numerical error when dividing some identical complex number

I wanted to calculate the angular difference (e.g. 155 deg vs. 20 deg) by dividing the exponential angular representation (exp(1j*theta1)./exp(1j*theta2) to achieve this goal. While doing this, I found some numerical error for certain complex number.

For instance, exp(1j*0.5)./exp(1j*0.5)=1 (as it should be be). However, exp(1j*0.5166)./exp(1j*0.5166)=1.000+ 4.8e-17j. The imaginary component is tiny, but non-zero. It seems like a numerical precision issue. I can fix the problem by applying some threshold, but I'm just curious why this problem occurs only in some number, and not all complex number. Does anyone know it? Just out of curiosity.

Here is my MATLAB version info

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MATLAB Version: 9.13.0.2049777 (R2022b)

Operating System: Linux 5.10.0-19-amd64 #1 SMP Debian 5.10.149-2 (2022-10-21) x86_64

Java Version: Java 1.8.0_202-b08 with Oracle Corporation Java HotSpot(TM) 64-Bit Server VM mixed mode

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Thanks!

Misun

[Paul]

Previously discussed: https://www.mathworks.com/matlabcentral/answers/548010-why-is-a-number-divided-by-itself-not-equal-to-unity

[Misun Kim]

Thanks a lot @Paul for directing me to the previous discussion. This answers to my question.

Answer 1

[Misun Kim]

@Paul provides the link to the previous discussion which answers to my question on dividing complex number. For the ease of finding it, I place the link here again. https://www.mathworks.com/matlabcentral/answers/548010-why-is-a-number-divided-by-itself-not-equal-to-unity

Answer 2

[Steven Lord]

Are you certain the two numbers you're using are identical down to the last bit, not just down to the last displayed digit?

x = 0.1234

x = 0.1234

y = 0.12341

y = 0.1234

Despite x and y being displayed the same, they are not the same stored value. False is the correct answer for this comparison that checks the stored value not the display.

areTheyIdentical = x == y

areTheyIdentical = logical
   0

Does that small difference between x and y make a difference in the exponentials?

ex = exp(1i*x)

ex = 0.9924 + 0.1231i

ey = exp(1i*y)

ey = 0.9924 + 0.1231i

They display the same, but is the answer exactly 1? No.

ex/ey

ans = 1.0000 - 0.0000i

Let's display the three values with a few more decimal places

format longg
[ex; ey; ex/ey]

ans = 
          0.992395876704891 +     0.123087058211376i
          0.992394645784689 +     0.123096982163989i
              0.99999999995 -  9.99999999984743e-06i

That small difference between x and y resulted in a difference in ex and ey.

[Misun Kim]

Yes, I already check whether two values are identical (not just shown identical on the display), but the exponential complex number is different, when divided by each other.

x=0.5166; y=0.5166;
x==y

ans = logical
   1

Then, I check exp(1i*x)./exp(1i*y)

ex=exp(1i*x)

ex = 0.8695 + 0.4939i

ey=exp(1i*y)

ey = 0.8695 + 0.4939i

ex./ey

ans = 1.0000 + 0.0000i

imag(ex./ey)

ans = 4.8267e-17

ex-ey

ans = 0

If I repeated the process with some other value, it returns clean 1.

x=0.5; y=0.5;
x==y

ans = logical
   1

ex=exp(1i*x)

ex = 0.8776 + 0.4794i

ey=exp(1i*y)

ey = 0.8776 + 0.4794i

ex./ey

ans = 1

imag(ex./ey)

ans = 0

So, it's some kind of numerical precision error with particular value.

[Walter Roberson]

More compactly:

format long g
x=0.5166

x = 
                    0.5166

ex = exp(1i*x)

ex = 
          0.869503552901398 +     0.493926686353193i

ex./ex

ans = 
                          1 +  4.82671432212255e-17i

This is not what I would have expected

[Paul]

Check the link in the comment to the Question.