An Ulm-like algorithm for generalized inverse eigenvalue problems

In this paper, we study the numerical solutions of the generalized inverse eigenvalue problem (for short, GIEP). Motivated by Ulm’s method for solving general nonlinear equations and the algorithm of Aishima (J. Comput. Appl. Math. 367 , 112485 2020 ) for the GIEP, we propose here an Ulm-like algori...

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Bibliographic Details
Published in:Numerical algorithms Vol. 98; no. 3; pp. 1611 - 1641
Main Authors: Luo, Yusong, Shen, Weiping
Format: Journal Article
Language:English
Published: New York Springer US 01.03.2025
Springer Nature B.V
Subjects:
Algebra
Algorithms
Approximation
Computer Science
Eigenvalues
Eigenvectors
Generalized inverse
Jacobi matrix method
Methods
Nonlinear equations
Numeric Computing
Numerical Analysis
Original Paper
Theory of Computation
Quadratic convergence
Newton’s method
65F15
Generalized inverse eigenvalue problem
65H17
Ulm’s method
15A18
15A29
ISSN:1017-1398, 1572-9265
Online Access:Get full text
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Summary:In this paper, we study the numerical solutions of the generalized inverse eigenvalue problem (for short, GIEP). Motivated by Ulm’s method for solving general nonlinear equations and the algorithm of Aishima (J. Comput. Appl. Math. 367 , 112485 2020 ) for the GIEP, we propose here an Ulm-like algorithm for the GIEP. Compared with other existing methods for the GIEP, the proposed algorithm avoids solving the (approximate) Jacobian equations and so it seems more stable. Assuming that the relative generalized Jacobian matrices at a solution are nonsingular, we prove the quadratic convergence property of the proposed algorithm. Incidentally, we extend the work of Luo et al. (J. Nonlinear Convex Anal. 24 , 2309–2328 2023 ) for the inverse eigenvalue problem (for short, IEP) to the GIEP. Some numerical examples are provided and comparisons with other algorithms are made.
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ISSN:1017-1398
1572-9265
DOI:10.1007/s11075-024-01845-5