Solving systems of nonlinear equations by means of an accelerated successive orthogonal projections method
The iterative method for calculating a zero point \(x^*\) of \(F:{\mathbb{R}}^ n\to {\mathbb{R}}^ n\) which is studied here consists of a block orthogonal projections method combined with an acceleration step. The first part involves a grouping of components of F into \(F=(F_ 1,...,F_ m)\) with \(F_ i:D\subset {\mathbb{R}}^ n\to R^{n_ i}\) such that \(n_ 1+...+n_ m=n\) and where it is assumed that the rows of the corresponding Jacobians \(J_ i(X^*)\) of \(F_ i\) at \(X^*\) are linearly independent. The block iterative part proceeds as follows. Set \(X_ 0=X^ k\) and for \(i=0,1,...,m-1\) set \[ U_{i+1}=- W_{i+1}[J_{i+1}(X_ i)J_{i+1}(X_ i)^ T]^{-1}F_{i+1}(x_ i) \] and \(x_{i+1}=x_ i+J_{i+1}(X_ i)^ TU_{i-1}\) where the \(W_ i's\) are relaxation parameters. Then \(X^{k+1}=X_ m\). The acceleration step proceeds by setting \(A(X^ k,X^{k+1})=X^ k+\lambda (X^{k+1}-X^ k)\) where \[ \lambda =-\sum^{m}_{i=1}<F_ i(x^ k),U_ i>/\| X^{k+1}-X^ k\|^ 2. \] A local convergence theorem is proved and some numerical examples are given.
scientific article; zbMATH DE number 3980381
scientific article