Sensitivity analysis for the generalized Cholesky factorization☆
Introduction
Let be the set of m × n real matrices and be the subset of 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 where is symmetric positive definite, and is symmetric positive semi-definite. The matrix K has the following factorization:wherewith and being lower triangular, and . From (1.2), it is easy to verify thatThe 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 its spectral norm and Frobenius norm are denoted by ‖A‖2 and ‖A‖F, respectively. For these two matrix norms, the following inequalities hold (see [10, pp. 80]),whenever the matrix product XYZ is well-defined.
For the given matrix we denote the vector of the last i elements of aj by
Condition numbers
For the generalized Cholesky factorization (1.2), we first define the following mappingIn 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 be as in (1.2). Then the Fréchet derivative of ϕL at vec(K) is given by 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 . 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 where is the m × m Pascal matrix, that is,
References (14)
The generalized Cholesky factorization method for saddle point problems
Appl. Math. Comput.
(1998)- et al.
Perturbation analysis for the generalized Cholesky factorization
Appl. Math. Comput.
(2004) - et al.
A note on the perturbation analysis for the generalized Cholesky factorization
Appl. Math. Comput.
(2010) - et al.
New rigorous perturbation bounds for the generalized Cholesky factorization
Appl. Math. Comput.
(2015) Perturbation bounds for the generalized Cholesky factorization
Nanjing Univ. J. Math Biquarterly
(2003)The generalized Cholesky factorization and perturbation of the real symmetric matrices
J. Zhaoqing Univ.
(2008)New perturbation analysis for the generalized Cholesky factorization
J. Zhanjiang Norm. Coll.
(2009)
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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).