Sparse Systems Of Nonlinear Equations Research Articles

Row update ABS methods for solving sparse nonlinear systems of equations

In [7], we proposed row update ABS methods for solving systems of nonlinear equations. A numerical comparison showed that the new methods are competitive with other methods. In this paper, we extend the row update ABS methods to sparse nonlinear systems of equations. We prove the local superlinear convergence of the proposed method.

Comparing Algorithms for Solving Sparse Nonlinear Systems of Equations

This paper describes implementations of eight algorithms of Newton and quasi-Newton type for solving large sparse systems of nonlinear equations. For linear algebra calculations, a symbolic manipulation is used, as well as a static data structure introduced recently by George and Ng, which allows a partial pivoting strategy for solving linear systems. A numerical comparison of the implemented methods is presented.

Successive column correction algorithms for solving sparse nonlinear systems of equations

This paper presents two algorithms for solving sparse nonlinear systems of equations: the CM-successive column correction algorithm and a modified CM-successive column correction algorithm. Aq-superlinear convergence theorem and anr-convergence order estimate are given for both algorithms. Some numerical results and the detailed comparisons with some previously established algorithms show that the new algorithms have some promise of being very effective in practice.

The Secant/Finite Difference Algorithm for Solving Sparse Nonlinear Systems of Equations

This paper presents an algorithm, the secant/finite difference algorithm, for solving sparse nonlinear systems of equations. This algorithm is a combination of a finite difference method and a secant method. A q-superlinear convergence result and an r-convergence rate estimate show that this algorithm has good local convergence properties. The numerical results indicate that this algorithm is competitive with some currently used algorithms.

A hybrid algorithm for solving sparse nonlinear systems of equations

This paper presents a hybrid algorithm for solving sparse nonlinear systems of equations. The algorithm is based on dividing the columns of the Jacobian into two parts and using different algorithms on each part. The hybrid algorithm incorporates advantages of both component algorithms by exploiting the special structure of the Jacobian to obtain a good approximation to the Jacobian, using as little effort as possible. A Kantorovich-type analysis and a locally q-superlinear convergence result for this algorithm are given.

A Hybrid Algorithm for Solving Sparse Nonlinear Systems of Equations

A Hybrid Algorithm for Solving Sparse Nonlinear Systems of Equations

Quasi-Newton methods with factorization scaling for solving sparse nonlinear systems of equations

In this paper we present a Quasi-Newton type method, which applies to large and sparse nonlinear systems of equations, and uses the Q-R factorization of the approximate Jacobians. This method belongs to a more general class of algorithms for which we prove a local convergence theorem. Some numerical experiments seem to confirm that the new algorithm is reliable.

CONKUB: A conversational path-follower for systems of nonlinear equations

Abstract We describe an interactive computer program to trace solutions of systems of nonlinear algebraic equations and illustrate its application to solve several difficult problems. Turning points and bifurcations are located and solution branches are identified and traced interactively. Of special interest is its application to study solutions of large, sparse systems of non-linear equations that result from the discretization of boundary value problems. Such systems arise in the description of physical, biological, and chemical phenomena. As an example, we show a model of urine formation in the mammalian kidney [13], where path-following in a subspace makes tracing the solution surface possible.

Secant methods for sparse systems of nonlinear equations with a special structure

Secant methods for sparse systems of nonlinear equations with a special structure are given, as they arise, e. g., in the solution of boundary value problems for systems of ordinary differential equations by multiple shooting. The presented methods are compared with an adapted Broyden method.

Piecewise-Linear Homotopy Algorithms for Sparse Systems of Nonlinear Equations

When piecewise-linear homotopy algorithms are applied to the problem of approximating a zero of a sparse function $f:R^n \to R^n $, a large piece of linearity can be traversed in one step by using a suitable linear system. The linear system has n rows and $n + 1$ columns, but is subject to a number of inequalities depending on the sparsity pattern of f. We show how an algorithm can be implemented using these large pieces; in particular, we demonstrate how to update the linear system corresponding to one large piece to obtain the appropriate system for an adjacent large piece. One measure of the complexity of such an implementation is the number of inequalities that may be required for any one piece. We prove that there can be no more than $O(n^{{3 / 2}} )$ such inequalities, and that this bound is essentially tight; the argument is purely combinatorial. Finally, we provide guidelines on when such a “large-piece implementation” should be used instead of much simpler “small-piece implementations” for piecewise-linear homotopy algorithms.

A comprehensive approach to chemical engineering computational problems

A general programme for the solution of large sparse systems of non-linear equations is presented that allows to afford in a general way most computational problems in chemical engineering. Equipments and plants can, in fact, be represented as networks of “unit cells” to which material and energy balance equations are applied in finite form. The lay-out of the network and the assessment of the equations is a matter of the engineer's judgment and depends upon the aims of the computation and the complexity and importance of the phenomena involved. The solution of the equations then becomes a straightforward procedure that may be committed to the computer. Examples of application to a variety of chemical equipments and to process computation are given.

Monoton einschließend konvergente iterationsprozesse vom gauß‐seidel‐typ zur lösung nichtlinearer gleichungssysteme im RN und anwendungen

AbstractIn this paper we describe a class of Gauß‐Seidel iterations for the solution of large sparse nonlinear systems of equations. For suitable starting vectors every method of this class is monotonically convergent with respect to both sides. The result will be applied to the numerical solution of nonlinear boundary‐value problems and boundary layer problems.

A note on solution of large sparse systems of nonlinear equations

A note on solution of large sparse systems of nonlinear equations

A note on the decomposition of systems of sparse non-linear equations

In this paper we point out some advantages of decomposing a system of non-linear equations into block-triangular form directly instead of decomposing the linearized systems.