  A novel orthogonalization-free method together with two specific algorithms
are proposed to solve extreme eigenvalue problems. On top of gradient-based
algorithms, the proposed algorithms modify the multi-column gradient such that
earlier columns are decoupled from later ones. Global convergence to
eigenvectors instead of eigenspace is guaranteed almost surely. Locally,
algorithms converge linearly with convergence rate depending on eigengaps.
Momentum acceleration, exact linesearch, and column locking are incorporated to
further accelerate both algorithms and reduce their computational costs. We
demonstrate the efficiency of both algorithms on several random matrices with
different spectrum distribution and matrices from computational chemistry.
