DOWNDATING THE SINGULAR-VALUE DECOMPOSITION

Let A be a matrix with known singular values and left and/or right singular vectors, and let A' be the matrix obtained by deleting a row from A. We present efficient and stable algorithms for computing the singular values and left and/or right singular vectors of A'. We also show that the problem of computing the singular values of A' is well conditioned when the left singular vectors of A are given, but can be ill conditioned when they are not. Our algorithms reduce the problem to computing the eigendecomposition or singular value decomposition of a matrix that has a simple structure, and solve the reduced problem via finding the roots of a secular equation. Previous algorithms of this type can be unstable and always solve the ill-conditioned problem.