Similarly, you can obtain the right polar decomposition (which is what you sought)
A = P U
P = W S W*
U = W V*
as before. Please note that the factors P in the left and right polar decompositions are different.
MKL/Lapack provides routines to perform the SVD. You may consider the intended application of the polar decomposition to see if you can use P and U in factorized form instead of forming them explicitly. In particular, if you are going to multiply some vector by P or U, you can accomplish that with two or three matrix-vector multiplications, avoiding the more costly matrix-matrix multiplications that are involved in explicitly forming P and U.