How to explain loss of significance in numerical analysis?

Question

As I have myself struggled a bit with this concept, I would like to present my own explanation of it. Context: Loss of significance is a loss of precision, not necessarily accuracy.

And for a long time, I didn't know the difference between the two concepts.

Explanation: Now first, I show them a calculation of subtracting two numbers close to each other in floating point. We then try to calculate the initial and final relative and absolute errors.

We then observe that the relative error has become much larger, and in addition to that, we have lost many significant figures.

Context: when I was presented this in my first course of numerical analysis, I said: "so? Who cares? We are still close to the number!","the absolute error is still bounded by the addition of the errors!"

But then when I realized the difference between precision and accuracy, I convinced myself that this was a loss of precision.

Explanation: the relativity of precision:

Intuitively, we can observe that as we increase the size of the Actual number to be estimated A, relative to the size of the absolute error, we see that the cluster of numbers (Whose average is a measure of precision) gets "closer together", and that might give us sufficient precision.

Example: Like trying to hit a ship with rockets, an error area of 1 meter is amazing.

But hitting an apple with a bow and arrow , and 1 meter is too imprecise, too big to hit the apple consistently.

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Last part of explanation: I then say," we want to be consistent with our data, and a loss of relative error takes that away from us. So we "care" about loss of significance.

I myself am only partially satisfied with this explanation, and find that there is something missing, or perhaps it is in part incorrect.

I would appreciate your comments or any guidance from your part.

Answer 1

If they're not impressed by small errors, point out catastrophic cancellation.

As a computer programmer, these issues are the kind of thing that we pay great attention to. We even modify the quadratic formula to avoid these problems.

Where you really get in trouble is when you use equations that are built around exact assumptions, like "energy is conserved." When you use equations where energy isn't conserved, you can get spectacularly absurd results because you were deponent on energy being the same before and after.

You may have to show them non-linear responses first. Linear responses to errors can be as boring as they think they are. Non-linear can be spectacular. Guidance and Control systems can be particularly unfriendly. You can mathematically show that they are stable, but due to numerical imprecision, the simulation version of the algorithms can be wildly unstable.

Answer 2

Significance is precision which is reproducability.

Accuracy is a degree of correctness.

Consider two 1 foot rulers lying on a table. Both are marked in nominal inches:

1. One ruler is made of Invar-42 low expansion nickel-iron alloy but is obviously only about 10 inches in length despite being numbered to 12.
2. The other looks to be 12 inches in length but is made of stretchy rubber.

Ruler #1 is very precise (albeit inaccurate) and reading off many significant figures from it has value and reproducability.

Ruler #2 is more accurate but measuring the same thing multiple times with it will give wildly varying precision, as in the significance is low.

Another thought approach is weighing a known high precision, to many significant figures, mass. If you are using a bathroom scale, anything more precise than a tenth pound would have no significance. On my bathroom scale it's more like 2 pounds.

Answer 3

1. In statistics, business and science, with reference to a set of repeated measurements, precision = consistency = low variability = closeness of the data points = reproducibility & repeatability.

Note that accuracy (closeness to the true value) and precision are orthogonal concepts.

2. In physics and numerical analysis, precision is typically thought of not as a measure of data dispersion but as a measure of the resolution of the instrument or the recordings or the calculations. As such, imprecision (lack of significant figures) is about uncertainty—and can ultimately lead to inaccuracy (loss of correctness), due to the approximation errors (rounding and truncation errors) being cumulatively propagated by an algorithm, its numerical instability, or a problem's ill-conditioning.

The following comment nicely links these two aspects/interpretations of precision: The closer the repeated results are to each other, the more confident you can be about expressing the result with a greater number of digits.

Answer 4

This is pretty much a scientific methods question (I guess math can concern itself also). But in any case, if you look at any first chapter of a college (or even high school) chemistry or physics text, you are very likely to get a great discussion of this topic.

FWIW, the analogy that "sings" to me is riflery or archery (or I guess darts). A marksmen will hit a "tight pattern". But if the sights are off, it will be displaced. The visual of the bullseye is very powerful here. A video example, from minute 1:00 to 1:45:

https://www.youtube.com/watch?v=nXrsB2RMo2w

Of course the same thing applies with artillery or NGFS.