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.