# what exactly norm(x) function do?

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Manu Chaudhary on 21 Sep 2022
Edited: James Tursa on 22 Sep 2022
Good Afternoon,
I have to convert a part of my matlab code into C++.
I would be great if you can explain me what exactly norm(x) do on a vector of complex number so that I can write a C++ code for it.
I just asked a similar question on stack overflow but could not get much response.
Manu Chaudhary on 22 Sep 2022
Edited: Manu Chaudhary on 22 Sep 2022 The formula of norm(x) is defined as above on your website. Just for confirmation, |vk| means a^2 + b^2?

James Tursa on 21 Sep 2022
Edited: James Tursa on 21 Sep 2022
The equivalent function to norm( ) for a complex column vector x is
sqrt(x' * x)
where x' is the complex conjugate transpose. Essentially sqrt(dot(x,x)). Note that dot( ) does the complex conjugate part of the calculation automatically.
E.g.,
x = [1;2;3;4] + [5;6;7;8]*1i
x =
1.0000 + 5.0000i 2.0000 + 6.0000i 3.0000 + 7.0000i 4.0000 + 8.0000i
norm(x)
ans = 14.2829
sqrt(x'*x)
ans = 14.2829
sqrt(dot(x,x))
ans = 14.2829
So in C/C++ just run a loop to sum up the individual element-by-element multiplies for the dot product result. For an element of the form a+b*i you will be summing a^2+b^2 and then taking the sqrt( ) of the final sum.
James Tursa on 22 Sep 2022
Edited: James Tursa on 22 Sep 2022
Yes this algorithm is correct, but I still advise changing your code to do the arithmetic in double to avoid potential pitfalls where the int calculations might overflow.

Matt J on 22 Sep 2022
Edited: Matt J on 22 Sep 2022
Assuming you only care about p=2, If you already have code that computes norm(x) for real x, you can extend it to complex x via,
norm(x) = norm( [norm(real(x)), norm(imag(x)) ] )
which is easily verified below,
x=complex(rand(5,1), rand(5,1));
norm(x)
ans = 1.2548
norm( [norm(real(x)), norm(imag(x)) ] )
ans = 1.2548
Alternatively, if you have an implementation of abs, you could do norm(abs(x),p) which will work for any p-norm:
p=3;
norm(x,p)
ans = 1.0042
norm(abs(x),p)
ans = 1.0042