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Volume 19, Issue 6
Picard Iteration for Nonsmooth Equations

Song-Bai Sheng & Hui-Fu Xu

J. Comp. Math., 19 (2001), pp. 583-590.

Published online: 2001-12

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  • Abstract

This paper presents an analysis of the generalized Newton method, approximate Newton methods, and splitting methods for solving nonsmooth equations from Picard iteration viewpoint. It is proved that the radius of the weak Jacobian (RGJ) of Picard iteration function is equal to its least Lipschitz constant. Linear convergence or superlinear convergence results can be obtained provided that RGJ of the Picard iteration function at a solution point is less than one or equal to zero. As for applications, it is pointed out that the approximate Newton methods, the generalized Newton method for piecewise $C^1$ problems and splitting methods can be explained uniformly with the same viewpoint.  

  • Keywords

Nonsmooth equations, Picard iteration, Weak Jacobian, Convergence.

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COPYRIGHT: © Global Science Press

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@Article{JCM-19-583, author = {Sheng , Song-Bai and Xu , Hui-Fu}, title = {Picard Iteration for Nonsmooth Equations}, journal = {Journal of Computational Mathematics}, year = {2001}, volume = {19}, number = {6}, pages = {583--590}, abstract = {

This paper presents an analysis of the generalized Newton method, approximate Newton methods, and splitting methods for solving nonsmooth equations from Picard iteration viewpoint. It is proved that the radius of the weak Jacobian (RGJ) of Picard iteration function is equal to its least Lipschitz constant. Linear convergence or superlinear convergence results can be obtained provided that RGJ of the Picard iteration function at a solution point is less than one or equal to zero. As for applications, it is pointed out that the approximate Newton methods, the generalized Newton method for piecewise $C^1$ problems and splitting methods can be explained uniformly with the same viewpoint.  

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/9010.html} }
TY - JOUR T1 - Picard Iteration for Nonsmooth Equations AU - Sheng , Song-Bai AU - Xu , Hui-Fu JO - Journal of Computational Mathematics VL - 6 SP - 583 EP - 590 PY - 2001 DA - 2001/12 SN - 19 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/9010.html KW - Nonsmooth equations, Picard iteration, Weak Jacobian, Convergence. AB -

This paper presents an analysis of the generalized Newton method, approximate Newton methods, and splitting methods for solving nonsmooth equations from Picard iteration viewpoint. It is proved that the radius of the weak Jacobian (RGJ) of Picard iteration function is equal to its least Lipschitz constant. Linear convergence or superlinear convergence results can be obtained provided that RGJ of the Picard iteration function at a solution point is less than one or equal to zero. As for applications, it is pointed out that the approximate Newton methods, the generalized Newton method for piecewise $C^1$ problems and splitting methods can be explained uniformly with the same viewpoint.  

Sheng , Song-Bai and Xu , Hui-Fu. (2001). Picard Iteration for Nonsmooth Equations. Journal of Computational Mathematics. 19 (6). 583-590. doi:
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