Solution of underdetermined nonlinear equations by stationary iteration methods

Abstract

Summary. Nonlinear stationary fixed point iterations in R" are considered. The Perron-Ostrowski theorem [23] guarantees convergence if the iteration function G possesses an isolated fixed point u. In this paper a sufficient condition for convergence is given if G possesses a manifold of fixed points.As an application, convergence of a nonlinear extension of the method of Kaczmarz is proved. This method is applicable to underdetermined equations; it is appropriate for the numerical treatment of large and possibly ill-conditioned problems with a sparse, nonsquare Jacobian matrix. A practical example of this type (nonlinear image reconstruction in ultrasound tomography) is included.

Presentation of a highly tuned multithreaded interval solver for underdetermined and well-determined nonlinear systems

“…x = x (1) 29: push (L, x (2) ) 30: end if 31: end loop Operations "push" and "pop" in the algorithm, mean inserting and removing elements to/from the set (the names will be used independently on how the set is represented -as a stack, queue or a more sophisticated data structure).…”

“…Many of them are not well-determined, but underdetermined, i.e., having fewer equations than unknowns (m < n), which means they have uncountably many solutions and their solution sets do not consist of isolated points, but are manifolds. In particular, we encounter such systems in robotics [19], stability theory of dynamical systems [35], differential equations solving [31] and multicriteria analysis [30].…”

Convergence Rates Results for Iterative Methods for Solving Nonlinear Ill-Posed Problems

“…by McCormick [86,87] and Meyn [88]. Natterer [94] has studied this method for the solution of (ill-posed) bilinear problems.…”

Solution of nonlinear systems of equations by an optimal projection method

“…In fact, only in 1983, Meyn [11] gave a sufficient condition for the convergence of nonlinear stationary processes of the type…”

“…Tompkins [16], McCormick [10], Meyn [11] and Martfnez [8,9] proposed generalizations of Kaczmarz's method [2] for nonlinear systems of equations. Kaczmarz's method and its generalizations (see [2] and references therein) make no changes in the original system, perform no operation on the system as a whole, and require access to only one component, or or a small group of components at a time.…”