Large-scale Systems Of Nonlinear Equations Research Articles

SOR-Secant Methods

A family of SOR-secant methods for solving large-scale nonlinear systems of equations is introduced. The components and the variables of the system are divided into m blocks. At each cycle of the method, the groups of components are changed one at a time using a quasi-Newton (least-change secant) step. Proofs of local convergence at an ideal rate are given, which use the theory of fixed-point quasi-Newton methods [J.M. Martínez, SIAM J. Numer. Anal., 29 (1992), pp. 1413–1434]. Numerical experiments are presented.

Dynamic and static load balancing for solving block bordered circuit equations on multiprocessors

A special class of large-scale nonlinear systems of equations of block bordered structure is generated in VLSI circuit simulation from the sparse and irregular nature of the circuit matrices. Several Newton-like parallel methods for solving the block bordered equations are presented. Static and dynamic load balancing problems that arise in solving the circuit equations on multiprocessors are discussed. Computational results obtained using the BBN GP1000, a shared memory multiprocessor through a distributed architecture, are presented to show the effectiveness of dynamic and static load balancing. These experiments include a simulation of the op-amp 741 amplifier and a large analog filter simulation that leads to a multilevel block bordered system of nonlinear equations.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

An inverse column-updating method for solving large–scale nonlinear systems of equations

In this paper we introduce a new quasi-Newton method for solving large-scale nonlinear systems of equations. The new method proceeds, at each iteration, changing one column of the inverse Jacobian approximation in an appropriate way. We prove a local convergence result and we present some encouraging numerical experiments.

Automatic computation of partial derivatives and rounding error estimates with applications to large-scale systems of nonlinear equations

Recently, a new approach has been proposed to efficiently compute the accurate values of partial derivatives of a function or functions, and simultaneously to estimate the rounding errors in the computed function values. In this paper the use of the method in the solution of nonlinear equations is investigated. The method makes use of the computational graph and, when applied to the evaluation of a function, traverses it from the top ( = the function node) down to the bottom ( = the input variable nodes). A remarkable analogy is observed between the partial derivatives and the shortest paths on the computational graph. The top-down traversing on the computational graph has the following advantages over the existing algorithms using the bottom-up traversing: (1) The gradient of a function can be computed within the same complexity as that of the evaluation of the function alone (the complexity being independent of the number of input variables); (2) A fairly sharp estimate of the rounding error in the function evaluation is obtained, on the basis of which a computationally meaningful norm may be introduced in the space of residuals to afford a convergence criterion for an iterative method of solving the system of nonlinear equations. As an example, a system of nonlinear equations with 108 variables for a distillation tower of a chemical plant is numerically analyzed in detail. It is shown that by the use of the proposed method we could satisfactorily resolve two main problems encountered in computing a numerical solution of the system of nonlinear equations, i.e., how to compute the accurate Jacobian matrix and when we should stop the iteration.

Menger-decomposition of a graph and its application to the structural analysis of a large-scale system of equations

Graph representation of a large-scale system of non-linear equations provides an efficient way of testing the structural solvability, detecting the inconsistencies in modelling and decomposing the whole system into partially ordered subsystems. In this paper, the M-decomposition is defined for a graph with specified ‘entrance’ and ‘exit’ vertices, in terms of the Menger-type linkings from the entrance to the exit. Some properties of the M-decomposition are shown; specifically it is noted that the M-decomposition agrees with the Dulmage-Mendelsohn decomposition of the associated bipartite graph. The M-decomposition is useful for the structural analysis of a large-scale system of equations; the M-decomposition leads to the finest block-triangularization and the resulting subproblems are structurally solvable. Also pointed out is the fact that among the cycles on the representation graph, only those which are contained in an M-irreducible component correspond to essential equations.