# Implicit casting overhead from real to complex when multiplying two matrices

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Daniele Giovannini on 11 Aug 2022
Answered: James Tursa on 10 Jul 2023
Is there an implicit casting overhead when multiplying a complex matrix by a real matrix to upcast the real matrix to complex or is it a natively-supported operation?
I have tried the following sample code but I get inconsistent results when changing the matrices dimension N.
N = 100;
A = rand(N);
B = rand(N);
C = complex(A);
tic
D1 = B*A;
toc
tic
D2 = B*C;
toc

Benjamin Thompson on 11 Aug 2022
Multiplying a complex matrix by a real matrix requires fewer calculations so should take less time. Your test is a little too simple since B would have already been cached into memory for the second part. If you do B*A twice, it is a lot faster the second time.
Daniele Giovannini on 23 Aug 2022
You are right. I have tried to eliminate the cache from playing a role in two ways:
• having variables small enough to fit in the cache (my CPU cache is 3 MB) at the same time and repeating many times each matrix product
• having variables so big that they cannot possibly be cached and perform each matrix product only once
Both ways seem to confirm your answer. I am reporting my test code for completeness.
% Test 1
N = 150;
C = crand(N);
A_R = rand(N);
A_C = complex(A_R);
whos
tic
for n = 1:1e4
D1 = C*A_R;
end
toc
tic
for n = 1:1e4
D2 = C*A_C;
end
toc
% Test 2
N = 3000;
C = crand(N);
A_R = rand(N);
A_C = complex(A_R);
whos
tic
D1 = C*A_R;
toc
tic
D2 = C*A_C;
toc

James Tursa on 10 Jul 2023
This is not a natively supported operation (to use your terms). Yes, the real matrix must be first upconverted to a complex matrix (deep data copy with interleaved 0's) in order to call the BLAS matrix multiply routines in the background, since the BLAS library has no mixed complex-real routines. This will be a performance hit because of the deep data copy and all those unnecessary 0 multiplies. See the related discussion and explanations here:

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