[SciPy-user] Gaussian quadrature error
Mon Oct 6 15:23:42 CDT 2008
Thanks for the clear explanation, Anne. If I understand the second take-home lesson -- feed integrators unity-order quantities -- then instead of calling quadrature as I did:
I should scale my function f so that my call looks more like
Was that what you meant?
Also, when you say "scipy.optimize.quad is a little more general-purpose" did you mean "scipy.integrate.quad"? Indeed, the call
Thanks, and sorry for the late followup.
> Date: Wed, 17 Sep 2008 12:00:09 -0400> From: firstname.lastname@example.org> To: email@example.com> Subject: Re: [SciPy-user] Gaussian quadrature error> > 2008/9/14 Dan Murphy <firstname.lastname@example.org>:> > I am trying out the integrate.quadrature function on the function f(x)=e**x> > to the left of the y-axis. If the lower bound in not too negative, I get a> > reasonable answer, but if the lower bound is too negative, I get 0.0 as the> > value of the integral. Here is the code:> >> >> >> > from scipy import *> >> >> >> > def f(x):> >> > return e**x> >> >> >> > integrate.quadrature(f,-10.0,0.0) # answer is (0.999954600065,> > 3.14148596026e-010)> >> >> >> > but> >> >> >> > integrate.quadrature(f,-1000.0,0.0) # yields (8.35116510531e-090,> > 8.35116510531e-090)> >> >> >> > Note that 'val' and 'err' are equal. Is this a bug in quadrature?> > No, unfortunately. It is a limitation of numerical quadrature in> general. Specifically, no matter how adaptive the algorithm is, it can> only base its result on a finite number of sampled points of the> function. If these points are all zero to numerical accuracy, then the> answer must be zero. So if you imagine those samples are taken at the> midpoints of 10 intervals evenly spaced between -1000 and 0, then the> rightmost one returns a value of e**(-50), which is as close to zero> as makes no nevermind. You might be all right if this were an adaptive> scheme and if it used the endpoints, since one endpoint is guaranteed> to give you one. But not using the endpoints is a design feature of> some numerical integration schemes.> > The take-home lesson is that you can't just use numerical quadrature> systems blindly; you have to know the features and limitations of the> particular one you're using. Gaussian quadrature can be very accurate> for smooth functions, but it has a very specific domain of> applicability. scipy.optimize.quad is a little more general-purpose by> intent (and necessarily a little less efficient when Gaussian> quadrature will do) but it can be tricked too.> > A more specific take-home lesson is to try to normalize your problem> as much as possible, so that all quantities you feed your integrator> are of order unity. Yes, it's a pain to have to handle scale factors> yourself, particularly in the normal case when you're solving a family> of related problems. But you'll get much more reliable performance.> > Anne> _______________________________________________> SciPy-user mailing list> SciPyemail@example.com> http://projects.scipy.org/mailman/listinfo/scipy-user
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