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 quasi-Newton 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 Family methods (using exact line searches) that skip p updates to the quasi-Newton matrix will terminate in no more than n+p steps on an n-dimensional strictly convex quadratic. We propose several new variations on the limited-memory BFGS method and test these on standard test problems.

We also introduce and test a new method for a process known as Latent Semantic Indexing (LSI) for information retrieval. The new method replaces the singular value matrix decomposition (SVD) at the heart of LSI with a semi-discrete matrix decomposition (SDD). We show several convergence results for the SDD and compare some strategies for computing it on general matrices. We also compare the SVD-based LSI to the SDD-based LSI and show that the SDD-based method has a faster query computation time and requires significantly less storage. We also propose and test several SDD-updating strategies for adding new documents to the collection.