When I'm using relaxation to find an eigenvalue, how do I make sure I relax to the right one? solvde seems intent on relaxing to the third eigenvector, but I want the first one!

A Google search turned up a reference to Castor (1971) and Osaki & Hansen (1973); the method in these papers seems to be a kind of hybrid between relaxation and shooting, but I wonder if there is any more modern advice on this subject.