adaptive Quadrature, TAQ in the following) is a modification of DAQ, obtained by trying Double Exponential Quadrature [8] when an integration difficulty is detected in an interval containing at least one endpoint. In [5] it has been proved that versions of these algorithms based on ClenshawCurtis rules achieve the best asymptotic performance for a large class of integrand functions. In this paper we develop a device that gives local error estimates (asymptotically very accurate), detecting the presence of singularities or local difficulties in the integrand. Such device does not affect the computational cost of the algorithms when a family of rules with suitable geometric properties (like RMS rules) is used. The implementation of this device into DAQ and TAQ algorithms is sketched, and the results of numerical experiments are presented. | |

Subject | Efficiency Algorithms Automatic integration Error estimates Local quadrature G.1.4 Quadrature and Numerical Differentiation G.4 Mathematical Software |

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