How do I ratio to very small values and avoid rounding to unity?

Questions
1

The title is not very precise.

I am ratioing the numerical outputs of two regular integrals, whose values are very small over the domain of interest. The denominator integral is a normalization integral. The calculated values differ in the seventh decimal place. The ratio always defaults to 1.0 in Mathematica. I investigated this at a detailed level because the parametric Plot function shows strange step functions in what should be smooth data.

However, a Table of values for the two integrals over the range shows the expected plots for both functions. Except, there seems to be a “noise” element with periodically spaced small step functions that should not be there. I am trying to understand what is real and what is noise and cannot find the expressions to force the accuracy needed for the ratio to supply smooth and consistent results. I tried to force various precision and accuracy goals and limits, but nothing gives me the correct values for the ratios except my hand held calculator!! Boo!

2

Impossible to suggest much without a concrete example.

3

I have attached an example…it took me some time to put it together, and it requires a bit of discussion.

The reason for using Table was to understand the source of the variations shown in the plot at the bottom of the attachment. The big step function should not be there. Also, the small ripples on the leading flat portion of the plot replicate if I “zoom” in on them by limiting the plot range. They should also not be in there. In fact, I had expected the plot to be a smooth continuation of the curved portion of the plot extrapolating back toward d->0…it never gets to zero but to a small value.

The Tables are for each integral separately and then for the difference between the two integrals and then the ratio of the two integrals. A plot of the Table points show smooth curves with an occasional discontinuity that should also not be in there. My hand calculations using the Table data show that the two integrals differ in the 7th or 8th decimal place of the already small values. In real life, there are common coefficients to the integrals that make them much larger, but those are constants that cancel in the ratio and are common to the other comparisons as simple multipliers.

I suspect my technique is faulty but I cannot find any way to improve the results. I am either seeing noise or the results of an error in performing the integrations and comparisons.

Does the attached file make sense?

4

[I wrote something that was badly incorrect and now do not know how to delete it. Sigh.]

Anyway, at least I can confirm I am seeing the same result. It is unexpected but that does not mean it is wrong. I simply do not know if that jump is legitimate.

5

Ack, it’s the AccuracyGoal. Bring it up to 12, say (even 8 seems to be enough for the range in question). Keeping in mind that the discrepancies are around the 6th digit, it’s safe to expect that AccuracyGoal=>6 may not be adequate.

6

Thanks for the suggestion. I had played with AccuracyGoal in the past but cannot locate the nb file I was using. I switched to the numerical integration because I could use the new parametric plot routine to get a better view of what I was looking at.

You have confused me with responding to two different threads! I am easily confused these days. Anyway, I cannot dig up an example for my other issue because I have moved away from Integrate to NIntegrate. With Integrate, sometimes Mathematica would run all night…Version 10. It was under some of these circumstances that my funky integral would give me these branch cut solutions. With NIntegrate, I don’t get them if I get an answer at all.

I will pursue your suggestions and look over your approach. I am learning more every day. I wonder how many people simply accept their first answers and move on without doing some “parametric evaluations” of their results?

7

Here are some remarks.

(1) I did some further checking. PrecisionGoal also needs tweaking. This is close to smooth, but slow.

fme[d_?NumericQ] := 
 NIntegrate[
   1000 (r1*r2*(d + z2 - z1)*
      d^2/(r1^2 + r2^2 + z1^2 + z2^2 - 2*z1*z2 - 
          2*r1*r2*Cos[t2 - t1] + d*(d + 2*z2 - 2*z1))^1.5), {t1, 0, 
    2 Pi}, {t2, 0, 2 Pi}, {z1, -1, 1}, {z2, -.27, .27}, {r1, 0, 
    Sqrt[1 - z1^2]}, {r2, 0, Sqrt[.27^2 - z2^2]}, PrecisionGoal -> 12,
    AccuracyGoal -> 15]/
  NIntegrate[
   r1*r2*1000, {t1, 0, 2 Pi}, {t2, 0, 2 Pi}, {z1, -1, 
    1}, {z2, -.27, .27}, {r1, 0, Sqrt[1 - z1^2]}, {r2, 0, 
    Sqrt[.27^2 - z2^2]}, PrecisionGoal -> 12, AccuracyGoal -> 15]

Plot[fme[d], {d, 100, 130}, AxesLabel -> {d, F (Normalized)}, 
 PlotLegends -> {"Sphere"}, PlotStyle -> {Thick}, PlotRange -> Full]

(2) Might or might not help to simplify the denominator and also not recompute it. Can even compute it exactly.

c2 = 27/100;
d2 = 16*Pi^2*
  Integrate[
   r1*r2, {z1, 0, 1}, {z2, 0, c2}, {r1, 0, Sqrt[1 - z1^2]}, {r2, 0, 
    Sqrt[c2^2 - z2^2]}]

Also you can use c2 in the numerator computations.

(3) At lower PrecisionGoal, maybe in the range 8-10, it is reasonably fast and not as jumpy but still shows jumps and smooth sections (at least 6 places out though, since the y axis points are all labeled as 1). I suspect numerical issues wherein error estimates cross a threshold and the routines either do further refinement, or do not, depending on which side of the threshold they are on.

(4) As for your other note, if you saw imaginary parts and jumps due to branch cuts in exact definite integrals involving elliptics, I am not surprised. Should you see them again, please politely remind them that they remain the bane of my existence.

8

Thank you for your efforts and insights.

As payment, I took #4 to heart and opened an elliptical integral and yelled at it " No more branch cuts!!!"

See if that helps. I will look into your suggestions as soon as I get my wife’s car running again.