Hanyu Li

Weighted UDV∗-decomposition and weighted spectral decomposition for rectangular matrices and their applications

In this paper, the weighted UDV∗-decomposition theorem and the weighted spectral decomposition theorem for rectangular matrices are proved. Their applications are discussed, in which the matrix functions, Cauchy’s formula for matrix functions, and the weighted resolvent equation of rectangular matrices are studied. In addition, we also present a solution of the matrix equation AXB = C.

Perturbation analysis for the hyperbolic QR factorization

The hyperbolic QR factorization is a generalization of the classical QR factorization and can be regarded as the triangular case of the indefinite QR factorization proposed by Sanja Singer and Sasa Singer. In this paper, the perturbation analysis for this factorization is considered using the classical matrix equation approach, the refined matrix equation approach, and the matrix-vector equation approach. The first order and rigorous normwise perturbation bounds with normwise or componentwise perturbations in the given matrix are derived. The obtained first order bounds can be much tighter than the corresponding existing ones. Each of the obtained rigorous bounds is composed of a small constant multiple of the corresponding first order bound and an additional term with simple form. In particular, for square matrix, the rigorous bounds for the factor R are just the 6+3 multiple of the corresponding first order bounds. These rigorous bounds can be used safely for all cases in comparison to the first order bounds.

A note on the perturbation analysis for the generalized Cholesky factorization

In this note, we consider the perturbation analysis for the generalized Cholesky factorization further. The conditions for the main theorems of the paper [W.-G. Wang, J.-X. Zhao, Perturbation analysis for the generalized Cholesky factorization, Appl. Math. Comput. 147 (2004) 601-606] are weakened by using an alternative method. Moreover, some new perturbation bounds are also derived.

New perturbation bounds and condition numbers for the hyperbolic QR factorization

Abstract Using the modified matrix-vector equation approach, the technique of Lyapunov majorant function and the Banach fixed point theorem, we obtain some new rigorous perturbation bounds for R factor of the hyperbolic QR factorization under normwise perturbation. These bounds are always tighter than the one given in the literature. Moreover, the optimal first-order perturbation bounds and the normwise condition numbers for the hyperbolic QR factorization are also presented.

Perturbation analysis for the symplectic QR factorization

In this paper, the perturbation analysis for the symplectic QR factorization is considered. Some first-order and rigorous normwise perturbation bounds with normwise or componentwise perturbations in the given matrix are presented.

New rigorous perturbation bounds for the generalized Cholesky factorization

Some new rigorous perturbation bounds for the generalized Cholesky factorization with normwise or componentwise perturbations in the given matrix are obtained, where the componentwise perturbation has the form of backward rounding error for the generalized Cholesky factorization algorithm. These bounds can be much tighter than some existing ones while the conditions for them to hold are simple and moderate.

A note on the condition number of the scaled total least squares problem

In this paper, we show that the normwise condition number of the scaled total least squares problem can be transformed into a new and compact form. Considering the relationship between the scaled total least squares problem and the total least squares problem, we obtain something new on the normwise condition number of the total least squares problem. The new forms of the normwise condition number are of particular interest in the following two aspects. Firstly, it is easy to use for the practitioners from applied disciplines. Secondly, the new forms enjoy great computational efficiency and require very little storage space compared with its original forms. Numerical examples are given to illustrate the results.

New perturbation bounds of partitioned generalized Hermitian eigenvalue problem

In this paper, some new absolute perturbation bounds of partitioned generalized Hermitian positive definite eigenvalue problem are established by two different ways. Numerical results show that our bounds are sharper than the ones in the literature.

New perturbation bounds for the subunitary polar factors

In this article, we present some new perturbation bounds for the (subunitary) unitary polar factors of the (generalized) polar decompositions. Two numerical examples are given to show the rationality and superiority of our results, respectively. In terms of the one-to-one correspondence between the weighted case and the non-weighted case, all these bounds can be applied to the weighted polar decomposition.

Relative and Absolute Perturbation Bounds for Weighted Polar Decomposition

Some new perturbation bounds for both weighted unitary polar factors and generalized nonnegative polar factors of the weighted polar decompositions are presented without the restriction that A and its perturbed matrix A˜ have the same rank. These bounds improve the corresponding recent results.