C++ implementation of the singular value decomposition(SVD)?

Question

C++ / C#

There was a need to look for a singular value decomposition of a matrix in C ++, I’ll immediately make a reservation that, unfortunately, I don’t understand the geometric ins and outs, I only know that for the decomposition found, the following relation must be satisfied in the matrix form A = U * W * V ^ T, where the matrices are U, W, V are the result of the decomposition of SVD(A), and ^T is the transposition operation.
Found a ready implementation here . For a matrix

, it produces the following
result .
However, mathcad disagrees and thinks a little differently, like this: The

question is - why does this implementation correctly calculate numbers, but confuse them in places and be smart with signs? If it is not working, then where can I find a working one?