. The focus of this dissertation is on matrix decompositions that use a limited amount of computer memory, thereby allowing problems with a very large number of variables to be solved. Specifically, we will focus on two applications areas: optimization and information retrieval. We introduce a general algebraic form for the matrix update in limited-memory quasiNewton methods. Many well-known methods such as limited-memory Broyden Family methods satisfy the general form. We are able to prove several results about methods which satisfy the general form. In particular, we show that the only limited-memory Broyden Family method (using exact line searches) that is guaranteed to terminate within n iterations on an n-dimensional strictly convex quadratic is the limited-memory BFGS method. Furthermore, we are able to introduce several new variations on the limited-memory BFGS method that retain the quadratic termination property. We also have a new result that shows that full-memory Broyden Fa..
