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Methods for Solution of Nonlinear Operator Equations - English
URL: http://www.vsppub.com/books/mathe/bk-MetSolNonOpeEqu.html

At present, the theory of linear ill-posed problems is rather well developed. However, the theory of nonlinear ill-posed problems --- of great importance for numerical applications, especially for solving inverse (coefficient) problems of mathematical physics --- is less well developed. Although there are a large number of works devoted to nonlinear theory of ill-posed problems, there are few applications of this theory.

[ eng ]

9.1 Newton's Iteration Method for Solution of Nonlinear Equations - English
URL: http://il.water.usgs.gov/proj/feq/feqdoc/chap_9_2.html

-------------------------------------------------------------------------------- In general, even a single nonlinear equation cannot be solved without some numerical method to approximate the solution to the equation. The example of a single equation illustrates some of the problems that are considered in FEQ simulation. Thus, Newton's iteration method for solution of nonlinear equations is initially described and illustrated for the case of a single nonlinear equation. The discussion of Newton's method is then expanded to the simultaneous solution of many equations.

[ eng ]

Numerical Solution of Systems of Nonlinear Equations - English
URL: http://wwwinfo.cern.ch/asdoc/shortwrupsdir/c201/top.html

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Systems of Nonlinear Equations - English
URL: http://www-fp.mcs.anl.gov/otc/Guide/OptWeb/continuous/unconstrained/nonlineareq/

Systems of nonlinear equations arise as constraints in optimization problems, but also arise, for example, when differential and integral equations are discretized. In solving a system of nonlinear equations, we seek a vector such that f(x)=0 where x is an n-dimensional --- of n variables. Most algorithms in this section are closely related to algorithms for unconstrained optimization and nonlinear least squares.

[ eng ]

Control Strategies for the Iterative Solution of Nonlinear Equations in ODE Solvers - English
URL: http://epubs.siam.org/sam-bin/dbq/article/28710

In the numerical solution of ODEs by implicit time-stepping methods, a system of (nonlinear) equations has to be solved each step. It is common practice to use fixed-point iterations or, in the stiff case, some modified Newton iteration. The convergence rate of such methods depends on the stepsize. Similarly, a stepsize change may force a refactorization of the iteration matrix in the Newton solver. This paper develops new strategies for handling the iterative solution of nonlinear equations in ODE solvers.

[ eng ]