NIntegrate preprocessing and rule selection

*To*: mathgroup at smc.vnet.net*Subject*: [mg81098] NIntegrate preprocessing and rule selection*From*: "Andrew Moylan" <andrew.j.moylan at gmail.com>*Date*: Wed, 12 Sep 2007 03:48:29 -0400 (EDT)

Here's an integrand that is highly oscillatory for large values of its first argument: f[w_, x_] = Sin[w*x]/(x^2); For small w, NIntegrate automatically selects GaussKronrodRule as usual: Reap[NIntegrate[f[1, x], {x, 1, 2}, EvaluationMonitor :> Sow[x]]] For large w, NIntegrate's preprocessor decides the integrand is highly oscillatory and uses its specialised Clenshaw-Curtis rule for oscillatory integrands: Reap[NIntegrate[f[10, x], {x, 1, 2}, EvaluationMonitor :> Sow[x]]] (One way to tell that the specialised rule was used is to notice that the endpoints (1. and 2.) were sampled.) Now consider this example: Reap[NIntegrate[f[1, x], {x, 1, 2}, Method -> "ClenshawCurtisRule", EvaluationMonitor :> Sow[x]]] As expected, NIntegrate's preprocessor does not decide to use the specialised Clenshaw-Curtis rule (because w is too low to be called "highly oscillatory"). However, the integration then apparently proceeds using GaussKronrodRule rather than the requested ClenshawCurtisRule. Why is this? One way to get around this for particular cases where you know it is going to happen is to use the option "SymbolicProcessing"->0. But what should we do if we *do* want symbolic processing, but want ClenshawCurtisRule to be used (instead of GaussKronrodRule) in those cases to which none of NIntegrate's specialised rules apply?