On the solution of highly structured nonlinear equations
The authors discuss a quasi-Newton method which is able to take advantage of the sparsity properties of the Jacobian, \(A\), of a large nonlinear equation system. Their method directly updates each of the sparse \(L\) and \(U\) factors of \(B\), a linearization of \(A\), in contrast to the method of \textit{J. E. Dennis jun.} and \textit{E. S. Marwil} [Math. Comput. 38, 459-474 (1982; Zbl 0482.65028)] in which one of the factors is updated in a quasi-Newton manner, the other being held constant in a modified Newton fashion. The numerical solution of two-point boundary value problems by multiple shooting leads to the need to solve a sparse linear system. Several methods of solution of these linear equations, including the two above, are compared over five problems, one of which is a second order partial differential equation. The results of these and more extensive tests demonstrate the superlinear convergence of the new method (which is proved in the second author's doctoral thesis) and show that it is robust.
scientific article; zbMATH DE number 89619
scientific article