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Assuming we have a specific equation involving two coefficients A and B.

If A is given and B is unknown, B can be numerically and uniquely obtained by performing root-finding on the above equation(process X hereafter).
If B is given and A is unknown, A can be analytically and uniquely obtained by the same equation(process Y hereafter).

The scenario I concerned:
If A is given arbitrarily, then B can be numerically obtained from A by root-finder. Afterwards, we can transform B into A' by process Y. Although A is theoretically equal to A', there is always a machine epsilon between original A and A' because no computer has infinite precision floating point number - in other words, if we had infinite precision floating point number on computer, then A would be exactly equal to A'.

Now, My question is how do I fluently rewrite the above scenario in the following pattern:
[The error between A and A'...]
or
[The error of A between ... and ...]

The sentence I have tried to write is as follows:

  1. The figure shows the error of A between the given and calculated values from B, which is computed through root-finder.
  2. The figure shows the error between given A and A calculated from B, which is computed through root-finder.

Is there still room for improvement? I will be grateful for any help you can provide.

To Jason Bassford: Both X and Y processes are actually carried out on computer. This is a major cause of why there is an error between A and A'.(the magnitude of error depend heavily on numerical precision)

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  • Your description of the scenario is inaccurate. If the conclusion of the scenario is that A' does not exactly equal A, then something is off. At best you need to either say that (1) a result similar to A can be obtained from B by Y process or (2) while A can be obtained be obtained from B by Y process, in this case it was not, and only A' was obtained. (The only other possibility is that Y process was not followed exactly.) – Jason Bassford Jun 2 '19 at 2:50
  • @JasonBassford Thank you. Both X and Y processes are actually carried out on computer. This is a major cause of why there is an error between A and A'. (the magnitude of error depend on finite numerical precision) – Rocky Tseng Jun 2 '19 at 3:18
  • In that case (assuming the computer always does exactly the same thing), either the computer isn't following process Y or it's simply not true that A can be obtained from B by following process Y. – Jason Bassford Jun 2 '19 at 3:24
  • @JasonBassford sorry for unclear description. Could you see modified question again? thanks. – Rocky Tseng Jun 2 '19 at 3:55
  • Can you distinguish between mathematically equal and machine epsilon? Although you're now using different terminology, either things actually are equal or they aren't. – Jason Bassford Jun 2 '19 at 4:17
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You don't. We don't use that pattern to talk formally about error terms.

You have two operators that can work on your sets. They are expected to be inverse operations such that X(A) yields B and Y(B) yields A, therefore Y(X(A)) yields A. But there is an error term. This error is usually thought of as an instance of a property of the chain of operations that is manifest in the output. It isn't regarded as a comparison, even though in this case it is simply A - A'.

Given the black box description of the problem, You can only associate the error with the sequence of operations as performed.

The error resulting from the operation chain Y(X(A)), as computed by root finder, ...

(None of your examples are complete sentences).

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