Ela Bidiagonal Decompositions, Minors and Applications

Bidiagonal decompositions, minors and applications

More Accurate Bidiagonal Reduction for Computing the Singular Value Decomposition

Bidiagonal reduction is the preliminary stage for the fastest stable algorithms for computing the singular value decomposition. However, the best error bounds on bidiagonal reduction methods are of the form A + A = UBV T ; kAk 2 " M f(n)kAk 2 where B is bidiagonal, U and V are orthogonal, " M is machine precision, and f(n) is a modestly growing function of the dimensions of A. A Givens-based bi...

Bidiagonal Decompositions of Oscillating Systems of Vectors

We establish necessary and sufficient conditions, in the language of bidiagonal decompositions, for a matrix V to be an eigenvector matrix of a totally positive matrix. Namely, this is the case if and only if V and V −T are lowerly totally positive. These conditions translate into easy positivity requirements on the parameters in the bidiagonal decompositions of V and V −T . Using these decompo...

Ela Zero Minors of Totally Positive Matrices

In this paper the structure of the zero minors of totally positive matrices is studied and applications are presented.

Ela Accurate Computations with Totally Positive Bernstein–vandermonde Matrices

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...