Direct secant updates of matrix factorizations

Abstract

Abstract. This paper presents a new context for using the sparse Broyden update method to solve systems of nonlinear equations. The setting for this work is that a Newton-like algorithm is assumed to be available which incorporates a workable strategy for improving poor initial guesses and providing a satisfactory Jacobian matrix approximation whenever required. The total cost of obtaining each Jacobian matrix, or the cost of factoring it to solve for the Newton step, is assumed to be sufficiently high to make it attractive to keep the same Jacobian approximation for several steps. This paper suggests the extremely convenient and apparently effective technique of applying the sparse Broyden update directly to the matrix factors in the iterations between réévaluations in the hope that fewer fresh factorizations will be required. The strategy is shown to be locally and ¡¡r-superlinearly convergent, and some encouraging numerical results are presented.

Calibration‐constrained Monte Carlo analysis of highly parameterized models using subspace techniques

“…It is particularly beneficial when it is expensive to compute X and/or when it is desirable to obtain as great a reduction in Φ as possible given one or perhaps two iterations. Indeed, Dennis and Marwil [1982] indicate that use of such an update may be most fruitful when the cost of obtaining a full new Jacobian is sufficiently high: as a result, they recommend that the use of such an update is most effective between reevaluations of X .…”

Novos resultados sobre formulas secantes e aplicações

“…Entretanto, nas últimas duas décadas, alguns autores introduziram algoritmos que, embora pareçam não ter esta propriedade, ainda assim apresentam bom desempenho prático. Entre eles, podemos citar os métodos introduzidos por Tewarson [54], Tewarson e Zhang [55], Dennis e Marwil [15) e Martínez [38). Gomes-Ruggiero e Martínez [27] mostraram que o "Column-Updating method" (CUM) introduzido em Martínez [39) é muito efetivo quando comparado com a implementação do primeiro método de Broyden com memória limitada (Gomes-Ruggiero, Martínez e Moretti [28), Griewank [30), Matthies e Strang [46]) na resolução de sistemas não lineares de grande porte.…”