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.