I have with a sparse matrix A and a vector x in Eigen, and I need to perform the following transposed matrix × matrix × vector operation: A^TAx. This can be decomposed into two matrix-vector multiplications: y=Ax followed by A^Ty. The first one is simple, but I don't know how to do the second one without the explicit construction of A^T. I can write something as:

y = A.transpose() * A * x;

However, it seems that A.transpose() explicitly constructs A^T in memory, which would not be efficient for large matrices.

Another option might be to work with a sparse A^TA, but in our case, this is a much more dense matrix than A itself.

Question: Is there any way how to multiply a vector by a transposed sparse matrix in Eigen without explicit construction of A^T or A^TA?