Daniel, your reply did not post in the User Group. I got your message and logged in on the User Group web site but your reply is not here. It seems you duplicated my issue, which is why I did not attach a worksheet, since I wanted an independent verification outside of my Mathematica application.

The issue is to me clear: somehow the simple numerical integration does not execute on the first attempt to evaluate but it does execute on the second attempt with the message you posted. Why does it execute the second time and throw out an inane comment? If I use the same process on a more complete numerical integration with the precision and accuracy goal and the working precision indicated, the same thing happens: it does not execute on the first iteration but does on the second, even though all parameters are not defined and so the numerical integration cannot proceed...yet on the second evalution is does with the second inane comment I posted.

Why does this happen and how do I keep it from happening? The reason I would like some clarity is that these integrals are defined as functions that are embedded in more complex expressions and I cannot assure that the second and unexpected execution has not created an expression that is incorrect in the more complete expression.

Now, my concern may not mean anything to you but to me it means that I cannot trust the results that I calculate. The reason is that I am preparing a worksheet to post in the physics section that identifies a particular integral that needs to be numerically evaluated and there are some issues I am addressing on how to interpret the results given how Mathematical performs the numerical calculations.

If I set a variable, call it d and evaluate d without giving it a value, I simply get d back....no matter how many times I evaluate d.

In the simple function I used as an example, when I evaluate that function without defining certain variables, I get the function back with no comment. If I evaluate this function a second time or more, I get the comment back that it evaluates with indeterminate values.

If a more complicated function with the accuracy and precision and working precision limits set but no variables defined, again, the first evaluation gives back the function with no comments. The second and later evaluations with the variable still not defined gives back the statement that it tried to evaluate the function and that the function's precision was less than the working precision. What can this possibly mean, since the function should not evaluate at all? And, I do not want some spurious evaluation to "contaminate" the function.

@GeorgeWoodrowIII The multi-point form is not heavily covered in the NIntegrate reference page documentation but it is shown under Scope > Basic Uses in the example in "Along a piecewise linear contour in the complex plane:". That said, the use in this post seems unlikely to be what was intended-- why would one integrate along a path from 1 to 2 and then back to 0.1, instead of just integrating from 1 to 0.1 to begin with?

I am not comfortable with working in the complex domain when the arguments to integrals are real and finite. I went through all the advanced mathematics and complex variable analysis coursework in my youth but in the pre-Mathematica days I just used Gratshteyn and Ryzhik and kept my fingers crossed. If that did not work out I resorted to writing numerical integral code using the trapezoidal or Simpson rules and stayed away from the complex plane. I simply do not want to have to understand the complex solutions that arise when what are finite real integrals throw off complex solutions.

OK, now that I have that off my chest, I have attached a worksheet that one of the Community members helped me develop. Do not pay attention to why I want to evaluate the particular integral, since I was going to post this to the Physics section. It is the real mutual attraction model that has never been solved....the classical point-mass integral we are all taught is a heuristic.

Consequently, I set out to prove my point. The issue became that the mutual attraction between sphere's is so close to the point-mass model that the solutions are really perturbations on the point-mass approach....which is why no issues have ever been identified. If the model is extended to cylinders and internal masses...a test mass within a distributed mass such as a sphere or disk, the results are smooth and totally unlike the point-mass results...the solutions are not simply perturbations. My goal has been to understand the detail of the spherical solutions. However, the perturbations are on the same order as the variances in the numerical results when I switch between coordinate-system descriptions of the spheres.

That being said, the worksheet models have been set up to explicitly evaluate at one point and I was attempting to generalize the approach to allow a parametric evaluation over the separation distance. It is actually easy to do and I have voluminous plots showing the result. However, when using the worksheet as a template and getting ready to post this material in the physics community, I got the behavior that I have been grumbling over and I do not understand what is going on. So far, I still don't understand the issue. To may simplistic way of looking at things, the integrals should not have evaluated at all until I defined the variable parameter...which it did in my original clunky methodology. I have no doubts about the validity of the results, but I wanted to create a more sophisticated approach and ran into these issues.

I am a bear about details, which is why I noticed that the Newtonian gravity model...or static electric interaction models as we use them...are heuristics. Now the issue is that I simply cannot understand why the numerical models evaluate at all when they are incomplete.

Regards and thanks for your forbearance.

Luther

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