A note on the Levenberg-Marquardt parameter

Jinyan Fan; Jianyu Pan

doi:10.1016/j.amc.2008.10.056

A note on the Levenberg-Marquardt parameter

Jinyan Fan^*, Jianyu Pan

^*Corresponding author for this work

School of Mathematical Sciences

Shanghai Jiao Tong University

Research output: Contribution to journal › Article › peer-review

79 Scopus citations

Abstract

In [C. Ma, L. Jiang, Some research on Levenberg-Marquardt method for the nonlinear equations, Appl. Math. Comput. 184 (2007) 1032-1040], the LM parameter at the kth iteration is chosen as λ_k = θ {norm of matrix} F_k {norm of matrix}₂ + (1 - θ) {norm of matrix} J_k^T F_k {norm of matrix}₂ where F is the residual function, J is the Jacobi of F, and θ ∈ [0, 1] is a constant. In this note, we point out that the LM parameter can be any combination of {norm of matrix} F_k {norm of matrix}₂ and {norm of matrix} J_k^T F_k {norm of matrix}₂ provided it is positive. Furthermore, we give a more general choice of the LM parameter, and show that the LM method still preserves the quadratic convergence under the local error condition which is weaker than nonsingularity. Crown

Original language	English
Pages (from-to)	351-359
Number of pages	9
Journal	Applied Mathematics and Computation
Volume	207
Issue number	2
DOIs	https://doi.org/10.1016/j.amc.2008.10.056
State	Published - 15 Jan 2009

Keywords

Levenberg-Marquardt method
Nonlinear equations
Quadratic convergence

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10.1016/j.amc.2008.10.056

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title = "A note on the Levenberg-Marquardt parameter",

abstract = "In [C. Ma, L. Jiang, Some research on Levenberg-Marquardt method for the nonlinear equations, Appl. Math. Comput. 184 (2007) 1032-1040], the LM parameter at the kth iteration is chosen as λk = θ \{norm of matrix\} Fk \{norm of matrix\}2 + (1 - θ) \{norm of matrix\} JkT Fk \{norm of matrix\}2 where F is the residual function, J is the Jacobi of F, and θ ∈ [0, 1] is a constant. In this note, we point out that the LM parameter can be any combination of \{norm of matrix\} Fk \{norm of matrix\}2 and \{norm of matrix\} JkT Fk \{norm of matrix\}2 provided it is positive. Furthermore, we give a more general choice of the LM parameter, and show that the LM method still preserves the quadratic convergence under the local error condition which is weaker than nonsingularity. Crown",

keywords = "Levenberg-Marquardt method, Nonlinear equations, Quadratic convergence",

author = "Jinyan Fan and Jianyu Pan",

year = "2009",

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doi = "10.1016/j.amc.2008.10.056",

language = "英语",

volume = "207",

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journal = "Applied Mathematics and Computation",

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T1 - A note on the Levenberg-Marquardt parameter

AU - Fan, Jinyan

AU - Pan, Jianyu

PY - 2009/1/15

Y1 - 2009/1/15

N2 - In [C. Ma, L. Jiang, Some research on Levenberg-Marquardt method for the nonlinear equations, Appl. Math. Comput. 184 (2007) 1032-1040], the LM parameter at the kth iteration is chosen as λk = θ {norm of matrix} Fk {norm of matrix}2 + (1 - θ) {norm of matrix} JkT Fk {norm of matrix}2 where F is the residual function, J is the Jacobi of F, and θ ∈ [0, 1] is a constant. In this note, we point out that the LM parameter can be any combination of {norm of matrix} Fk {norm of matrix}2 and {norm of matrix} JkT Fk {norm of matrix}2 provided it is positive. Furthermore, we give a more general choice of the LM parameter, and show that the LM method still preserves the quadratic convergence under the local error condition which is weaker than nonsingularity. Crown

AB - In [C. Ma, L. Jiang, Some research on Levenberg-Marquardt method for the nonlinear equations, Appl. Math. Comput. 184 (2007) 1032-1040], the LM parameter at the kth iteration is chosen as λk = θ {norm of matrix} Fk {norm of matrix}2 + (1 - θ) {norm of matrix} JkT Fk {norm of matrix}2 where F is the residual function, J is the Jacobi of F, and θ ∈ [0, 1] is a constant. In this note, we point out that the LM parameter can be any combination of {norm of matrix} Fk {norm of matrix}2 and {norm of matrix} JkT Fk {norm of matrix}2 provided it is positive. Furthermore, we give a more general choice of the LM parameter, and show that the LM method still preserves the quadratic convergence under the local error condition which is weaker than nonsingularity. Crown

KW - Levenberg-Marquardt method

KW - Nonlinear equations

KW - Quadratic convergence

UR - https://www.scopus.com/pages/publications/58349084113

U2 - 10.1016/j.amc.2008.10.056

DO - 10.1016/j.amc.2008.10.056

M3 - 文章

AN - SCOPUS:58349084113

SN - 0096-3003

VL - 207

SP - 351

EP - 359

JO - Applied Mathematics and Computation

JF - Applied Mathematics and Computation

IS - 2

ER -

A note on the Levenberg-Marquardt parameter

Abstract

Keywords

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