Splitting iteration method for simple singular points and simple bifurcation points
type: Item
instance of: scholarly article
title: Splitting iteration method for simple singular points and simple bifurcation points
zbMATH Open document ID: 0689.65037
DOI: 10.1007/BF02238731
author: Mei Zhen
published in: Computing
publication date: 1989-01-01T00:00:00Z
review text: To avoid the large computational cost or loss of information caused by solving the linearized equation required in Newton's iteration, different independent blocks are introduced for singular points and the null vector of the derivative. Let E be a Hilbert space and G: \(E\to E\) be a \(C^ 3\)-mapping. Assme that the equation \(G(u)=0(*)\) has a solution \(u_ 0\in E\) satisfying hypotheses H1 and H2. H1) \(DG_ 0:=DG(u_ 0)\) is a Fredholm operator with index 0, and zero is a simple eigenvalue of it with \(\dim (N(DG_ 0))=1\). Then there exist \(\phi,\phi^*\in E\) such that \(N(DG_ 0)=Span[\phi]\), \(N(DG_ 0^*)=Span[\phi^*]\), \(<\phi,\phi^*>=<\phi^*,\phi^*>=1.\) H2) \(<\phi^*,D^ 2G_ 0\phi \phi >\neq 0\). Then \(u_ 0\) is a simple singular point of (*), which is to be found. Define \(E_ i:=(E\times {\mathbb{R}})^ 3\), \(x\in_ 1\), \(x:=(u,c,u_ 1,c_ 1,u_ 2,c_ 2)\), and \(C^ 2\)-continuous mappings \(H_ i: E_ 1\to E\times {\mathbb{R}}\) by \[ H_ 1(x):=\left( \begin{matrix} G(u)+cu_ 2/m\\ <u_ 2,DG(u)u_ 1>/m^ 3\end{matrix} \right),\quad H_ 2(x):=\left( \begin{matrix} DG(u)u_ 1+c_ 1u_ 2/m\\ [<u_ 2,u_ 1>-m^ 3]/m\end{matrix} \right), \] \[ H_ 3(x):=\left( \begin{matrix} DG(u)^*u_ 2+c_ 2u_ 2/m\\ [<u_ 2,u_ 2>- m^ 2]/2m\end{matrix} \right), \] where \(m>0\) is a normalization parameter. Let \(x^*:=(u_ 0,0,m^ 2\phi,0,m\phi^*,0)\in E_ 1\). Then \(x^*\) may be approximated by the followingsplitting iteration: Algorithm: For \(x^ 0\) in the neighbourhood of \(x^*\), \(k=1,2,...\), do \(x^{k+1}=F(x^ k):=(f_ 1(x^ k),f_ 2(x^ k),f_ 3(x^ k))\) where \(f_ i(x):=x_ i[D_{xi}H_ i(\tilde x)]^{-1}H_ i(x)\), \(x\in E_ 1\), and \(\tilde x:=(u,0,u_ 1,0,u_ 2,0)\in E_ 1\) is a restriction of x on \(E_ 1.\) Convergence with error proportional to \((1/2)^ k\) after k iterations is proved, and the precise rate depends on m. Faster convergence is obtained by using a Gauss-Seidel form of iteration in which the 3 components of \(x\in E_ 1\equiv (E\times {\mathbb{R}})^ 3\) are updated sequentially rather…
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 47J05
zbMATH DE Number: 4129899
zbMATH Keywords: splitting iteration method
zbMATH Keywords: singular points
zbMATH Keywords: null vector
zbMATH Keywords: Hilbert space
zbMATH Keywords: Fredholm operator
zbMATH Keywords: Algorithm
zbMATH Keywords: Convergence
zbMATH Keywords: bifurcation point
zbMATH Keywords: numerical examples
reviewed by: Francis Wright
describes a project that uses: ALCON
MaRDI profile type: Publication
cites work: Expanded Convergence Domains for Newton’s Method at Nearly Singular Roots
cites work: Efficient Numerical Pathfollowing Beyond Critical Points
cites work: On Solving Nonlinear Equations with Simple Singularities or Nearly Singular Solutions
cites work: Characterization and Computation of Generalized Turning Points
cites work: Convergence Cones Near Bifurcation
cites work: Q3703206
cites work: The numerical treatment of non-trivial bifurcation points
cites work: Computing simple bifurcation points using a minimally extended system of nonlinear equations
cites work: Newton's method and high order singularities
cites work: Tensor Methods for Nonlinear Equations
cites work: On the accurate determination of nonisolated solutions of nonlinear equations
description: scientific article; zbMATH DE number 4129899