Re: Mathematica Weirdness

*To*: mathgroup at smc.vnet.net*Subject*: [mg116689] Re: Mathematica Weirdness*From*: Christopher Henrich <chenrich at monmouth.com>*Date*: Thu, 24 Feb 2011 06:21:36 -0500 (EST)*References*: <ik2nab$9kk$1@smc.vnet.net>

In article <ik2nab$9kk$1 at smc.vnet.net>, Steve Heston <sheston at rhsmith.umd.edu> wrote: > My question is why I get a negative integral of a positive > function? > > Integrate[1000000*Exp[x^2-12*x]*x^14,{x,0,1}]//N > Integrate[1000000*Exp[x^2-12*x]*x^14,{x,0.,1}]//N > NIntegrate[1000000*Exp[x^2-12*x]*x^14,{x,0,1}] > > The first line gives a negative answer, while the second two lines give > identical positive answers. Something is strange here. If you strip "//N" from the first line, you get an expression with three terms (I am counting as "one term" a product that involves (Erfi[5]-Erfi[6]).) Two of these terms are much larger than the third, with opposite signs and nearly equal magnitudes. Their sum seems to be of opposite sign to the other term, and *very* nearly equal magnitudes. In short there is some massive cancellation going on. I think the number of extra digits of precision carried along in the numerical evaluation implied by "//N" was not enough to get a good result. I do not know why the second line came out differently. I suspect that Mathematica used a different strategy for working out the analytical form of the integration, because of the machine-precision number "0." . The third line does not attempt to find an analytical form for the integral, but applies numerical integration methods from the outset. A plot of the integrand shows that it is not very "pathological" in the sense of having a tall narrow spike, so the numerical result is probably good. The online documentation of N and NIntegrate is helpful for giving one a sense of the limitations of the numerical accuracy of these functions, and for suggestions of how to experiment with the precision of their internal operations. -- Christopher J. Henrich chenrich at monmouth.com http://www.mathinteract.com "A bad analogy is like a leaky screwdriver." -- Boon