Numerical precision in subtracting two almost identical variables?

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Marco 2024-2-28

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https://ww2.mathworks.cn/matlabcentral/answers/2088091-numerical-precision-in-subtracting-two-almost-identical-variables

编辑： Stephen23 2024-2-28

As far as I understand it, Matlab uses 16 digits of precision. However, I have a program that subtracts two variables and very often gives results much smaller than 1e-16. For example. When asked it displays a=0.000000006129562, b=0.000000006129562 but a-b=1.0e-24 * 0.827180612553028 How does it distinguish two variables that differ by a quantity much smaller than the 15th decimal place, if it doesn't have an internal representation of it no larger than 16 digits? Shouldn't be it a-b=0 always?

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Stephen23 2024-2-28

编辑：Stephen23 2024-2-28

In your question you incorrectly conflate two related but different topics. Here, I highlighted them for you:

"How does it distinguish two variables that differ by a quantity much smaller than the 15th decimal place, if it doesn't have an internal representation of it no larger than 16 digits?"

Decimal places is not the same thing as significant digits. Mixing up those two different things (like you are doing) is not going to help you do any computational mathematics, let alone do any science. There are many many many tutorials online which explain the difference, so I will not waste my time duplicating tutorial content here.

"Shouldn't be it a-b=0 always?"

No.

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Answer 1

Aquatris 2024-2-28

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https://ww2.mathworks.cn/matlabcentral/answers/2088091-numerical-precision-in-subtracting-two-almost-identical-variables#answer_1418166

在 MATLAB Online 中打开

Here is your answer link

Its all in the low level things. If you display your values with 'format hex', you will see they differ by 1 bit, and that bits gives you the 1e-16 difference.

Example:

x = 0.6;
y = 6*0.1;
delta = x-y
delta = -1.1102e-16
format long
[x y]
ans = 1×2
   0.600000000000000   0.600000000000000
format hex
[x y]
ans = 1×2
   3fe3333333333333   3fe3333333333334

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John D'Errico 2024-2-28

编辑：John D'Errico 2024-2-28

在 MATLAB Online 中打开

@Marco

You are missing that MATLAB does not store this number 0.00000123456789 as those 15 decimal digits, in cluding the leading zeros. That would be a bad thing, because it would fail due to exactly what you are worried about.

Instead, you store numbers using a scientific notation.

format long g
0.00000123456789
ans = 
            1.23456789e-06

Of course, this shows the number using powers of 10. MATLAB stores the number internally in a binary form, but the two forms are very similar. Because there is always an exponent attached to each number, this form is called FLOATING POINT. Essentially, the decimal point is allowed to float.

Th nice thing is, now a number like 0.00000123456789

is stored with a relative error only down in the least significant bits of that number. That is given by eps.

eps(0.00000123456789)
ans = 
      2.11758236813575e-22

So if you subtract two numbers of that size, you can now start to see differences with an order of magnitude 1e-22.

Marco 2024-2-28

Ok, so if I understand it, the FP internal representation does not include the leading zeros. This makes sense. Is there a way to visualize the value of the two variables which difference is delta to more than 15 decimal places? Format long g does not.

For example:

Displayed are a=0.000000006129562 and b=0.000000006129562, but delta=a-b=1.0e-24 * 0.827180612553028

Then I expect it to be internally a=6.129562xxxxxxxxxe-9 and b= a=6.129562yyyyyyyyye-9

Could those decimal places in x and y made be visible?