I am working on a numerical method for the least-squares solution of a linear system. I know that I can approximate the solution to $Ax=b$ with $x=A^+b$, where $A^+$ is the Moore-Penrose pseudoinverse of $A$. In my method, I have to solve such a linear system repeatedly, where the matrices I need to (pseudo)invert are column subsets of a common matrix.
So my question is: Suppose I know the pseudoinverse of $A$ ($A$ is not invertible). Suppose $B$ is another matrix, whose columns are columns of $A$ (a subset, but not permuted). Is there an efficient way to compute the pseudoinverse of $B$ from that of $A$?