Structured condition numbers and statistical condition estimation for the LDU factorization
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Structured condition numbers and statistical condition estimation for the LDU factorization Mahvish Samar∗
Aamir Farooq
MU Chun-lai
Abstract. In this article, we consider the structured condition numbers for LDU, factorization by using the modified matrix-vector approach and the differential calculus, which can be represented by sets of parameters. By setting the specific norms and weight parameters, we present the expressions of the structured normwise, mixed, componentwise condition numbers and the corresponding results for unstructured ones. In addition, we investigate the statistical estimation of condition numbers of LDU factorization using the probabilistic spectral norm estimator and the small-sample statistical condition estimation method, and devise three algorithms. Finally, we compare the structured condition numbers with the corresponding unstructured ones in numerical experiments.
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Introduction
As a real n × n matrix A whose first n − 1 leading principal submarines are all nonsingular there exists a unique unit lower triangular matrix L, unit upper triangular matrix U and diagonal matrix D such that A ∈ Rn×n have the following unique LDU factorization A = LDU,
(1.1)
Since L, D and U in (1.1) are uniquely determined by A. where L is a unit lower triangular and U is a unit upper triangular and D is diagonal matrix. The difference between LDU and LU factorizations in upper triangular matrix U , i.e. U is unit upper triangular matrix in LDU factorization. The LDU factorization is one of the most important matrix factorizations and has many applications, such as solving systems of linear equations, inverting matrices, and computing determinants [1,2]. The componentwise perturbation bounds were first discussed by Gal´antai [3]. Later, the acquired first-order bounds for LDU factorization were enhanced by Wenjun Received: 2018-11-10. Revised: 2019-10-03. MR Subject Classification: 65F35, 15A23, 15A57. Keywords: LDU factorization, Structured condition number, Normwise condition number, Mixed condition number, Componentwise condition number. Digital Object Identifier(DOI): https://doi.org/10.1007/s11766-020-3659-4. Supported by the National Natural Science Foundation of China (11671060). ∗ Corresponding author.
Mahvish Samar, et al.
Structured condition numbers and statistical condition...
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[4]. The obtained bounds [4] are optimal, which leads to the normwise condition numbers for LDU factorization. The structured perturbation theory for the LDU factorization of diagonally dominant matrices was presented by Dopico and Koev [5] and later it was extended by Dailey et. al [6]. It is necessary to mention that the systematic theory for normwise condition number was first given by Rice [7] and the terminologies of mixed and componentwise condition numbers were first introduced by Gohberg and Koltracht [8]. The normwise condition numbers for LU , Cholesky, and QR factorizations can be found in [9-11]. As we know that the normwise condition numbers may overestimate the illness of problem because they ignore th
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