DIRECTION-PRESERVING AND SCHUR-MONOTONIC SEMISEPARABLE APPROXIMATIONS OF SYMMETRIC POSITIVE DEFINITE MATRICES∗ MING GU† , XIAOYE S. LI‡ , AND PANAYOT S. VASSILEVSKI§ Abstract. For a given symmetric positive definite matrix A ∈ RN×N , we develop a fast and backward stable algorithm to approximate A by a symmetric positive definite semiseparable matrix, accurate to a constant multiple of any pres...

An analytic proof is proposed of Wiener’s theorem on factorization of positive definite matrix-functions.

The modified Cholesky factorization of Gill and Murray plays an important role in optimization algorithms. Given a symmetric but not necessarily positive definite matrix A, it computes a Cholesky factorization ofA +E, where E= if A is safely positive definite, and E is a diagonal matrix chosen to make A +E positive definite otherwise. The factorization costs only a small multiple of n 2 operati...

Sparse linear equations Kd r are considered, where K is a specially structured symmetric indefinite matrix that arises in numerical optimization and elsewhere. Under certain conditions, K is quasidefinite. The Cholesky factorization PKP T LDL T is then known to exist for any permutation P, even though D is indefinite. Quasidefinite matrices have been used successfully by Vanderbei within barrie...

We present an approximate structured factorization method which is efficient, robust, and also relatively insensitive to ill conditioning, high frequencies, or wavenumbers for some discretized PDEs. Given a sparse symmetric positive definite discretized matrix A, we compute a structured approximate factorization A ≈ LLT with a desired accuracy, where L is lower triangular and data sparse. This ...

The accurate solution of some of the main problems in numerical linear algebra (linear system solving, eigenvalue computation, singular value computation and the least squares problem) for a totally positive Bernstein–Vandermonde matrix is considered. Bernstein–Vandermonde matrices are a generalization of Vandermonde matrices arising when considering for the space of the algebraic polynomials o...

The accurate solution of some of the main problems in numerical linear algebra (linear system solving, eigenvalue computation, singular value computation and the least squares problem) for a totally positive Bernstein–Vandermonde matrix is considered. Bernstein–Vandermonde matrices are a generalization of Vandermonde matrices arising when considering for the space of the algebraic polynomials o...

Nonnegative matrix factorization (NMF) aims at factoring a data matrix into low-rank latent factor matrices with nonnegativity constraints on (one or both of) the factors. Specifically, given a data matrix X ∈ RM×N and a target rank R, NMF seeks a factorization model X ≈WH>, W ∈ RM×R, H ∈ RN×R, to ‘explain’ the data matrix X, where W ≥ 0 and/or H ≥ 0 and R ≤ min{M,N}. At first glance, NMF is no...