Two-step inexact Newton-type method for inverse singular value problems
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Two-step inexact Newton-type method for inverse singular value problems Wei Ma1 · Xiao-shan Chen2 Received: 24 January 2018 / Accepted: 22 July 2019 / © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract In this paper, based on the two-step Newton iterative procedure, we propose a twostep inexact Newton-type method for solving inverse singular value problems. Under some mild assumptions, our results show that the two-step inexact Newton-type method is super quadratically convergent. Numerical implementations demonstrate the effectiveness of the new method. Keywords Inverse singular value problems · Two-step inexact Newton-type method · Super quadratically convergent Mathematics Subject Classification (2010) 65F18 · 65F15 · 15A18
1 Introduction The inverse singular value problem has attracted increasing interest in the past decade which is defined as follows. Let {Ai }ni=0 be n + 1 real m × n matrices with m ≥ n and c = (c1 , c2 , . . . , cn )T ∈ Rn . We define A(c) := A0 + c1 A1 + c2 A2 + · · · + cn An The research of Wei Ma was partially supported by the Special Project Grant of Nanyang Normal University (ZX2014078). The research of Xiao-shan Chen was partially supported by National Natural Science Foundations of China (11771159, U1811464). Xiao-shan Chen
[email protected] Wei Ma [email protected] 1
School of Mathematics and Statistics, Nanyang Normal University, Nanyang, 473061, Henan, People’s Republic of China
2
School of Mathematical Sciences, South China Normal University, Guangzhou, 530631, People’s Republic of China
(1)
Numerical Algorithms
and denote its singular values by {σi (c)}ni=1 with the ordering σ1 (c) ≥ σ2 (c) ≥ · · · ≥ σn (c) ≥ 0. With given real numbers {σi∗ }ni=1 such that σ1∗ ≥ σ2∗ ≥ · · · ≥ σn∗ ≥ 0, an inverse singular value problem (ISVP) is to find c∗ ∈ Rn such that σi (c∗ ) = σi∗ ,
i = 1, . . . , n.
This is a special kind of inverse singular value problems, which was originally proposed by Chu [8], and the ISVP arises in a variety of applications such as the optimal sequence design for direct-spread code division multiple access [23], the passivity enforcement in nonlinear circuit simulation [21], the constructions of Toeplitz-related matrices from prescribed singular values [1, 3, 11], the inverse problem in some quadratic groups [18], and others [2, 9, 10, 15, 16, 19, 26]. In [6], a sufficient condition is given such that the inverse singular value problems are unsolvable almost everywhere. To our knowledge, there is no literature on the existence and uniqueness questions for the ISVP. The ISVP is equivalent to finding a solution c∗ ∈ Rn of the following nonlinear equation: f(c) := (σ1 (c) − σ1∗ , σ2 (c) − σ2∗ , . . . , σn (c) − σn∗ )T = 0.
(2)
It is natural that Newton’s method can be applied to solve the nonlinear equation (2). The advantage of Newton’s method lies in its quadratic convergence [12, 14, 25]. To relax the implementation cost of Newton’s method, different Newton-type methods have also been proposed [4, 5, 8, 12, 24, 25], which inv
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