We consider the Broyden-like method for a nonlinear mapping \(F:\mathbb {R}^{n}\rightarrow \mathbb {R}^{n}\) that has some affine component functions, using an initial matrix B₀ that agrees with the Jacobian of F in the rows that correspond to affine components of F. We show that in this setting, the iterates belong to an affine subspace and can be viewed as outcome of the Broyden-like method applied to a lower-dimensional mapping \(G:\mathbb {R}^{d}\rightarrow \mathbb {R}^{d}\), where d is the dimension of the affine subspace. We use this subspace property to make some small contributions to the decades-old question of whether the Broyden-like matrices converge: First, we observe that the only available result concerning this question cannot be applied if the iterates belong to a subspace because the required uniform linear independence does not hold. By generalizing the notion of uniform linear independence to subspaces, we can extend the available result to this setting. Second, we infer from the extended result that if at most one component of F is nonlinear while the others are affine and the associated n − 1 rows of the Jacobian of F agree with those of B₀, then the Broyden-like matrices converge if the iterates converge; this holds whether the Jacobian at the root is invertible or not. In particular, this is the first time that convergence of the Broyden-like matrices is proven for n > 1, albeit for a special case only. Third, under the additional assumption that the Broyden-like method turns into Broyden’s method after a finite number of iterations, we prove that the convergence order of iterates and matrix updates is bounded from below by \(\frac {\sqrt {5}+1}{2}\) if the Jacobian at the root is invertible. If the nonlinear component of F is actually affine, we show finite convergence. We provide high-precision numerical experiments to confirm the results.

我们考虑使用非线性矩阵\（F：\ mathbb {R} ^ {n} \ rightarrow \ mathbb {R} ^ {n} \）的类Broyden方法，该方法使用初始矩阵B ₀与上述雅可比同意˚F在行对应于仿射部件˚F。我们表明，在这种情况下，迭代属于仿射子空间，可以看作是应用于低维映射\（G：\ mathbb {R} ^ {d} \ rightarrow \ mathbb的类Broyden方法的结果{R} ^ {d} \），其中d是仿射子空间的尺寸。我们使用此子空间属性对数十年前存在的类Broyden矩阵是否收敛的问题做出了一些小贡献：首先，我们观察到，如果迭代属于子空间，则无法应用与该问题有关的唯一可用结果，因为需要统一线性独立性不成立。通过将统一线性独立性的概念概括为子空间，我们可以将可用结果扩展到此设置。其次，我们从扩展结果推断出，如果F的一个分量最多是非线性的，而其他分量是仿射的，并且F的雅可比行列的相关n − 1行与B _0的那些一致，则如果迭代收敛，则类似Broyden的矩阵收敛；这决定了根的雅可比定律是否可逆。特别是，这是首次证明类Broyden矩阵的收敛在n > 1的情况下进行，尽管仅在特殊情况下如此。第三，在类似的Broyden方法经过有限次迭代后变成Broyden方法的附加假设下，我们证明了迭代和矩阵更新的收敛顺序从下限为\（\ frac {\ sqrt {5} + 1} {2} \）（如果根的雅可比行列是可逆的）。如果F的非线性分量实际上是仿射的，则表明有限收敛。我们提供高精度的数值实验来确认结果。