Modifiable low-rank approximation to a matrix

A truncated ULV decomposition (TULVD) of an m × n matrix X of rank k is a decomposition of the form X=ULV^T+E, where U and V are left orthogonal matrices, L is a k × k non-singular lower triangular matrix, and E is an error matrix. Only U,V, L, and ∥E∥_F are stored, but E is not stored. We propose algorithms for updating and downdating the TULVD. To construct these modification algorithms, we also use a refinement algorithm based upon that in (SIAM J. Matrix Anal. Appl. 2005; 27(1):198-211) that reduces ∥E∥_F, detects rank degeneracy, corrects it, and sharpens the approximation.