Given a matrix $A$ in SVD, $A = U \Sigma V^{\ast}$, I am interested in approximating the SVD of the matrix $$\hat{A} = A + \Delta X $$ where $\Delta \in \mathbb{R}$ is a small parameter and $X$ is a matrix. Having looked around I have found bounds on the angles between subspaces of left and right singular vectors, but no way of approximating singular vectors. I have also found ways of approximating the SVD of $\Delta X$ given the SVD of $\hat{A}$ but this is not what I am interested in.
Is anyone aware of a way to approximate the SVD of $\hat{A}$ given the SVD of $A$, perhaps with some assumptions about the structure of A?