Re: NIntegrate preprocessing and rule selection

*To*: mathgroup at smc.vnet.net*Subject*: [mg81488] Re: NIntegrate preprocessing and rule selection*From*: antononcube <antononcube at gmail.com>*Date*: Wed, 26 Sep 2007 06:31:01 -0400 (EDT)*References*: <fc85un$ll0$1@smc.vnet.net>

The specialized integrator for finite oscillatory integrals uses modified Clenshaw-Curtis rule. When the integral is not oscillatory enough the modified Clenshaw-Curtis rule switches to a Gauss-Kronrod rule. Consider this an integral part of the modified Clenshaw-Curtis rule (which is applied only to finite oscillatory integrals). So if we want to use the non-modified Cleshaw-Curtis rule, we should preclude the OscillatorySelection strategy. Here is one way to do this: In[8]:= f[w_, x_] = Sin[w*x]/(x2); In[9]:= Reap[NIntegrate[f[1, x], {x, 1, 2}, Method -> {"SymbolicPreprocessing", "OscillatorySelection" -> False, Method -> "ClenshawCurtisRule"}, EvaluationMonitor :> Sow[x]]] Out[9]= {0.472399, {{1., 1.03806, 1.14645, 1.30866, 1.5, 1.69134, 1.85355, 1.96194, 2., 1.01903, 1.07322, 1.15433, 1.25, 1.34567, 1.42678, 1.48097, 1.98097, 1.92678, 1.84567, 1.75, 1.65433, 1.57322, 1.51903}}} Compare with: In[13]:= Reap[NIntegrate[x6, {x, 1, 2}, Method -> "ClenshawCurtisRule", EvaluationMonitor :> Sow[x]]] Out[13]= {18.1429, {{1., 1.03806, 1.14645, 1.30866, 1.5, 1.69134, 1.85355, 1.96194, 2., 1.01903, 1.07322, 1.15433, 1.25, 1.34567, 1.42678, 1.48097, 1.98097, 1.92678, 1.84567, 1.75, 1.65433, 1.57322, 1.51903}}} Anton Antonov Wolfram Research, Inc On Sep 12, 2:53 am, "Andrew Moylan" <andrew.j.moy... at gmail.com> wrote: > 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,NIntegrateautomatically selects GaussKronrodRule as usual: > > Reap[NIntegrate[f[1, x], {x, 1, 2}, EvaluationMonitor :> Sow[x]]] > > For large w,NIntegrate'spreprocessor 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'spreprocessor 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 ofNIntegrate'sspecialised rules apply?