Geodesic
Inexact Newton-type method for solving large-scale absolute value equation $Ax-|x| = b$
Applications of Mathematics, Tome 69 (2024) no. 1, pp. 49-66

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Newton-type methods have been successfully applied to solve the absolute value equation $Ax-|x| = b$ (denoted by AVE). This class of methods usually solves a system of linear equations exactly in each iteration. However, for large-scale AVEs, solving the corresponding system exactly may be expensive. In this paper, we propose an inexact Newton-type method for solving the AVE. In each iteration, the proposed method solves the corresponding system only approximately. Moreover, it adopts a new line search technique, which is well-defined and easy to implement. We prove that the proposed method has global and local superlinear convergence under the condition that the interval matrix $[A - I,A + I]$ is regular. This condition is much weaker than those used in some Newton-type methods. Numerical results show that our method has fairly good practical efficiency for solving large-scale AVEs.
DOI : 10.21136/AM.2023.0171-22
Classification : 90C05, 90C33
Keywords: absolute value equation; inexact Newton method; regularity of interval matrices; superlinear convergence 
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     title = {Inexact {Newton-type} method for solving large-scale absolute value equation $Ax-|x| = b$},
     journal = {Applications of Mathematics},
     pages = {49--66},
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Tang, Jingyong. Inexact Newton-type method for solving large-scale absolute value equation $Ax-|x| = b$. Applications of Mathematics, Tome 69 (2024) no. 1, pp. 49-66. doi: 10.21136/AM.2023.0171-22

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