Sensitivity analysis for the generalized Cholesky factorization

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Abstract

The explicit expressions of the normwise, mixed, and componentwise condition numbers and their upper bounds for the generalized Cholesky factorization are first obtained. Then, some improved rigorous perturbation bounds with normwise or componentwise perturbation in the given matrix are derived by bringing together the modified matrix-vector equation approach with the method of Lyapunov majorant function and the Banach fixed point theorem. Theoretical and experimental results show that these new bounds are always tighter than the corresponding ones in the literature.

Introduction

Let Rm×n be the set of m × n real matrices and Rrm×n be the subset of Rm×n consisting of matrices with rank r. Let Ir be the identity matrix of order r and denote the transpose of a matrix A by AT.

Consider the following symmetric quasi-definite matrix KR(m+n)×(m+n)K=[ABTBC],where ARmm×m is symmetric positive definite, BRnn×m, and CRn×n is symmetric positive semi-definite. The matrix K has the following factorization:K=LJm+nLT,whereL=[L110L21L22],Jm+n=[Im00In],with L11Rmm×m and L22Rnn×n being lower triangular, and L21Rnn×m. From (1.2), it is easy to verify thatA=L11L11T,B=L21L11T,C+L21L21T=L22L22T.The factorization (1.2) is called the generalized Cholesky factorization and L is referred to as the generalized Cholesky factor [1]. If the diagonal entries of the matrices L11 and L22 are positive, the factorization is unique.

Some scholars have discussed the algorithms, stability of algorithms, sensitivity analysis, and applications for the generalized Cholesky factorization (see, e.g., [1], [2], [3], [4], [5], [6], [7]). The first-order perturbation bounds can be found in [2], [4], [5], [6] and the rigorous perturbation bounds are given in [3], [4], [6], [7]. In this paper, we continue the research of sensitivity analysis for this factorization. We will mainly focus our attention on the condition numbers and new rigorous perturbation bounds. Specifically, we will present the explicit expressions of the normwise, mixed and componentwise condition numbers and their upper bounds in Section 3, and obtain the improved rigorous perturbation bounds in Section 4 by combining the matrix-vector equation approach, the method of Lyapunov majorant function (e.g., [8, Chapter 5]), and the Banach fixed point theorem (e.g., [8, Appendix D]). In addition, in Section 2, we introduce some notations and preliminaries and in Section 5, we give some numerical experiments.

Section snippets

Notations and preliminaries

Most of the notations and preliminaries can be found in [9]. We still present them here to make easier for readers.

In a given matrix A=(aij)Rm×n, its spectral norm and Frobenius norm are denoted by ‖A2 and ‖AF, respectively. For these two matrix norms, the following inequalities hold (see [10, pp. 80]),XYZ2X2Y2Z2,XYZFX2YFZ2,whenever the matrix product XYZ is well-defined.

For the given matrix A=[a1,a2,,an]=(aij)Rn×n, we denote the vector of the last i elements of aj by aj(i

Condition numbers

For the generalized Cholesky factorization (1.2), we first define the following mappingϕL:vec(K)lvec(L).In the following, we will present the Fréchet derivative of ϕL at vec(K), from which we can derive the condition numbers for generalized Cholesky factorization.

Theorem 3.1

Let the unique generalized Cholesky factorization of KR(m+n)×(m+n) be as in (1.2). Then the Fréchet derivative of ϕL at vec(K) is given byDϕL(vec(K))=Mlvec(Jm+nL)Mlow(L1L1).

Proof

Assume that the perturbed generalized Cholesky

Improved rigorous perturbation bounds

Using the definitions and explicit expressions of condition numbers, we can obtain the optimal first-order perturbation bounds, that is, these bounds are attainable. However, it is well known that the first-order perturbation bound may lead to erroneous conclusions because they neglect the high-order terms. So it is necessary to discuss the rigorous perturbation bound. Next we consider the rigorous perturbation bounds for the factor L when the original matrix has normwise or componentwise

Numerical examples

In this section, we will provide three numerical examples to illustrate the results obtained in Sections 3 and 4. All computations are carried out in MATLAB R2016a, with precision 2.22×1016.

Example 5.1

In this example, we compare the normwise, mixed and componentwise condition numbers and their upper bounds given in Theorem 3.2 and Corollary 3.3 respectively. To construct the test matrix K, we set Am=Pm+ImRm×m, where Pm=[pij] is the m × m Pascal matrix, that is, p1j=pi1=1, pij=pi(j1)+p(i1)j, B=[bij]=0.3

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The work is supported by the National Natural Science Foundation of China (Grant nos. 11671060, 11571062, and 11771062), the Basic and Advanced Research Project of CQC-STC (Grant no. cstc2015jcyjBX0007) and the Fundamental Research Funds for the Central Universities (Grant nos. 06112016CDJXZ238826).

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