I have to solve a generalized eigenvalue problem, $A x = \lambda B x $. Here $A,B$ are both $n\times n $ symmetric and in particular, $B$ is positive definite. I just need the smallest generalized eigenvalue.
Now, suppose that $B $ has the simple form $B = I + uu^T $. Similarly, $A$ is also simple and its action on a vector $x$ can be done in $O(n)$ operations.
For such special cases, is there a particular fast algorithm?