# Riemannian inexact Newton method for structured inverse eigenvalue and singular value problems

Chun Yueh Chiang, Matthew M. Lin, Xiao Qing Jin

Research output: Contribution to journalArticle

### Abstract

Inverse eigenvalue and singular value problems have been widely discussed for decades. The well-known result is the Weyl-Horn condition, which presents the relations between the eigenvalues and singular values of an arbitrary matrix. This result by Weyl-Horn then leads to an interesting inverse problem, i.e., how to construct a matrix with desired eigenvalues and singular values. In this work, we do that and more. We propose an eclectic mix of techniques from differential geometry and the inexact Newton method for solving inverse eigenvalue and singular value problems as well as additional desired characteristics such as nonnegative entries, prescribed diagonal entries, and even predetermined entries. We show theoretically that our method converges globally and quadratically, and we provide numerical examples to demonstrate the robustness and accuracy of our proposed method.

Original language English 675-694 20 BIT Numerical Mathematics 59 3 https://doi.org/10.1007/s10543-019-00754-7 Published - 2019 Sep 1

### Fingerprint

Inexact Newton Methods
Newton-Raphson method
Singular Values
Eigenvalue
Inverse problems
Geometry
Differential Geometry
Inverse Problem
Non-negative
Robustness
Converge
Numerical Examples
Arbitrary
Demonstrate

### All Science Journal Classification (ASJC) codes

• Software
• Computer Networks and Communications
• Computational Mathematics
• Applied Mathematics

### Cite this

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abstract = "Inverse eigenvalue and singular value problems have been widely discussed for decades. The well-known result is the Weyl-Horn condition, which presents the relations between the eigenvalues and singular values of an arbitrary matrix. This result by Weyl-Horn then leads to an interesting inverse problem, i.e., how to construct a matrix with desired eigenvalues and singular values. In this work, we do that and more. We propose an eclectic mix of techniques from differential geometry and the inexact Newton method for solving inverse eigenvalue and singular value problems as well as additional desired characteristics such as nonnegative entries, prescribed diagonal entries, and even predetermined entries. We show theoretically that our method converges globally and quadratically, and we provide numerical examples to demonstrate the robustness and accuracy of our proposed method.",
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Riemannian inexact Newton method for structured inverse eigenvalue and singular value problems. / Chiang, Chun Yueh; Lin, Matthew M.; Jin, Xiao Qing.

In: BIT Numerical Mathematics, Vol. 59, No. 3, 01.09.2019, p. 675-694.

Research output: Contribution to journalArticle

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AU - Chiang, Chun Yueh

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AU - Jin, Xiao Qing

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